diff --git a/-tAyT4oBgHgl3EQfqfjJ/content/tmp_files/2301.00545v1.pdf.txt b/-tAyT4oBgHgl3EQfqfjJ/content/tmp_files/2301.00545v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..22fdc9bbd31d0d1037d6f4b6d5dc160ec7b6b964 --- /dev/null +++ b/-tAyT4oBgHgl3EQfqfjJ/content/tmp_files/2301.00545v1.pdf.txt @@ -0,0 +1,3319 @@ +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +1 +Knockoffs-SPR: Clean Sample Selection in +Learning with Noisy Labels +Yikai Wang, Yanwei Fu, and Xinwei Sun. +Abstract—A noisy training set usually leads to the degradation of the generalization and robustness of neural networks. In this paper, +we propose a novel theoretically guaranteed clean sample selection framework for learning with noisy labels. Specifically, we first +present a Scalable Penalized Regression (SPR) method, to model the linear relation between network features and one-hot labels. In +SPR, the clean data are identified by the zero mean-shift parameters solved in the regression model. We theoretically show that SPR +can recover clean data under some conditions. Under general scenarios, the conditions may be no longer satisfied; and some noisy +data are falsely selected as clean data. To solve this problem, we propose a data-adaptive method for Scalable Penalized Regression +with Knockoff filters (Knockoffs-SPR), which is provable to control the False-Selection-Rate (FSR) in the selected clean data. To +improve the efficiency, we further present a split algorithm that divides the whole training set into small pieces that can be solved in +parallel to make the framework scalable to large datasets. While Knockoffs-SPR can be regarded as a sample selection module for a +standard supervised training pipeline, we further combine it with a semi-supervised algorithm to exploit the support of noisy data as +unlabeled data. Experimental results on several benchmark datasets and real-world noisy datasets show the effectiveness of our +framework and validate the theoretical results of Knockoffs-SPR. Our code and pre-trained models will be released. +Index Terms—Learning with Noisy Labels, Knockoffs Method, Type-Two Error Control. +! +1 +INTRODUCTION +D +EEP learning has achieved remarkable success on +many supervised learning tasks trained by millions +of labeled training data. The performance of deep models +heavily relies on the quality of label annotation since neural +networks are susceptible to noisy labels and even can easily +memorize randomly labeled annotations [1]. Such noisy +labels can lead to the degradation of the generalization +and robustness of such models. Critically, it is expensive +and difficult to obtain precise labels in many real-world +scenarios, thus exposing a realistic challenge for supervised +deep models to learn with noisy data. +There are many previous efforts in tackling this challenge by +making the models robust to noisy data, such as modifying +the network architectures [2]–[5] or loss functions [6]–[9]. +This paper addresses the challenge by directly selecting +clean samples. Inspired by the dynamic sample selection +methods [9]–[16], we construct a “virtuous” cycle between +sample selection and network training: the selected clean +samples can improve the network training; and on the +other hand, the improved network has a more powerful +ability in picking up clean data. As this cycle evolves, the +performance can be improved. To well establish this cycle, a +key question remains: how to effectively differentiate clean data +from noisy ones? +Preliminary. Typical principles in existing works [9]–[16] +to differentiate clean data from noisy data include large +loss [11], inconsistent prediction [17], and irregular feature +• +Yikai Wang and Yanwei Fu contribute equally. +• +Xinwei Sun is the corresponding author. +• +Yikai Wang, Yanwei Fu and Xinwei Sun are with the School of +Data Science, Fudan University. E-mail: {yikaiwang19, yanweifu, +sunxinwei}@fudan.edu.cn +representation [18]. The former two principles identify +irregular behaviors in the label space, while the last one +analyzes the instance representations of the same class in +the feature space. In this paper, we propose unifying the +label and feature space by making the linear relationship, +yi = x⊤ +i β + ε, +(1) +between feature-label pair (xi ∈ Rp: feature vector; yi ∈ Rc: +one-hot label vector) of data i. We also have the fixed +(unknown) coefficient matrix β ∈ Rp×c, and random noise +ε ∈ Rc. Essentially, the linear relationship here is an ideal +approximation, as the networks are trained to minimize +the divergence between a (soft-max) linear projection of +the feature and a one-hot label vector. For a well-trained +network, the output prediction of clean data is expected +to be as similar to a one-hot vector as possible, while the +entropy of the output of noisy data should be large. Thus if +the underlying linear relation is well-approximated without +soft-max operation, the corresponding data is likely to be +clean. In contrast, the feature-label pair of noisy data may +not be approximated well by the linear model. +The simplest way to measure the goodness of the linear +model in fitting the feature-label pair is to check the +prediction error, or residual, ri = yi − x⊤ +i ˆβ, where ˆβ is the +estimate of β. The larger ∥r∥ indicates a larger fitting error +and thus more possibility for instance i to be outlier/noisy +data. Many methods have been proposed to test whether ri +is non-zero. Particularly, we highlight the classical statistical +leave-one-out approach [19] that computes the studentized +residual as, +ti = +yi − x⊤ +i ˆβ−i +ˆσ−i +� +1 + x⊤ +i +�X⊤ +−iX−i +�−1 xi +�1/2 , +(2) +arXiv:2301.00545v1 [cs.LG] 2 Jan 2023 + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +2 +Images +Features +Noisy Labels +Stage 1: +Network Learning +Classifier +Stage 2: +Sample Selection +β ∈ Rd×c +Noisy Labels +Y ∈ Rn×c +X ∈ Rn×d +Features +Noisy Data Indicator += ++ +γ ∈ Rn×c +Yi (1, 0, 0) +(0, 1, 0) ˜Yi +Permute +Compare +Selected Clean Data +Fig. 1. Knockoffs-SPR runs a cycle between network learning and sample selection, where clean data are selected via the comparison of the +mean-shift parameters between its original label and permuted label. +where ˆσ is the scale estimate and the subscript −i indicates +estimates based on the n − 1 observations, leaving out +the i-th data for testing. Equivalently, the linear regression +model can be re-formulated into explicitly representing the +residual, +Y = Xβ + γ + ε, +εi,j ∼ N(0, σ2), +(3) +by introducing a mean-shift parameter γ as in [20] with the +feature X ∈ Rn×p, and label Y ∈ Rn×c paired and stacked +by rows. For each row of γ ∈ Rn×c, γi represents the predict +residual of the corresponding data. This formulation has +been widely studied in different research topics, including +economics [21]–[24], robust regression [20], [25], statistical +ranking [26], face recognition [27], semi-supervised few-shot +learning [28], [29], and Bayesian preference learning [30], +to name a few. This formulation is differently focused on +the specific research tasks. For example, for the robust +regression problem [20], [25], the target is to get a robust +estimate ˆβ against the influence of γ. Here for solving the +problem of learning with noisy labels, we are interested +in recovering zeros elements of γ, since these elements +correspond to clean data. +SPR [31]. To this end, from the statistical perspective, +our conference report [31] starts from Eq. (3) to build up +a sample selection framework, dubbed Scalable Penalized +Regression (SPR). With a sparse penalty P(γ; λ) on γ, the +SPR obtains a regularization solution path of γ(λ) by +evolving λ from ∞ to 0. Then it identifies those samples +that are earlier (or at larger λ) selected to be non-zeros +as noisy data and those later selected as clean data, with +a manually specified ratio of selected data. Under the +irrepresentable condition [4], [33], the SPR enjoys model +selection consistency in the sense that it can recover the set +of noisy data. By feeding only clean data into next-round +training, the trained network is less corrupted by the noisy +data and hence performs well empirically. +Knockoffs-SPR. However, the irrepresentable condition +demands the prior of the ground-truth noisy set, which +is not accessible in practice. When this condition fails, +the trained network with SPR may be still corrupted by +a large proportion of noisy data, leading to performance +degradation as empirically verified in our experiments. To +amend this problem, we provide a data-adaptive sample +selection algorithm, in order to well control the expected +rate of noisy data in the selected data under the desired level +q, e.g., q = 0.05. As the goal is to identify clean data for the +next-round training, we term this rate as the False-Selection- +Rate (FSR). The FSR is the expected rate of the type-II error +in sparse regression, as non-zero elements correspond to +the noisy data. Our method to achieve the FSR control is +inspired by the ideas of Knockoffs in Statistics, which is +a recently developed framework for variable selection [1], +[2], [34], [35]. The Knockoffs framework aims at selecting +non-null variables and controlling the False-Discovery-Rate +(FDR), by taking as negative controls knockoff features ˜ +X, +which are constructed as a fake copy for the original features +X. Here, the FDR corresponds to the expectation of the +type-I error rate in sparse regression. Therefore, the vanilla +Knockoffs cannot be directly applied to our SPR framework, +since FSR is the expected rate of the type-II error and there +is no theoretical guarantee in Knockoffs for this control. To +achieve the FSR control, we propose Knockoffs-SPR, which +turns to construct the knockoff labels ˜Y via permutation for +the original label Y , and incorporates it into a data-partition +strategy for FSR control. +Formally, we repurpose the knockoffs in Statistics in our +SPR method; and propose a novel data-adaptive sample +selection algorithm, dubbed Knockoffs-SPR. It extends SPR +in controlling the ratio of noisy data among the selected +clean data. With this new property, Knockoffs-SPR ensures +that the clean pattern is dominant in the data and hence +leads to better network training. Specifically, we partition +the whole noisy training set into two random subsets and +apply the Knockoffs-SPR to two subsets separately. For each +time, we use one subset to estimate the intercept β and the +other to select the clean data by comparing between the +solution paths of γ(λ) and ˜γ(λ) that respectively obtained +via regression on noisy labels and the permuted labels. With +such a decoupled structure between β and γ, we prove +that the FSR can be controlled by any prescribed level. +Compared with the original theory of SPR, our new theory +enables us to effectively select clean data under general +conditions. Besides, Knockoffs-SPR also enjoys a superior +performance over the original SPR. +Together with network training, the whole framework is +illustrated in Fig. 1 in which the sample selection and +the network learning are well incorporated into each +other. Specifically, we run the network learning process +and sample selection process iteratively and repeat this +cycle until convergence. To incorporate Knockoffs-SPR into +the end-to-end training pipeline of deep architecture, the +simplest way is to directly solve Knockoffs-SPR for each + +featureA +B +C +DJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +3 +training mini-batch or training epoch to select clean data. +Solving Knockoffs-SPR for each mini-batch is efficient but +suffers from the identifiability issue. The sample size in a +mini-batch may be too small to distinguish clean patterns +from noisy ones among all classes, especially for large +datasets with small batch size. Solving Knockoffs-SPR for +the whole training set is powerful but suffers from the +complexity issue, leading to an unacceptable computation +cost. To resolve these two problems, we strike a balance +between complexity and identifiability by proposing a +splitting strategy that divides the whole data into small +pieces such that each piece is class-balanced with the proper +sample size. In this regard, the sample size of each piece is +small enough to be solved efficiently and large enough to +distinguish clean patterns from noisy ones. Then Knockoffs- +SPR runs on each piece in parallel, making it scalable to +large datasets. +As the removed noisy data still contain useful information +for network training, we adopt the semi-supervised training +pipeline with CutMix [38] where the noisy data are utilized +as unlabeled data. We conduct extensive experiments to +validate the effectiveness of our framework on several +benchmark datasets and real-world noisy datasets. The +results show the efficacy of our Knockoffs-SPR algorithm. +Contributions. Our contributions are as follows: +• Ideologically, we propose to control the False-Selection- +Rate in selecting clean data, under general scenarios. +• Methodologically, we propose Knockoffs-SPR, a data- +adaptive method to control the FSR. +• Theoretically, we prove that the Knockoffs-SPR can +control the FSR under any desired level. +• Algorithmically, we propose a splitting algorithm for +better sample selection with balanced identifiability and +complexity in large datasets. +• Experimentally, we demonstrate the effectiveness and +efficiency of our method on several benchmark datasets +and real-world noisy datasets. +Extensions. Our conference version of this work, SPR, was +published in [31]. Compared with SPR [31], we have the +following extensions. +• We identify the limitation of the SPR and consider the FSR +control in selecting clean data. +• We propose a new framework: Knockoffs-SPR which is +effective in selecting clean data under general scenarios, +theoretically and empirically. +• We apply our method on Clothing1M and achieve better +results than compared baselines. +Logistics. The rest of this paper is organized as follows: +• In Section 9, we introduce our SPR algorithm with its +noisy set recovery theory. +• In Section 3, the Knockoffs-SPR algorithm is introduced +with its FSR control theorem. +• In Section 4, several training strategies are proposed to +well incorporate the Knockoffs-SPR with the network +training. +• In Section 5, connections are made between our proposed +works and several previous works. +• In Section 6, we conduct experiments on several synthetic +and real-world noisy datasets. +• Section 7 concludes this paper. +2 +CLEAN SAMPLE SELECTION +2.1 +Problem Setup +We are given a dataset of image-label pairs {(imgi, yi)}n +i=1, +where the noisy label yi is corrupted from the ground- +truth label y∗ +i . The ground-truth label y∗ +i and the corruption +process are unknown. Our target is to learn a recognition +model f(·) such that it can recognize the true category y∗ +i +from the image imgi, i.e., f(imgi) = y∗ +i , after training on the +noisy label yi. +In this paper, we adopt deep neural networks as the +recognition model and divide the f(·) into fc(g(·)) where +g(·) is the deep model for feature extraction and fc(·) is +the final fully-connected layer for classification. For each +input image imgi, the feature extractor g(·) is used to encode +the feature xi := g(imgi). Then the fully-connected layer is +used to output the score vector ˆyi = fc(xi) which indicates +the chance it belongs to each class and the prediction is +provided with ˆyi = argmax(ˆyi). +As the training data contain many noisy labels, simply +training from all the data leads to severe degradation +of generalization and robustness. Intuitively, if we could +identify the clean labels from the noisy training set, and +train the network with the clean data, we can reduce the +influence of noisy labels and achieve better performance and +robustness of the model. To achieve this, we thus propose a +sample selection algorithm to identify the clean data in the +noisy training set with theoretical guarantees. +Notation. In this paper, we will use a to represent scalar, a +to represent a vector, and A to represent a matrix. We will +annotate a∗ to denote the ground-truth value of a. We use +∥ · ∥F to denote the Frobenius norm. +2.2 +Clean Sample Selection via Penalized Regression +Motivated +by +the +leave-one-out +approach +for +outlier +detection, we introduce an explicit noisy data indicator γi +for each data and assume a linear relation between extracted +feature xi and one-hot label yi with noisy data indicator as, +yi = x⊤ +i β + γi + εi, +(4) +where yi ∈ Rc is one-hot vector; and xi ∈ Rp, β ∈ +Rp×c, γi ∈ Rc, εi ∈ Rc. The noisy data indicator γi can be +regarded as the correction of the linear prediction. For clean +data, yi ∼ N(x⊤ +i β∗, σ2Ic) with γ∗ +i = 0, and for noisy data +y∗ +i = yi −γ∗ +i ∼ N(x⊤ +i β∗, σ2). We denote C := {i : γ∗ +i = 0} +as the ground-truth clean set. +To select clean data for training, we propose Scalable +Penalized Regression (SPR), designed as the following sparse +learning paradigm, +argmin +β,γ +1 +2 ∥Y − Xβ − γ∥2 +F + P(γ; λ), +(5) +where we have the matrix formulation X ∈ Rn×p, and +Y +∈ +Rn×c of {xi, yi}n +i=1; and P(·; λ) is a row-wise + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +4 +Fig. 2. Solution Path of SPR. Red lines indicate noisy data while blue +lines indicate clean data. As λ decreases, the γi gradually solved with +non-zero values. +sparse penalty with coefficient parameter λ. So we have +P(γ; λ) = �n +j=1 P(γi; λ), e.g., group-lasso sparsity with +P(γ; λ) = λ � +i ∥γi∥2. +To estimate C, we only need to solve γ with no need to +estimate β. Thus to simplify the optimization, we substitute +the Ordinary Least Squares (OLS) estimate for β with γ +fixed into Eq. (5). To ensure that ˆβ is identifiable, we apply +PCA on X to make p ≪ n so that the X has full-column +rank. Denote +˜ +X = I − X +�X⊤X +�† X⊤, ˜Y += +˜ +XY , the +Eq. (5) is transformed into +argmin +γ +1 +2 +��� ˜Y − ˜ +Xγ +��� +2 +F + P(γ; λ), +(6) +which is a standard sparse linear regression for γ. Note that +in practice we can hardly choose a proper λ that works well +in all scenarios. Furthermore, from the equivalence between +the penalized regression problem and Huber’s M-estimate, +the solution of γ is returned with soft-thresholding. Thus +it is not worth finding the precise solution of a single γ. +Instead, we use a block-wise descent algorithm [39] to solve +γ with a list of λs and generate the solution path. As +λ changes from ∞ to 0, the influence of sparse penalty +decreases, and γi are gradually solved with non-zero values, +in other words, selected by the model, as visualized in +Fig. 2. Since earlier selected instance is more possible to be +noisy, we rank all samples in the descendent order of their +selecting time defined as: +Zi = sup {λ : γi (λ) ̸= 0} . +(7) +A large Zi means that the γi is earlier selected. Then the top +samples are identified as noisy data and the other samples +are selected as clean data. In practice, we select 50% of the +data as clean data. +2.3 +The Theory of Noisy Set Recovery in SPR +The SPR enjoys theoretical guarantees that the noisy data +set can be fully recovered with high probability, under +the irrepresentable condition [33]. Formally, consider the +vectorized version of Eq. (6): +argmin +⃗γ +1 +2 +���⃗y − ˚ +X⃗γ +��� +2 +2 + λ ∥⃗γ∥1 , +(8) +where ⃗y, ⃗γ is vectorized from Y , γ in Eq. (6); ˚ +X = Ic ⊗ ˜ +X +with ⊗ denoting the Kronecker product operator. Denote +S := supp(⃗γ∗), which is the noisy set Cc. We further denote +˚ +XS (resp. ˚ +XSc) as the column vectors of ˚ +X whose indexes +are in S (resp. Sc) and µ ˚ +X = maxi∈Sc ∥ ˚ +X∥2 +2. Then we have +Theorem 1 (Noisy set recovery). Assume that: +C1, Restricted eigenvalue: λmin( ˚ +X⊤ +S ˚ +XS) = Cmin > 0; +C2, Irrepresentability: there exists a η ∈ (0, 1], such that +∥ ˚ +X⊤ +Sc ˚ +XS( ˚ +X⊤ +S ˚ +XS)−1∥∞ ≤ 1 − η; +C3, Large error: +⃗γ∗ +min := mini∈S |⃗γ∗ +i | > h(λ, η, ˚ +X, ⃗γ∗); +where ∥A∥∞ +:= +maxi +� +j |Ai,j|, and h(λ, η, ˚ +X, ⃗γ∗) += +λη/�Cminµ ˚ +X + λ∥( ˚ +X⊤ +S ˚ +XS)−1sign(⃗γ∗ +S)∥∞. +Let λ ≥ +2σ√µ ˚ +X +η +√log cn. Then with probability greater than +1 − 2(cn)−1, model Eq. (8) has a unique solution ˆ⃗γ such that: 1) +If C1 and C2 hold, ˆ +Cc ⊆ Cc;2) If C1, C2 and C3 hold, ˆ +Cc = Cc. +We present the proof in the appendix, following the +treatment in [4], [40]. In this theorem, C1 is necessary to get +a unique solution, and in our case is mostly satisfied with +the natural assumption that the clean data is the majority +in the training data. If C2 holds, the estimated noisy data +is the subset of truly noisy data. This condition is the key +to ensuring the success of SPR, which requires divergence +between clean and noisy data such that we cannot represent +clean data with noisy data. If C3 further holds, the estimated +noisy data is exactly all the truly noisy data. C3 requires the +error measured by γi is large enough to be identified from +random noise. If the conditions fail, SPR will fail in a non- +vanishing probability, not deterministic. +3 +CONTROLLED CLEAN SAMPLE SELECTION +In the last section, we stop the solution path at λ such +that 50% samples are selected as clean data. If this happens +to be the rate of clean data, Thm. 1 shows that our SPR +can identify the clean data C under the irrepresentable +condition. However, the irrepresentable condition and the +information of the ground-truth clean set C are practically +unknown, making this theory hard to be used in the real +life. Particularly, with |Cc| unknown, the algorithm can stop +at an improper time such that the noisy rate of the selected +clean data ˆC can be still high, making the next-round trained +model corrupted a lot by noisy patterns. +To resolve the problem of false selection in SPR , we in this +section propose a data-adaptive early stopping method for +the solution path, that targets controlling the expected noisy +rate of the selected data dubbed as False-Selection-Rate (FSR) +under the desired level q (0 < q < 1): +FSR = E +� +�# +� +j : j ̸∈ H0 ∩ ˆC +� +# +� +j : j ∈ ˆC +� +∨ 1 +� +� , +(9) +where ˆC = {j : ˆγj = 0} is the recovered clean set of +γ, and H0 : γ∗ +i = 0 denotes the null hypothesis, i.e., the +sample i belonging to the clean dataset. Therefore, the FSR +in Eq. (9) targets controlling the false rate among selected +null hypotheses, which is also called the expected rate of +the type-II error in hypothesis testing. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +5 +3.1 +Knockoffs-SPR +To achieve the FSR control, we propose the Knockoffs- +SPR for clean sample selection. Our method is inspired +by knockoff methods [1], [2], [34], [35], [41] with the +different focus that we target selecting clean labels via +permutation instead of constructing knockoff features to +select explanatory variables. Specifically, under model (4) +we permute the label for each data and construct the +permutation ˜y. Then model (4) can be solved for y and +˜y to obtain the solution paths γ(λ) and ˜γ(λ), respectively. +We will show that this construction can pick up clean data +from noisy ones, by comparing the selecting time (Eq. (7)) +between γ(λ) and ˜γ(λ) for each data. On the basis of this +construction, we propose to partition the whole dataset +into two disjoint parts, with one part for estimating β +and the other for learning γ(λ) and ˜γ(λ). We will show +that the independent structure with such a data partition +enables us to construct the comparison statistics whose sign +patterns among alternative hypotheses (noisy data) are the +independent Bernoulli processes, which is crucial for FSR +control. +Formally speaking, we split the whole data D into D1 := +(X1, Y1) and D2 +:= +(X2, Y2) with ni +:= +|Di|, and +implement Knockoffs-SPR on both D1 and D2. In the +following, we only introduce the procedure on D2, as the +procedure for D1 shares the same spirit. Roughly speaking, +the procedure is composed of three steps: i) estimate β on +D1; ii) estimate ˜γ(λ)) on D2; and iii) construct the comparison +statistics and selection filters. We leave detailed discussions for +each step in Sec. 3.2. +Step i): Estimating β on D1. Our target is to provide +an estimate of β that is independent of D2. The simplest +strategy is to use the standard OLS estimator to obtain +ˆβ1. However, this estimator may not be accurate since it +is corrupted by noisy samples. For this consideration, we +first run SPR on D1 to get clean data and then solve β via +OLS on the estimated clean data. +Step ii): Estimating (γ(λ), ˜γ(λ)) on D2. After obtaining the +solution ˆβ1 on D1 , we learn the γ(λ) on D2: +1 +2 +���Y2 − X2 ˆβ1 − γ2 +��� +2 +F + P(γ2; λ). +(10) +For each one-hot encoded vector y2,j, we randomly permute +the position of 1 and obtain another one-hot vector ˜y2,j ̸= +y2,j. For clean data j, the ˜y2,j turns to be a noisy label; +while for noisy data, the ˜y2,j is switched to another noisy +label with probability c−2 +c−1 or clean label with probability +1 +c−1. After obtaining the permuted matrix as ˜Y2, we learn +the solution paths (γ2(λ), ˜γ2(λ)) using the same algorithm +as SPR via: +� +� +� +� +� +1 +2 +���Y2 − X2 ˜β1 − γ2 +��� +2 +F + � +j P(γ2,j; λ), +1 +2 +��� ˜Y2 − X2 ˜β1 − ˜γ2 +��� +2 +F + � +j P(˜γ2,j; λ). +(11) +Step iii): Comparison statistics and selection filters. +After obtaining the solution path (γ2(λ), ˜γ2(λ)), we define +sample significance scores with respect to y2,i and ˜y2,i of +each i respectively, as the selection time: Zi := sup{λ : +Algorithm 1 Knockoffs-SPR +Input: subsets D1 and D2. +Output: clean set of D2. +1: Use D1 to fit an linear regression model and get β(D1); +2: Generate permuted label of each sample i in D2; +3: Solve Eq. (26) for D2 and generate {Wi} by Eq. (12); +4: Initialize q = 0.02 and T = 0; +5: while q < 0.5 and T = 0 do +6: +Compute T by Eq. (13); +7: +q = q + 0.02; +8: end while +9: if T is 0 then +10: +Construct clean set via half of the samples with largest +Wi in Eq. (14) with T = ∞; +11: else +12: +Construct clean set via samples in Eq. (14); +13: end if +14: return clean set. +∥γ2,i(λ)∥2 ̸= 0} and ˜Zi := sup{λ : ∥˜γ2,i(λ)∥2 ̸= 0}. With +Zi, ˜Zi, we define the Wi as: +Wi := Zi · sign(Zi − ˜Zi). +(12) +Based on these statistics, we define a data-dependent +threshold T as +T = max +� +t > 0 : 1 + # {j : 0 < Wj ≤ t} +# {j : −t ≤ Wj < 0} ∨ 1 ≤ q +� +, +(13) +or T = 0 if this set is empty, where q is the pre-defined upper +bound. Our algorithm will select the clean subset identified +by +C2 := {j : −T ≤ Wj < 0}. +(14) +Empirically, T may be equal to 0 if the threshold q is +sufficiently small. In this regard, no clean data are selected, +which is meaningless. Therefore, we start with a small q and +iteratively increase q and calculate T, until an attainable T +such that T > 0 to bound the FSR as small as possible. In +practice, when the FSR cannot be bounded by q = 50%, +we will end the selection and simply select half of the most +possible clean examples via {Wj}. The whole procedure of +Knockoffs-SPR is shown in Algorithm 1. +3.2 +Statistical Analysis about Knockoffs-SPR +In this part, we present the motivations and intuitions of +each step in Knockoffs-SPR. +Data Partition. Knockoffs-SPR partitions the dataset D +into two subset D1 and D2. This step decomposes the +dependency of the estimate of β and γ in that we use D1/D2 +to estimate β/γ, respectively. Then ˆβ(D1) is independent of +ˆγ(D2) if D1 and D2 are disjoint. The independent estimation +of β and γ makes it provable for FSR control on D2. +Permutation. As we discussed in step ii, when the original +label is clean, its permuted label will be a noisy label. On +the other hand, if the original label is noisy, its permuted +label changes to clean with probability +1 +c−1 and noisy with + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +6 +probability +c−2 +c−1, where c denotes the number of classes. +Note that γ of noisy data is often selected earlier than that +of clean data in the solution path. This implies larger Z +values for noisy data than those for clean data. As a result, +according to the definition of W, a clean sample will ideally +have a small negative of W := Z · sign(Z − ˜Z), where Z +and ˜Z respectively correspond to the clean label and noisy +label. In contrast for a noisy sample, the W tends to have +a large magnitude and has approximately equal probability +to be positive or negative. Such a different behavior of W +between clean and noisy data can help us to identify clean +samples from noisy ones. +Asymmetric comparison statistics W. The classical way to +define comparison statistics is in a symmetric manner, i.e., +Wi := Zi ∨ ˜Zi · sign(Zi − ˜Zi). In this way, a clean sample +with a noisy permuted label tends to have a large |Wi|, as +we expect the noisy label to have a large ˜Zi. However, this is +against our target as we only require clean samples to have +a small magnitude. For this purpose, we design asymmetric +comparison statistics that only consider the magnitude of +the original labels. +To see the asymmetric behavior of W +for noisy and +clean data, we consider the Karush–Kuhn–Tucker (KKT) +conditions of Eq. (26) with respect to (γ2,i, ˜γ2,i) +γ2,i + ∂P(γ2,i; λ) +∂γ2,i += x⊤ +2,i(β∗ − ˆβ1) + γ∗ +2,i + ε(2),i, +(15a) +˜γ2,i + ∂P(˜γ2,i; λ) +∂˜γ2,i += x⊤ +2,i(β∗ − ˆβ1) + ˜γ∗ +2,i + ˜ε(2),i, +(15b) +where ε(2),i ∼i.i.d ˜ε(2),i, |γ∗ +2,i| = |˜γ∗ +2,i| if both y2,i and +˜y2,i are noisy, and P(γ2,i; λ) := λ|γ2,i| as an example. By +conditioning on ˆβ1 and denoting ai := x⊤ +2,i(β∗ − ˆβ1), we +have that +P(Wi > 0) = P(|ai+γ∗ +2,i+ε2,i| > |ai+ ˜γ∗ +2,i+ ˜ε(2),i|). (16) +Then it can be seen that if i is clean, we have γ∗ +2,i = 0. +Then Zi tends to be small and besides, it is probable to have +Zi < ˜Zi if ˆβ1 can estimate β∗ well. As a result, Wi tends +to be a small negative. On the other hand, if i is noisy, then +Zi tends to be large for γi to account for the noisy pattern, +and besides, it has equal probability between Zi < ˜Zi and +Zi ≥ ˜Zi when ˜y2,i is switched to another noisy label, with +probability +c−2 +c−1. So Wi tends to have a large value and +besides, +P(Wi > 0) = P(Wi > 0|˜y2,i is noisy)P(˜y2,i is noisy) ++ P(Wi > 0|˜y2,i is clean)P(˜y2,i is clean) = 1 +2 · c − 2 +c − 1 ++ P(Wi > 0|˜y2,i is clean) · +1 +c − 1, +(17) +which falls in the interval of +� +c−2 +c−1 · 1 +2, +c +c−1 · 1 +2 +� +. That is to say, +P(Wi > 0) ≈ 1 +2. In this regard, the clean data corresponds +to small negatives of W in the ideal case, which can help +to discriminate noisy data with large W with almost equal +probability to be positive or negative. +Remark. For noisy y2,i, we have P(Wi > 0|˜y2,i is noisy) = +1/2 by assuming |γ∗ +2,i| = |˜γ∗ +2,i|. However, it may not hold +in practice when y2,i corresponds to the noisy pattern that +has been learned by the model. In this regard, it may +have |γ∗ +2,i| < |˜γ∗ +2,i| for a randomly permuted label ˜y2,i. +To resolve this problem, we instead set the permutation +label as the most confident candidate of the model, please +refer to Sec. 4.1 for details. Besides, if ˆβ1 can accurately +estimate β∗, according to KKT conditions in Eq. (15), we +have P(Wi > 0) < 1/2. That is Wi tends to be negative for +the clean data, which is beneficial for clean sample selection. +Data-adaptive +threshold. +The +proposed +data-adaptive +threshold T +is directly designed to control the FSR. +Specifically, the FSR defined in Eq. (9) is equivalent to +FSR(t) = E +�# {j : γj ̸= 0 and − t ≤ Wj < 0} +# {j : −t ≤ Wj < 0} ∨ 1 +� +, +(18) +where the denominator denotes the number of selected +clean data according to Eq. (14) and the nominator denotes +the number of falsely selected noisy data. This form of +Eq. (18) can be further decomposed into, +E +� # {γj ̸= 0, −t ≤ Wj < 0} +1 + # {γj ̸= 0, 0 < Wj ≤ t} · 1 + # {γj ̸= 0, 0 < Wj ≤ t} +# {−t ≤ Wj < 0} ∨ 1 +� +≤ E +� # {γj ̸= 0, −t ≤ Wj < 0} +1 + # {γj ̸= 0, 0 < Wj ≤ t} +1 + # {0 < Wj ≤ t} +# {−t ≤ Wj < 0} ∨ 1 +� +≤ E +� # {γj ̸= 0, −t ≤ Wj < 0} +1 + # {γj ̸= 0, 0 < Wj ≤ t}q +� +, +(19) +where the last inequality comes from the definition of +T in Eq. (13). To control the FSR, it suffices to bound +E +� +#{γj̸=0, −t≤Wj<0} +1+#{γj̸=0, 0 0) +among noisy data, which is independent of the magnitude +|W| due to data partition. We leave these details in the +appendix. +3.3 +FSR Control of Knockoffs-SPR +Our target is to show that FSR ≤ q with our data-adaptive +threshold T in Eq. (13). Our main result is as follows: +Theorem 2 (FSR control). For c-class classification task, and for +all 0 < q ≤ 1, the solution of Knockoffs-SPR holds +FSR(T) ≤ q +(20) +with the threshold T for two subsets defined respectively as +Ti = max +� +t ∈ W : 1 + # {j : 0 < Wj ≤ t} +# {j : −t ≤ Wj < 0} ∨ 1 ≤ c − 2 +2c q +� +. +We present the proof in the appendix. The coefficient +1/2 comes from the subset-partition strategy that we run +Knockoffs-SPR on two D1 and D2, and the term +c−2 +c +comes from the upper-bound of the first part in Eq. (19). +This theorem tells us that FSR can be controlled by the +given threshold q using the procedure of Knockoffs-SPR. +Compared to SPR, this procedure is more practical and +useful in real-world experiments and we demonstrate its +utility in Sec. 6.3 for more details. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +7 +Algorithm 2 Knockoffs-SPR on full training set +Input: Noisy feature-label pairs {(xi, yi)}n +i=1, group class +size N sample size m, (Optional) clean set. +Output: clean set. +1: if Number of classes > N then +2: +Compute class prototypes using Eq. (22); +3: +Divide classes into groups using Eq. (21); +4: else +5: +Use all classes as a single group; +6: end if +7: Construct pieces with uniformly sampled m examples +for each class (total=N × m); +8: for each piece do +9: +Randomly partition the piece into two sub-pieces A and +B (each contains Nm/2 examples); +10: +Run Algorithm 1 (B, A) on A to get clean-set-A; +11: +Run Algorithm 1 (A, B) on B to get clean-set-B; +12: +Concat clean-set-A and clean-set-B to get clean-set-piece; +13: end for +14: Concat clean-set-pieces to get clean set; +15: return clean set. +4 +LEARNING WITH KNOCKOFFS-SPR +In this section, we introduce how to incorporate Knockoffs- +SPR into the training of neural networks. We first introduce +several implementation details of Knockoffs-SPR, then we +introduce a splitting algorithm that makes Knockoffs-SPR +scalable to large-scale datasets. Finally, we discuss some +training strategies to better utilize the selected clean data. +4.1 +Knockoffs-SPR in Practice +We introduce several strategies to improve FSR control and +the power of selecting clean samples, which are inspired by +different behaviors of W between noisy and clean samples. +Ideally, for a clean sample i, Wi is expected to be a small +negative; if i is noisy data, Wi tends to be large and is +approximately 50% to be positive or negative, as shown +in Eq. (17). To achieve these properties for better clean +sample selection, the following strategies are proposed, in +the procedure of feature extractor, data-preprocessing, label +permutation strategy, estimating β on D1, and clean data +identification in Eq. (13), (14). +Feature Extractor. A good feature extractor is essential for +clean sample selection algorithms. In our experiments, we +adopt the self-supervised training method SimSiam [42] to +pre-train the feature extractor, to make X well encode the +information of the training data in the early stages. +Data Preprocessing. We implement PCA on the features +extracted by neural network for dimension reduction. This +can make X of full rank, which ensures the identifiability of +ˆβ in SPR. Besides, such a low dimensionality can make the +model estimate β more accurately. According to the KKT +conditions Eq. (15), we have that Wi of clean data i tends +to be negative with small magnitudes. In this regard, the +model can have better power of clean sample selection, i.e., +selecting more clean samples while controlling FSR. +Label +Permutation +Strategy. +Instead +of +the +random +permutation strategy, our Knckoff-SPR permutes the label +as the most-confident candidate provided by the model at +each training stage, for FSR consideration especially when +the noise rate is high or some noisy pattern is dominant +in the data. Specifically, if the pattern of some noisy label +y2,i is learned by the model, then γ∗ +2,i may have a smaller +magnitude than that of ˜γ∗ +2,i for a randomly permuted +label ˜y2,i that may not be learned by the model, violating +P(Wi > 0|˜y2,i) = 1/2 and hence P(Wi > 0) ≈ 1/2 in +practice. In contrast, the most confident permutation can +alleviate this problem, because the most confident label ˜y2,i +can naturally have a small magnitude of ˜γ∗ +2,i. +Estimating β on D1. We implement SPR as the first step +to learn β on D1. Compared to vanilla OLS, the SPR +can remove some noisy patterns from data, and hence +can achieve an accurate estimate of β. Similar to the data +processing step, such an accurate estimation can improve the +power of selecting clean samples. +Clean data identification in Eq. (13), (14). We calculate T +among W for each class, and identify the clean subset for +each class, to improve the power of clean data for each +class. In practice, since some classes may be easier to learn +than others, the Wi for i in these classes have smaller +magnitudes. Therefore, data from these classes will take the +main proportion if we calculate T and identify C2 among all +classes. With this design, the clean data are more balanced, +which facilitates the training in the next epochs. +4.2 +Scalable to Large Dataset +The computation cost of the sample selection algorithm +increases with the growth of the training sample, making +it not scalable to large datasets. To resolve this problem, +we propose to split the total training set into many pieces, +each of which contains a small portion of training categories +with a small number of training data. With the splitting +strategy, we can run the Knockoffs-SPR on several pieces +in parallel and significantly reduce the running time. For +the splitting strategy, we notice that the key to identifying +clean data is leveraging different behavior in terms of the +magnitude and the sign of W. Such a difference can be +alleviated if the patterns from clean classes are similar to the +noisy ones, which may lead to unsatisfactory recall/power +of identifying the clean set. This motivates us to group +similar categories together, to facilitate the discrimination +of clean data from noisy ones. +Formally speaking, we define the similarity between the +class i and j as +s(i, j) = p⊤ +i pj, +(21) +where p represents the class prototype. To obtain pi for the +class i, we take the clean features xi of each class extracted +by the network along the training iteration, and average +them to get the class prototype pc after the current training +epoch ends, as +pc = +�n +i=1 xi1(yi = c, i ∈ C) +�n +i=1 1(yi = c, i ∈ C) , +(22) +Then the most similar classes are grouped together. In +the initialization step when the clean set has not been + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +8 +Algorithm 3 Training with Knockoffs-SPR. +Input: Noisy dataset {(imgi, xi, yi)}n +i=1, p. +Output: Trained network. +Initialization : +1: Model: A self-supervised pre-trained backbone with +a random initialized fully-connected layer, an EMA +model; +2: Initial clean set: Run Algorithm 2 with self-supervised +pre-trained feature and noisy labels; +Training Process: +3: for ep = 0 to max epochs do +4: +for each mini-batch do +5: +Sample r from U(0, 1); +6: +if r > p then +7: +Train the network using Eq. (25); +8: +else +9: +Train the network using Eq. (24); +10: +end if +11: +Update features x visited in current mini-batch; +12: +Update EMA model; +13: +end for +14: +Run Algorithm 2 on {(xi, yi)}n +i=1 to get clean set; +15: end for +16: return Trained network. +estimated yet, we simply use all the data to calculate the +class prototypes. In our experiments, each group is designed +to have 10 classes. +For the instances in each group, we split the training data +of each class in a balanced way such that each piece contains +the same number of instances for each class. The number +is determined to ensure that the clean pattern remains the +majority in the piece, such that optimization can be done +easily. In practice, we select 75 training data from each +class to construct the piece. When the class proportion +is imbalanced in the original dataset, we adopt the over- +sampling strategy to sample the instance of each class with +less training data multiple times to ensure that each training +instance is selected once in some piece. The pipeline of our +splitting algorithm is described in Algorithm 2. +4.3 +Network Learning with Knockoffs-SPR +When training with Knockoffs-SPR, we can further exploit +the support of noisy data by incorporating Knockoffs- +SPR with semi-supervised algorithms. In this paper, we +interpolate part of images between clean data and noisy +data as in CutMix [38], +˜ +img = M ⊙ imgclean + (1 − M) ⊙ imgnoisy +(23a) +˜y = λyclean + (1 − λ)ynoisy +(23b) +where M ∈ {0, 1}W ×H is a binary mask, ⊙ is element- +wise multiplication, λ ∼ Beta(0.5, 0.5) is the interpolation +coefficient, and the clean and noisy data are identified +by Knockoffs-SPR. Then we train the network using the +interpolated data using +L +� +˜ +img, ˜y +� += LCE +� +˜ +img, ˜y +� +. +(24) +Empirically, we could switch between this semi-supervised +training with standard supervised training on estimated +clean data. +L (imgi, yi) = 1i/∈O · LCE (imgi, yi) , +(25) +where 1i/∈O is the indicator function, which means that +only the cross-entropy loss of estimated clean data is +used to calculate the loss. We further store a model with +EMA-updated weights. Our full algorithm is illustrated in +Algorithm 3. Neural networks trained with this pipeline +enjoy powerful recognition capacity in several synthetic and +real-world noisy datasets. +5 +RELATED WORK +Here we make the connections between our Knockoffs-SPR +and previous research efforts. +5.1 +Learning with Noisy Labels +The target of Learning with Noisy Labels (LNL) is to +train a more robust model from the noisy dataset. We can +roughly categorize LNL algorithms into two groups: robust +algorithm and noise detection. A robust algorithm does not +focus on specific noisy data but designs specific modules +to ensure that networks can be well-trained even from the +noisy datasets. Methods following this direction includes +constructing robust network [2]–[5], robust loss function [6]– +[9] robust regularization [43]–[46] against noisy labels. +The noise detection method aims to identify the noisy +data and design specific strategies to deal with the noisy +data, including down-weighting the importance in the loss +function for the network training [47], re-labeling them to +get correct labels [48], or regarding them as unlabeled data +in the semi-supervised manner [49], etc. +For the noise detection algorithm, noisy data are identified +by some irregular patterns, including large error [14], +gradient directions [50], disagreement within multiple +networks [15], inconsistency along the training path [17] and +some spatial properties in the training data [18], [51]–[53]. +Some algorithms [50], [54] rely on the existence of an extra +clean set to detect noisy data. +After detecting the clean data, the simplest strategy is to +train the network using the clean data only or re-weight +the data [55] to eliminate the noise. Some algorithms [49], +[56] regard the detected noisy data as unlabeled data to +fully exploit the distribution support of the training set in +the semi-supervised learning manner. There are also some +studies of designing label-correction module [2], [48], [54], +[57]–[59] to further pseudo-labeling the noisy data to train +the network. Few of these approaches are designed from the +statistical perspective with non-asymptotic guarantees, in +terms of clean sample selection. In contrast, our Knockoffs- +SPR can theoretically control the false-selected rate in +selecting clean samples under general scenarios. +5.2 +Mean-Shit Parameter +Mean-shift +parameters +or +incidental +parameters +[21] +originally tackled to solve the robust estimation problem + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +9 +via penalized estimation [60] . With a different focus on +specific parameters, this formulation address wide attention +in different research topics, including economics [21]–[24], +robust regression [20], [25], statistical ranking [26], face +recognition [27], semi-supervised few-shot learning [28], +[29], and Bayesian preference learning [30], to name a +few. Previous work usually uses this formulation to solve +robust linear models, while in this paper we adopt this to +select clean data and help the training of neural networks. +Furthermore, we design an FSR control module and a +scalable sample selection algorithm based on mean-shift +parameters with theoretical guarantees. +5.3 +Knockoffs +Knockoffs was first proposed in [34] as a data-adaptive +method to control FDR of variable selection in the sparse +regression problem. This method was then extended to +high-dimension regression [1], [61], multi-task regression +[35], outlier detection [41] and structural sparsity [2]. The +core of Knockoffs is to construct a fake copy of X as +negative controls of original features, in order to select true +positive features with FDR control. Our Knockoffs-SPR is +inspired by but different from the classical knockoffs in the +following aspects: i) the Knockoff is to control the FDR, +i.e., the expected rate of the type-I error while our goal is +to control the expected rate of the type-II error, a.k.a, FSR, +in the noisy data scenario; ii) instead of constructing copy +for X, we construct the copy ˜Y via permutation. Equipped +with a calibrated data-partitioning strategy, our method can +control the FER under any desired level. +6 +EXPERIMENTS +Datasets. We validate the effectiveness of Knockoffs- +SPR on synthetic noisy datasets CIFAR-10 and CIFAR- +100 [62], and real-world noisy datasets WebVision [63] +and Clothing1M [2]. We consider two types of noisy +labels for CIFAR: (i) Symmetric noise: Every class is +corrupted uniformly with all other labels; (ii) Asymmetric +noise: Labels are corrupted by similar (in pattern) classes. +WebVision has 2.4 million images collected from the +internet with the same category list as ImageNet ILSVRC12. +Clothing1M has 1 million images collected from the internet +and labeled by the surrounding texts. Thus, the WebVision +and Clothing1M datasets can be regarded as real-world +challenges. +Backbones. For CIFAR, we use ResNet-18 [64] as our +backbone. For WebVision we use Inception-ResNet [65] to +extract features to follow previous works. For Clothing1M +we use ResNet-50 as backbones. For CIFAR and WebVision, +we respectively self-supervised pretrain for 100 epochs and +350 epochs using SimSiam [42]. For Clothing1M, we use +ImageNet pre-trained weights to follow previous works. +Hyperparameter setting. We use SGD to train all the +networks with a momentum of 0.9 and a cosine learning +rate decay strategy. The initial learning rate is set as 0.01. +The weight decay is set as 1e-4 for Clothing1M, and 5e- +4 for other datasets. We use a batch size of 128 for all +experiments. We use random crop and random horizontal +flip as augmentation strategies. The network is trained +for 180 epochs for CIFAR, 300 epochs for WebVision, and +5 epochs for Clothing1M. Network training strategy is +selected with p = 0.5 (line 6 in Alg. 3) for Clothing1M, while +for other datasets we only use CutMix training. For features +used in Knockoffs-SPR, we reduce the dimension of X to +the number of classes. For Clothing1M, this is 14 and for +other datasets the reduced dimension is 10 (each piece of +CIFAR-100 and WebVision contains 10 classes). We also run +SPR with our new network training algorithm (Alg. 3). +6.1 +Evaluation on Synthetic Label Noise +Competitors. We use cross-entropy loss (Standard) as the +baseline algorithm for two datasets. We compare Knockoffs- +SPR with algorithms that include Forgetting [66] with +train the network using dropout strategy, Bootstrap [67] +which trains with bootstrapping, Forward Correction [55] +which corrects the loss function to get a robust model, +Decoupling +[68] +which +uses +a +meta-update +strategy +to +decouple +the +update +time +and +update +method, +MentorNet [12] which uses a teacher network to help train +the network, Co-teaching [11] which uses two networks to +teach each other, Co-teaching+ [15] which further uses an +update by disagreement strategy to improve Co-teaching, +IterNLD [51] which uses an iterative update strategy, +RoG [52] which uses generated classifiers, PENCIL [59] +which uses a probabilistic noise correction strategy, GCE [7] +and SL [8] which are extensions of the standard cross- +entropy loss function, and TopoFilter [18] which uses feature +representation to detect noisy data. For each dataset, all the +experiments are run with the same backbone to make a +fair comparison. We randomly run all the experiments five +times and calculate the average and standard deviation of +the accuracy of the last epoch. The results of competitors are +reported in [18]. +As in Table 1, Knockoffs-SPR enjoys a higher performance +compared with other competitors on CIFAR, validating the +effectiveness of Knockoffs-SPR on different noise scenarios. +SPR enjoys better performance on higher symmetric noise +rate of CIFAR-100. This may contributes to the manual +selection threshold of 50% of the data. Then SPR will +select more data than Knockoffs-SPR, for example in Sym. +80% noise scenario SPR will select 24816 clean data while +Knockoffs-SPR will select 18185 clean data. This leads to +a better recovery of clean data (recall of 94.22% while +Knockoffs-SPR is 81.20%) and thus a better recognition +capacity. +6.2 +Evaluation on Real-World Noisy Datasets +In this part, we compare Knockoffs-SPR with other methods +on real-world noisy datasets: WebVision and Clothing1M. +We follow previous work to train and test on the first 50 +classes of WebVision. We also evaluate models trained on +WebVision to ILSVRC12 to test the cross-dataset accuracy. +Competitors. +For +WebVision, +we +compare +with +CE +that +trains +with +cross-entropy +loss +(CE), +as +well +as +Decoupling [68], D2L [69], MentorNet [12], Co-teaching [11], +Iterative-CV [13], and DivideMix [49]. For clothing1M, we + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +10 +TABLE 1 +Test accuracies(%) on several benchmark datasets with different settings. +Dataset +Method +Sym. Noise Rate +Asy. Noise Rate +0.2 +0.4 +0.6 +0.8 +0.2 +0.3 +0.4 +CIFAR-10 +Standard +85.7 ± 0.5 +81.8 ± 0.6 +73.7 ± 1.1 +42.0 ± 2.8 +88.0 ± 0.3 +86.4 ± 0.4 +84.9 ± 0.7 +Forgetting +86.0 ± 0.8 +82.1 ± 0.7 +75.5 ± 0.7 +41.3 ± 3.3 +89.5 ± 0.2 +88.2 ± 0.1 +85.0 ± 1.0 +Bootstrap +86.4 ± 0.6 +82.5 ± 0.1 +75.2 ± 0.8 +42.1 ± 3.3 +88.8 ± 0.5 +87.5 ± 0.5 +85.1 ± 0.3 +Forward +85.7 ± 0.4 +81.0 ± 0.4 +73.3 ± 1.1 +31.6 ± 4.0 +88.5 ± 0.4 +87.3 ± 0.2 +85.3 ± 0.6 +Decoupling +87.4 ± 0.3 +83.3 ± 0.4 +73.8 ± 1.0 +36.0 ± 3.2 +89.3 ± 0.3 +88.1 ± 0.4 +85.1 ± 1.0 +MentorNet +88.1 ± 0.3 +81.4 ± 0.5 +70.4 ± 1.1 +31.3 ± 2.9 +86.3 ± 0.4 +84.8 ± 0.3 +78.7 ± 0.4 +Co-teaching +89.2 ± 0.3 +86.4 ± 0.4 +79.0 ± 0.2 +22.9 ± 3.5 +90.0 ± 0.2 +88.2 ± 0.1 +78.4 ± 0.7 +Co-teaching+ +89.8 ± 0.2 +86.1 ± 0.2 +74.0 ± 0.2 +17.9 ± 1.1 +89.4 ± 0.2 +87.1 ± 0.5 +71.3 ± 0.8 +IterNLD +87.9 ± 0.4 +83.7 ± 0.4 +74.1 ± 0.5 +38.0 ± 1.9 +89.3 ± 0.3 +88.8 ± 0.5 +85.0 ± 0.4 +RoG +89.2 ± 0.3 +83.5 ± 0.4 +77.9 ± 0.6 +29.1 ± 1.8 +89.6 ± 0.4 +88.4 ± 0.5 +86.2 ± 0.6 +PENCIL +88.2 ± 0.2 +86.6 ± 0.3 +74.3 ± 0.6 +45.3 ± 1.4 +90.2 ± 0.2 +88.3 ± 0.2 +84.5 ± 0.5 +GCE +88.7 ± 0.3 +84.7 ± 0.4 +76.1 ± 0.3 +41.7 ± 1.0 +88.1 ± 0.3 +86.0 ± 0.4 +81.4 ± 0.6 +SL +89.2 ± 0.5 +85.3 ± 0.7 +78.0 ± 0.3 +44.4 ± 1.1 +88.7 ± 0.3 +86.3 ± 0.1 +81.4 ± 0.7 +TopoFilter +90.2 ± 0.2 +87.2 ± 0.4 +80.5 ± 0.4 +45.7 ± 1.0 +90.5 ± 0.2 +89.7 ± 0.3 +87.9 ± 0.2 +SPR +92.0 ± 0.1 +94.6 ± 0.2 +91.6 ± 0.2 +80.5 ± 0.6 +89.0 ± 0.8 +90.3 ± 0.8 +91.0 ± 0.6 +Knockoffs-SPR +95.4 ± 0.1 +94.5 ± 0.1 +93.3 ± 0.1 +84.6 ± 0.8 +95.1 ± 0.1 +94.5 ± 0.2 +93.6 ± 0.2 +CIFAR-100 +Standard +56.5 ± 0.7 +50.4 ± 0.8 +38.7 ± 1.0 +18.4 ± 0.5 +57.3 ± 0.7 +52.2 ± 0.4 +42.3 ± 0.7 +Forgetting +56.5 ± 0.7 +50.6 ± 0.9 +38.7 ± 1.0 +18.4 ± 0.4 +57.5 ± 1.1 +52.4 ± 0.8 +42.4 ± 0.8 +Bootstrap +56.2 ± 0.5 +50.8 ± 0.6 +37.7 ± 0.8 +19.0 ± 0.6 +57.1 ± 0.9 +53.0 ± 0.9 +43.0 ± 1.0 +Forward +56.4 ± 0.4 +49.7 ± 1.3 +38.0 ± 1.5 +12.8 ± 1.3 +56.8 ± 1.0 +52.7 ± 0.5 +42.0 ± 1.0 +Decoupling +57.8 ± 0.4 +49.9 ± 1.0 +37.8 ± 0.7 +17.0 ± 0.7 +60.2 ± 0.9 +54.9 ± 0.1 +47.2 ± 0.9 +MentorNet +62.9 ± 1.2 +52.8 ± 0.7 +36.0 ± 1.5 +15.1 ± 0.9 +62.3 ± 1.3 +55.3 ± 0.5 +44.4 ± 1.6 +Co-teaching +64.8 ± 0.2 +60.3 ± 0.4 +46.8 ± 0.7 +13.3 ± 2.8 +63.6 ± 0.4 +58.3 ± 1.1 +48.9 ± 0.8 +Co-teaching+ +64.2 ± 0.4 +53.1 ± 0.2 +25.3 ± 0.5 +10.1 ± 1.2 +60.9 ± 0.3 +56.8 ± 0.5 +48.6 ± 0.4 +IterNLD +57.9 ± 0.4 +51.2 ± 0.4 +38.1 ± 0.9 +15.5 ± 0.8 +58.1 ± 0.4 +53.0 ± 0.3 +43.5 ± 0.8 +RoG +63.1 ± 0.3 +58.2 ± 0.5 +47.4 ± 0.8 +20.0 ± 0.9 +67.1 ± 0.6 +65.6 ± 0.4 +58.8 ± 0.1 +PENCIL +64.9 ± 0.3 +61.3 ± 0.4 +46.6 ± 0.7 +17.3 ± 0.8 +67.5 ± 0.5 +66.0 ± 0.4 +61.9 ± 0.4 +GCE +63.6 ± 0.6 +59.8 ± 0.5 +46.5 ± 1.3 +17.0 ± 1.1 +64.8 ± 0.9 +61.4 ± 1.1 +50.4 ± 0.9 +SL +62.1 ± 0.4 +55.6 ± 0.6 +42.7 ± 0.8 +19.5 ± 0.7 +59.2 ± 0.6 +55.1 ± 0.7 +44.8 ± 0.1 +TopoFilter +65.6 ± 0.3 +62.0 ± 0.6 +47.7 ± 0.5 +20.7 ± 1.2 +68.0 ± 0.3 +66.7 ± 0.6 +62.4 ± 0.2 +SPR +72.5 ± 0.2 +75.0 ± 0.1 +70.9 ± 0.3 +38.1 ± 0.8 +71.9 ± 0.2 +72.4 ± 0.3 +70.9 ± 0.5 +Knockoffs-SPR +77.5 ± 0.2 +74.3 ± 0.2 +67.8 ± 0.4 +30.5 ± 1.0 +77.3 ± 0.4 +76.3 ± 0.3 +73.9 ± 0.6 +TABLE 2 +Test accuracies(%) on WebVision and ILSVRC12 (trained on +WebVision). +Method +WebVision +ILSVRC12 +top1 +top5 +top1 +top5 +F-correction +61.12 +82.68 +57.36 +82.36 +Decoupling +62.54 +84.74 +58.26 +82.26 +D2L +62.68 +84.00 +57.80 +81.36 +MentorNet +63.00 +81.40 +57.80 +79.92 +Co-teaching +63.58 +85.20 +61.48 +84.70 +Iterative-CV +65.24 +85.34 +61.60 +84.98 +DivideMix +77.32 +91.64 +75.20 +90.84 +SPR +77.08 +91.40 +72.32 +90.92 +Knockoffs-SPR +78.20 +92.36 +74.72 +92.88 +compare with F-correction [55], M-correction [56], Joint- +Optim [48], Meta-Cleaner [70], Meta-Learning [71], P- +correction [59], TopoFilter [18] and DivideMix [49]. +The results of real-world datasets are shown in Table 3 +and Table 2, where the results of competitors are reported +in [49]. Our algorithm Knockoffs-SPR enjoys superior +performance to almost all the competitors, showing the +ability of handling real-world challenges. Compared with +SPR, Knockoffs-SPR also achieves better performance, +indicating the beneficial of FSR control in real-world +problems of learning with noisy labels. +TABLE 3 +Test accuracies(%) on Clothing1M. +Method +Accuracy +Cross-Entropy +69.21 +F-correction +69.84 +M-correction +71.00 +Joint-Optim +72.16 +Meta-Cleaner +72.50 +Meta-Learning +73.47 +P-correction +73.49 +TopoFiler +74.10 +DivideMix +74.76 +SPR +71.16 +Knockoffs-SPR +75.25 +6.3 +Evaluation of Sample Selection Quality +To test whether Knockoffs-SPR leads to better sample +selection quality, we test the following statistics on CIFAR- +10 with different noise scenarios, including Sym. 40%, Sym. +80%, and Asy. 40%. (1) FSR: the ratio of falsely selected +noisy data in the estimated clean data, which is the target +that Knockoffs-SPR aims to control; (2) Recall: the ratio of +selected ground-truth clean data in the full ground-truth +clean data, which indicates the power of sample selection +algorithms; (3) F1-score: the harmonic mean of precision (1- +FSR) and recall, which measures the balanced performance +of FSR control and power. We plot the corresponding + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +11 +0 +25 +50 +75 +100 +125 +150 +175 +0 +2 +4 +6 +8 +10 +12 +14 +16 +FSR +Symmetric-40% +0 +25 +50 +75 +100 +125 +150 +175 +0 +10 +20 +30 +40 +50 +60 +70 +80 +Symmetric-80% +0 +25 +50 +75 +100 +125 +150 +175 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +Asymmetric-40% +Knockoff-SPR +Estimated q +SPR +TopoFilter +0 +25 +50 +75 +100 +125 +150 +175 +60 +65 +70 +75 +80 +85 +90 +95 +100 +Recall +0 +25 +50 +75 +100 +125 +150 +175 +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +0 +25 +50 +75 +100 +125 +150 +175 +55 +60 +65 +70 +75 +80 +85 +90 +95 +100 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Training Epochs +70 +75 +80 +85 +90 +95 +100 +F score +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Training Epochs +10 +20 +30 +40 +50 +60 +70 +80 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Training Epochs +65 +70 +75 +80 +85 +90 +95 +100 +Fig. 3. Performance(%) comparison on sample selection along the +training path on CIFAR 10 with different noise scenarios. In the FSR, +we also visualize the estimated FSR (q) by Knockoffs-SPR, which is the +threshold we use to select clean data. +statistics of each algorithm along the training epochs in +Fig. 3. We further visualize the estimated FSR, q, of +Knockoffs-SPR to compare with the ground-truth FSR. As +we use the splitting algorithm, where each piece contains +10 classes with each class containing a subset of data, we +estimate FSR for each piece and report their average and +standard deviation. +FSR control in practice. (1) When the noise rate is not +high, for example in Sym. 40% and Asy. 40% scenarios, the +ground-truth FSR is well upper-bounded by the estimated +FSR (with no larger than a single standard deviation). When +the noise rate is high, for example in Sym. 80% noise +scenario, the FSR cannot get controlled in the early stage. +However, as the training goes on, FSR can be well-bounded +by Knockoffs-SPR. +(2) When the training set is not very noisy, for example in +Sym. 40% scenario, the true FSR is far below the estimated +q. This gap can be explained by a good estimation of +β due to the small noisy rate. When ˆβ1 can accurately +estimate β∗, the ˜γ∗ +2,i dominate in Eq. (15). Therefore, the +P(Wi > 0|˜y2,i is clean) > 1 +2, making P(Wi > 0) > 1/2 > +c−2 +2(c−1). Since the true FSR bound is inversely proportional to +P(Wi > 0) (FSR ∝ maxi∈Cc 1/P(Wi > 0) − 1), it is smaller +than the theoretical bound q. +Sample selection quality comparison. We compare the +sample selection quality of Knockoffs-SPR with SPR and +TopoFilter [18]. (1) Knockoffs-SPR enjoys the (almost) best +FSR control capacity in all noise scenarios, especially in the +high noise rate setting. Other algorithms can suffer from +failure in controlling the FSR (for example in Sym. 80% +scenario). (2) The power of Knockoffs-SPR is comparable to +the best algorithms in Sym. 40% and Asy. 40% scenarios. For +the Sym. 80% case, Knockoffs-SPR sacrifices some power for +FSR control. (3) Compared together, Knockoffs-SPR enjoys +the best F1 score on sample selection quality, which well- +establishes its superiority in selecting clean data with FSR +control. +TABLE 4 +Ablation(%) of Knockoffs-SPR on CIFAR-10. +SPR +∗-random +∗-multi +∗-noPCA +Knockoffs-SPR +Sym. 40% +Acc. +94.0 +92.0 +94.4 +81.7 +94.7 +FSR +0.82 +23.04 +1.31 +11.51 +1.27 +q +- +4.31±0.73 +2.00±0.00 +14.18±7.62 +5.59±1.11 +Sym. 80% +Acc. +78.0 +84.6 +83.0 +10.0 +84.3 +FSR +60.47 +49.76 +25.77 +78.06 +26.72 +q +- +9.47±4.39 +2.22±0.62 +25.95±11.88 +19.52±12.77 +Asy. 40% +Acc. +89.5 +84.4 +93.4 +93.7 +93.5 +FSR +2.19 +16.94 +2.97 +7.62 +2.84 +q +- +4.15±2.59 +2.00±0.00 +5.22±2.85 +4.45±2.68 +6.4 +Further Analysis +Influence +of +Knockoffs-SPR +strategies. +We +compare +Knockoffs-SPR +with +several +variants, +including: +SPR +(The original SPR algorithm), ∗-random (Knockoffs-SPR +with +randomly +permuted +labels), +∗-multi +(Knockoffs- +SPR without class-specific selection), ∗-noPCA (Knockoffs- +SPR without using PCA to pre-process the features). +Experiments are conducted on CIFAR-10 with different +noise scenarios, as in Table 4. We observe the following +results: +(1) As also shown in Fig. 3, the SPR can control the FSR in +Sym. 40% and Asy. 40% but fails in Sym. 80%. This may +be due to that when the noisy pattern is not significant, +the collinearity is weak between noisy samples and clean +ones, as shown by the distribution of irrepresentable value +{∥(X⊤ +S XS)−1X⊤ +S Xj∥1}j∈Sc +in Fig. 1 in the appendix. +In this regard, most of the earlier (resp. later) selected +samples in the solution path tend to be noisy (resp. clean) +samples. When there is strong multi-collinearity and the +irrepresentable condition is violated seriously, our proposed +Knockoff procedure can help to control the FSR. The higher +accuracy of Knockoffs-SPR over SPR can be explained by +consistent improvements in terms of the F1 score of sample +selection capacity, as shown in Fig. 3. +(2) Compared with the random permutation strategy, +Knockoffs-SPR with most-confident permutation enjoys +much better FSR control and works much better in Sym. +40% and Asy. 40% noise scenarios. In Sym. 80% noise +scenario, the accuracy is comparable, but the most-confident +permutation still enjoys much better FSR control. This +result empirically demonstrates the advantage of the most- +confident permutation over the random permutation. +(3) Running Knockoffs-SPR on each class separately is +beneficial for the FSR control capacity and recognition +capacity. When the noise rate is high, running Knockoffs- +SPR on multiple classes cannot control the FSR properly by +the estimated q. +(4) Using PCA on features as the pre-processing is beneficial +for FSR control in all cases and will increase the recognition +capacity in some cases, especially when the noise rate is +high. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +12 +TABLE 5 +Ablation of the splitting algorithm in computation efficiency (for one +epoch) on CIFAR-10. +Model +Training Time +Knockoffs-SPR w/o split algorithm +about 6h +Knockoffs-SPR w/ split algorithm +66s +TABLE 6 +Ablation(%) of training strategies on CIFAR-10. +Method +Sym. 40% +Sym. 80% +Asy. 40% +Knockoffs-SPR - Self +92.5 +24.3 +92.2 +Knockoffs-SPR - Semi +91.3 +54.0 +88.5 +Knockoffs-SPR - EMA +94.5 +83.8 +93.2 +Knockoffs-SPR +94.7 +84.3 +93.5 +Influence of Scalable. In our framework, we propose a split +algorithm to divide the whole training set into small pieces +to run Knockoffs-SPR in parallel. In this part, we compare +the running time between using the split algorithm and not +using it. Results are shown in Tab. 5. We can see that the +splitting algorithm can significantly reduce the computation +time. This is important in large-scale applications. +Influence of network training strategies. To better train +the network, we adopt the self-supervised pre-trained +backbone and the semi-supervised learning framework with +an EMA update model. In this part, we test the influence of +these strategies on CIFAR-10 with different noise scenarios. +Concretely, we compare the full framework with Knockoffs- +SPR - Self which uses a randomly initialized backbone, +Knockoffs-SPR - Semi which uses supervised training, and +Knockoffs-SPR - EMA which does not use the EMA update +model. Results are summarized in table. 6. We can find +that: (1) The self-supervised pre-training is important for +high noise rate scenarios, while for other settings, it is +not so essential; (2) Semi-supervised training consistently +improves the recognition capacity, indicating the utility of +leveraging the support of noisy data; (3) The EMA model +will slightly improve the recognition capacity. +airplane +frog +automobile +truck +bird +deer +cat +dog +deer +cat +dog +cat +frog +deer +horse +dog +ship +airplane +truck +ship +Fig. 4. Qualitative results of falsely selected examples by Knockoffs- +SPR. The black words are the labeled classes while the real classes +are denoted by red words. +Qualitative visualization. We randomly visualize some +falsely selected examples of CIFAR-10 in Fig. 4. Most of +these cases have some patterns that confuse the noisy +label and the true label, thus making Knockoffs-SPR falsely +identify them as clean samples. +7 +CONCLUSION +This +paper +proposes +a +statistical +sample +selection +framework – Scalable Penalized Regression with Knockoff +Filters (Knockoffs-SPR) to identify noisy data with a +controlled false selection rate. Specifically, we propose an +equivalent leave-one-out t-test approach as a penalized +linear model, in which non-zero mean-shift parameters can +be induced as an indicator for noisy data. We propose a +delicate Knockoff-SPR algorithm to identify clean samples +in a way that the false selection rate is controlled by +the user-provided upper bound. Such an upper bound is +proved theoretically and works well in empirical results. +Experiments on several synthetic and real-world datasets +demonstrate the effectiveness of Knockoff-SPR. +REFERENCES +[1] +C. Zhang, S. Bengio, M. Hardt, B. Recht, and O. 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Erfani, S. Xia, +S. Wijewickrema, and J. Bailey, “Dimensionality-driven learning +with noisy labels,” in ICML, 2018. 6.2 +[70] W. Zhang, Y. Wang, and Y. Qiao, “Metacleaner: Learning +to hallucinate clean representations for noisy-labeled visual +recognition,” in Proceedings of the IEEE/CVF Conference on Computer +Vision and Pattern Recognition, 2019, pp. 7373–7382. 6.2 +[71] J. Li, Y. Wong, Q. Zhao, and M. S. Kankanhalli, “Learning to learn +from noisy labeled data,” in Proceedings of the IEEE/CVF Conference +on Computer Vision and Pattern Recognition, 2019, pp. 5051–5059. 6.2 +Yikai +Wang +is +a +PhD +candidate +at +the +School +of +Data +Science, +Fudan +University, +under the supervision of Prof. Yanwei Fu. He +received a Bachelor’s degree in mathematics +from the School of Mathematical Sciences, +Fudan University, in 2019. He published 1 +IEEE TPAMI paper and 2 CVPR papers. His +current research interests include theoretically +guaranteed machine learning algorithms and +applications to computer vision. +Yanwei Fu received his PhD degree from the +Queen Mary University of London, in 2014. He +worked as a post-doctoral researcher at Disney +Research, Pittsburgh, PA, from 2015 to 2016. +He is currently a tenure-track professor at Fudan +University. He was appointed as the Professor +of Special Appointment (Eastern Scholar) at +Shanghai Institutions of Higher Learning. He +published +more +than +80 +journal/conference +papers including IEEE TPAMI, TMM, ECCV, +and CVPR. His research interests are one- +shot/meta-learning, learning-based 3D reconstruction, and image and +video understanding in general. +Xinwei Sun is currently an assistant professor +at the School of Data Science, Fudan University. +He +received +his +Ph.D. +in +the +school +of +mathematical sciences, at Peking University in +2018. His research interests mainly focus on +high-dimensional statistics and causal inference, +with their applications in machine learning and +medical imaging. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +15 +Supplementary Material +Yikai Wang, Yanwei Fu, and Xinwei Sun. +In this supplementary material, we formally present the +proof of FSR control theorem of knockoff-SPR in Sec. 8. +For consistency, we also provide the proof of the noisy +set recovery theorem of SPR in Sec. 9. Some additional +experimental results are provided in Sec. 10. +8 +FSR CONTROL THEOREM OF KNOCKOFF-SPR +Recall that we are solving the problem of +� +� +� +� +� +1 +2 +���Y2 − X2 ˜β1 − γ2 +��� +2 +F + � +j P(γ2,j; λ), +1 +2 +��� ˜Y2 − X2 ˜β1 − ˜γ2 +��� +2 +F + � +j P(˜γ2,j; λ). +(26) +We introduce the following lemma from [1] and [2]. +Lemma 3. Suppose that B1, . . . , Bn are indenpendent variables, +with Bi ∼ Bernoulli(ρi) for each i, where mini ρi ≥ ρ > 0. Let +J be a stopping time in reverse time with respect to the filtration +{Fj}, where +Fj = σ ({B1 + · · · + Bj, Bj+1, . . . , Bn}) . +Then +E +� +1 + J +1 + B1 + · · · + BJ +� +≤ ρ−1. +Proof. We first follow [1] to prove the case when {Bi} are +i.i.d. variables with Bi ∼ Bernoulli(ρ), where ρ > 0. Then +we follow [2] to generalize the conclusion to non-identical +case. +Define the stochastic process +Mj := 1 + j +1 + Sj +with +Sj := B1 + · · · + Bj +(27) +We show that {Mj} is a super-martingale with respect to +the reverse filtration {Fj}. It is trivial that {Mj} is {Fj}- +adapted and {Fj} is reverse filtration, that is a decreasing +sequence +Fj ⊂ Fj−1 · · · ⊂ {Bi}n +i=1 +(28) +with each Fj be a sub-σ-algebras of σ({Bi}n +i=1). Further, +we have E [|Mj|] ≤ 1 + j ≤ 1 + n < ∞ with fixed n. Now +we bound the conditional expectation E[Mj | Fj+1]. Note +that since {Bj}i+1 +j=1 are i.i.d. variable and thus exchangeable +when conditioned on Fj+1, then we have +P(Bj+1 = 1 | Fj+1) = Sj+1 +j + 1 +(29) +When Sj+1 = 0, it is natural that Sj = 0 thus Mj = 1 + j < +1 + (j + 1) = Mj+1. When Sj+ > 0, we have +E[Mj | Fj+1] = +1 + j +1 + Sj+1 − 1 · P(Bj+1 = 1 | Fj+1) ++ +1 + j +1 + Sj+1 +· P(Bj+1 = 0 | Fj+1) +=1 + j +Sj+1 +· Sj+1 +j + 1 + +1 + j +1 + Sj+1 +· j + 1 − Sj+1 +j + 1 +=1 + (j + 1) +1 + Sj+1 +=Mj+1. +(30) +Hence we have E[Mj | Fj+1] ≤ Mj+1, which finishes the +proof for the super-martingale {Mj}. Then by the Doob’s +optional sampling theorem [3], we have +E[Mj] ≤ E[Mn]. +(31) +Finally, we have +E[Mn] = E[ 1 + n +1 + Sn +] += (1 + n) +n +� +m=0 +1 +1 + m · +n! +m!(n − m)!ρm(1 − ρ)n−m += ρ−1(1 − (1 − ρ)n+1) +≤ ρ−1. +(32) +Now it suffices to show that the conclusion also holds for +non-identical Bernoulli variables. Following [2], for each +Bi ∼ Bernoulli(ρi), we construct the following disjoint +Borel sets {Ai +j}4 +j=1 such that R = ∪4 +j=1Aj with +P(Ai +1) = 1 − ρi; +P(Ai +2) = ρ1 − ρi +1 − ρ ; +P(Ai +3) = ρρi − ρ +1 − ρ ; +P(Ai +4) = ρi − ρ. +(33) +Define Ui = Ai +1∪Ai +2, Vi = Ai +2∪Ai +3, Gi = Ai +2∪Ai +3∪Ai +4. Based +on the specific construction we can set Gi = Bi. Further +define Qi = 1{ξi ∈ Vi} and a random set A = {i : ξi ∈ Ui}. +Then we have +Qi · 1{i ∈ A} + 1{i /∈ A} += 1{{ξi ∈ Vi ∩ Ui} ∪ {ξi ∈ U C +i }} += 1{{ξi ∈ Ai +2} ∪ {ξi ∈ Ai +3 ∪ Ai +4}} += 1{{ξi ∈ Ai +2 ∪ Ai +3 ∪ Ai +4}} = Bi. +(34) +Hence +1 + j +1 + Sj += 1 + |i ≤ j : i ∈ A| + |i ≤ j : i /∈ A| +1 + � +i≤j,i∈A Qi + |i ≤ j : i /∈ A| +≤ 1 + |i ≤ j : i ∈ A| +1 + � +i≤j,i∈A Qi +. +(35) + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +16 +The inequality holds because a+c +b+c ≤ a +b for 0 < b ≤ a, c ≥ 0. +Note that by definition +P(Qi = 1 | i ∈ A) = P(ξi ∈ Vi | ξi ∈ Ui) += +P(Ai +2) +P(Ai +1 ∪ Ai +2) += ρ = P(Qi = 1), +P(Qi = 1 | i ̸∈ A) = P(ξi ∈ Vi | ξi /∈ Ui) += +P(Ai +3) +P(Ai +3 ∪ Ai +4) += ρ = P(Qi = 1). +(36) +indicating that Qi and A are independent. +For any fixed A, define ˜Qi = Qi · 1{i ∈ A} and the reverse +filtration ˜Fj = σ({�j +k=1 ˜Qk, ˜Qj+1, . . . , ˜Qn, A}). Then when +conditioned on A, the established result suggests that +E +� +1 + |i ≤ j : i ∈ A| +1 + � +i≤j,i∈A Qi +����A +� +≤ ρ−1. +(37) +Take expectation over A finishes the proof. +8.1 +Proof of Theorem 2 +Proof. We first control the FSR rate of the second subset. +Specifically, we have +FSR(T) ≤ E +� # {j : γj ̸= 0 and − T ≤ Wj < 0} +1 + # {j : γj ̸= 0 and 0 < Wj ≤ T} +· 1 + # {j : 0 < Wj ≤ T} +# {j : −T ≤ Wj < 0} ∨ 1 +� +≤ q · E +� # {j : γj ̸= 0 and − T ≤ Wj < 0} +1 + # {j : γj ̸= 0 and 0 < Wj ≤ T} +� +. +(38) +The second inequality holds by the definition of T. Now it +suffices to show that +E +� # {j : γj ̸= 0 and − T ≤ Wj < 0} +1 + # {j : γj ̸= 0 and 0 < Wj ≤ T} +� +≤ +c +c − 2. +(39) +For γj ̸= 0, we have a probability of +1 +c−1 to get a clean +˜γj, leading to Zj > Zj+n with non-zero probability, and +a probability of c−2 +c−1 to get a noisy ˜γj, where we have no +information and hence assume a equal probability of Zj > +Zj+n and Zj < Zj+n. Then we have +P(Wi > 0) =P(Wi > 0|˜γ∗ +j ̸= 0)P(˜γ∗ +j ̸= 0) ++ P(Wi > 0|˜γ∗ +j = 0)P(˜γ∗ +j = 0) +≥1 +2 × c − 2 +c − 1 + 0 = +c − 2 +2(c − 1) +(40) +Hence the random variable Bj := 1{Wj>0} ∼ Bernoulli(ρj) +for γj ̸= 0 with ρj ≥ (c − 2)/(2(c − 1)). +Now we consider all the Wj of non-null variables, and +assumes |W1| ≤ · · · ≤ |Wn| with the abuse of subscripts. +We have +γj ̸= 0 and − T ≤ Wj < 0 +⇐⇒ +j ≤ J and Bj = 0 +and +γj ̸= 0 and 0 < Wj ≤ T +⇐⇒ +j ≤ J and Bj = 1 +Hence +# {j : γj ̸= 0 and − T ≤ Wj < 0} +1 + # {j : γj ̸= 0 and 0 < Wj ≤ T} += (1 − B1) + · · · + (1 − BJ) +1 + B1 + · · · + BJ += +1 + J +1 + B1 + · · · + BJ +− 1. +(41) +If we can use Lemma 3, then +E +� # {j : γj ̸= 0 and − T ≤ Wj < 0} +1 + # {j : γj ̸= 0 and 0 < Wj ≤ T} +� +≤ ρ−1−1 ≤ +c +c − 2 +(42) +Then we finally get +FSR(T) ≤ q +c +c − 2. +(43) +as long as c > 2. Now it suffices to show that our +random variables {Bj} are mutually independent. This is +straightforward as we set P(α2; λ) as a sparse penalty for +each row α2,j in Eq. (26), respectively. Then problem of +Eq. (26) now is a combination of independent sub-problems +for each row α2,j, and the solution only depends on +(x2,j, y2,j, β(λ; D1)). Then with fixed β(λ; D1), the mutual +independence naturally exist. +Finally, after we control the FSR rate for the second subset, +we can get the estimate of β(λ; D2) based on the identified +clean data in the second subset, and return to run knockoff- +SPR on the first subset in a similar pipeline. Then we have +for the whole dataset: +FSR = E +�|S1 ∩ C1| + |S2 ∩ C2| +|C1| + |C2| +� +≤ E +�|S1 ∩ C1| +|C1| +� ++ E +�|S2 ∩ C2| +|C2| +� +≤ 2 +c +c − 2q. +(44) +To control the FSR with q, the threshold of T should be +defined as c−2 +2c q, which leads to Theorem 2. +9 +NOISY SET RECOVERY THEOREM OF SPR +Recall that we are solving the problem of +min +⃗γ +���⃗y − ˚ +X⃗γ +��� +2 +2 + λ ∥⃗γ∥1 . +(45) +Proposition 4. Assume that ˚ +X⊤ ˚ +X is invertible. If +����λ ˚ +X⊤ +Sc ˚ +XS +� +˚ +X⊤ +S ˚ +XS +�−1 +ˆvS + ˚ +X⊤ +Sc (I − IS) ( ˚ +Xε) +���� +∞ +< λ +(46) +holds for all ˆvS ∈ [−1, 1]S, where IS = ˚ +XS +� +˚ +X⊤ +S ˚ +XS +�−1 ˚ +X⊤ +S , +then the estimator ˆ⃗γ of Eq. (45) satisfies that +ˆS = supp +�ˆ⃗γ +� +⊆ supp (⃗γ∗) = S. +Moreover, if the sign consistency +sign +�ˆ⃗γS +� += sign (⃗γ∗ +S) +(47) +holds, Then ˆ⃗γ is the unique solution of (45) with the same sign as +ˆ⃗γ∗. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +17 +Proof. Note that Eq. (45) is convex that has global minima. +Denote Eq. (45) as L, the solution of ∂L/∂⃗γ = 0 is the +unique minimizer. Hence we have +∂L +∂⃗γ = − ˚ +X⊤ � +⃗y − ˚ +X⃗γ +� ++ λv = 0 +(48) +where v = ∂ ∥⃗γ∥1 /∂⃗γ. Note that ∥⃗γ∥1 is non-differentiable, +so we instead compute its sub-gradient. Further note that +vi = ∂ ∥⃗γ∥1 /∂⃗γi = ∂ |⃗γi| /∂γi. Hence vi = sign (⃗γi) if ⃗γi ̸= +0 and vi ∈ [−1, 1] if ⃗γi = 0. To distinguish between the two +cases, we assume vi ∈ (−1, 1) if ⃗γi = 0. Hence there exists +ˆv ∈ Rn×1 such that +− ˚ +X⊤ � +⃗y − ˚ +X ˆ⃗γ +� ++ λˆv = 0, +(49) +ˆvi = sign +�ˆ⃗γi +� +if i ∈ ˆS and ˆvi ∈ (−1, 1) if i ∈ ˆSc. +To obtain ˆS ⊆ S, we should have ˆ⃗γi = 0 for i ∈ Sc, that is, +∀i ∈ Sc, |ˆvi| < 1, i.e., +��� ˚ +X⊤ +Sc +� +⃗y − ˚ +XS ˆ⃗γS +���� +∞ < λ, +(50) +For i ∈ S, we have +− ˚ +X⊤ +S +� +⃗y − ˚ +XS ˆ⃗γS +� ++ λˆvS = 0. +(51) +If ˚ +X⊤ ˚ +X is invertible then +ˆ⃗γS = +� +˚ +X⊤ +S ˚ +XS +�−1 � +˚ +X⊤ +S ⃗y − λˆvS +� +(52) +Recall that we have +⃗y = ˚ +XS⃗γ∗ +S + ˚ +X⃗ε +(53) +Hence +ˆ⃗γS = ⃗γ∗ +S+δS, +δS := +� +˚ +X⊤ +S ˚ +XS +�−1 � +˚ +X⊤ +S ˚ +X⃗ε − λˆvS +� +. (54) +Plugging (54) and (53) into (50) we have +���� ˚ +X⊤ +Sc ˚ +X⃗ε − ˚ +X⊤ +Sc ˚ +XS +� +˚ +X⊤ +S ˚ +XS +�−1 � +˚ +X⊤ +S ˚ +X⃗ε − λˆvS +����� +∞ +< λ, +(55) +or equivalently +����λ ˚ +X⊤ +Sc ˚ +XS +� +˚ +X⊤ +S ˚ +XS +�−1 +ˆvS + ˚ +X⊤ +Sc (I − IS) ˚ +X⃗ε +���� +∞ +< λ, +(56) +where IS += +˚ +XS +� +˚ +X⊤ +S ˚ +XS +�−1 ˚ +X⊤ +S . To ensure the sign +consistency, replacing ˆvS += sign (⃗γ∗ +S) in the inequality +above leads to the final result. +Lemma 5. Assume that ⃗ε is indenpendent sub-Gaussian with +zero mean and bounded variance Var (⃗εi) ≤ σ2. +Then with probability at least +1 − 2cn exp +� +� +�− +λ2η2 +2σ2 maxi∈Sc +��� ˚ +Xi +��� +2 +2 +� +� +� +(57) +there holds +��� ˚ +X⊤ +Sc (I − IS) +� +˚ +X⃗ε +���� +∞ ≤ λη +(58) +and���� +� +˚ +X⊤ +S ˚ +XS +�−1 ˚ +X⊤ +S ˚ +X⃗ε +���� +∞ +≤ +λη +√Cmin maxi∈Sc +��� ˚ +Xi +��� +2 +. (59) +Proof. Let zc = ˚ +X⊤ +Sc (I − IS) +� +˚ +X⃗ε +� +, for each i ∈ Sc the +variance can be bounded by +Var (zc +i ) ≤ σ2 ˚ +X⊤ +i (I − IS)2 ˚ +Xi ≤ σ2 max +i∈Sc +��� ˚ +Xi +��� +2 +2 . +Hoeffding inequality implies that +P +���� ˚ +X⊤ +Sc (I − IS) +� +˚ +X⃗ε +���� +∞ ≥ t +� +≤ 2 |Sc| exp +� +� +�− +t2 +2σ2 maxi∈Sc +��� ˚ +Xi +��� +2 +2 +� +� +� , +Setting t = λη leads to the result. +Now let z = +� +˚ +X⊤ +S ˚ +XS +�−1 ˚ +X⊤ +S ˚ +X⃗ε, we have +Var (z) = +� +˚ +X⊤ +S ˚ +XS +�−1 ˚ +X⊤ +S ˚ +XVar (⃗ε) ˚ +X⊤ ˚ +XS +� +˚ +X⊤ +S ˚ +XS +�−1 +≤ σ2 � +˚ +X⊤ +S ˚ +XS +�−1 +≤ +σ2 +Cmin +I. +Then +P +����� +� +˚ +X⊤ +S ˚ +XS +�−1 ˚ +X⊤ +S ˚ +X⃗ε +���� +∞ +≥ t +� +≤ 2 |S| exp +� +−t2Cmin +2σ2 +� +. +Choose +t = +λη +√Cmin maxi∈Sc +��� ˚ +Xi +��� +2 +, +(60) +then there holds +P +� +� +�∥ +� +˚ +X⊤ +S ˚ +XS +�−1 ˚ +X⊤ +S ˚ +X⃗ε∥∞ ≥ +λη +√Cmin maxi∈Sc +��� ˚ +Xi +��� +2 +� +� +� +≤ 2 |S| exp +� +� +�− +λ2η2 +2σ2 maxi∈Sc +��� ˚ +Xi +��� +2 +2 +� +� +� . +9.1 +Proof of Theorem 1 +Proof. The proof essentially follows the treatment in [4]. +The results follow by applying Lemma 5 to Proposition 4. +Inequality (46) holds if condition C2 and the first bound (58) +hold, which proves the first part of the theorem. The +sign consistency (47) holds if condition C3 and the second +bound (59) hold, which gives the second part of the theorem. +It suffices to show that ˆS ⊆ S implies ˆCc ⊆ Cc. Consider +one instance i, there are three possible cases for γ∗ +i ∈ R1×c: +(1) γ∗ +i,j ̸= 0, ∀j ∈ [c]; (2) γ∗ +i,j = 0, ∀j ∈ [c]; (3) ∃j, k ∈ +[c] , s.t. γ∗ +i,j = 0, γ∗ +i,k ̸= 0. If instance i follows case (1) or +case (3), then i ∈ Cc. If it follows case (2), then i ∈ C, and +the indexes of all elements of γi are in Sc. Since we have +ˆS ⊆ S, all elements of γi is in ˆSc, hence i ∈ ˆC. Then we +have ˆCc ⊆ Cc. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +18 +10 +MORE EXPERIMENTAL RESULTS +Histogram of the median value of IRR condition of +SPR. We visualize the median value of the irrepresentable +(IRR) value, i.e., {∥(X⊤ +S XS)−1X⊤ +S Xj∥1}j of SPR final epoch +on CIFAR10 with various noisy scenarios in Fig. 5. As +SPR is running on each piece split from the training set, +we calculate matrix ˚ +X⊤ +Sc ˚ +XS( ˚ +X⊤ +S ˚ +XS)−1 in irrepresentable +condition (C2 in Theorem 1) for each piece at the final +epoch. Then the L1 norm of each row of the matrix is the +IRR value of corresponding clean data. The median value +of IRR values in a single piece is used to construct the +histogram. For the noise scenario of Asy. 40% and Sym. 40%, +the median IRR value is small, indicating weak collinearity +between clean data and noisy data. In these cases, SPR +has more chance to distinguish noisy data from clean data +and thus leads to a good FSR control capacity. For the +noise scenario of Sym. 80%, the median IRR values are +much larger, indicating a strong multi-collinearity. Thus SPR +can hardly distinguish between clean data and noisy data, +leading to a high FSR rate. +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +Symmetric-40% +4 +6 +8 +10 +12 +14 +16 +0 +5 +10 +15 +20 +25 +Symmetric-80% +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +0 +10 +20 +30 +40 +50 +60 +Asymmetric-40% +Fig. 5. Histogram of the median value of the IRR value of SPR on +CIFAR10 with various noisy scenarios. +REFERENCES +[1] Rina Foygel Barber and Emmanuel J. Cand‘es. A knockoff filter +for high-dimensional selective inference. The Annals of Statistics, +47(5):2504 – 2537, 20 (document), 3.1, 5.3, 8, 8 +[2] Yang Cao, Xinwei Sun, and Yuan Yao. Controlling the false +discovery rate in transformational sparsity: Split knockoffs. In +arXiv, 2021. (document), 3.1, 5.3, 8, 8, 8 +[3] Joseph L Doob. Stochastic processes. Wiley New York, 195 8 +[4] M. +J. +Wainwright, +“Sharp +thresholds +for +high-dimensional +and noisy sparsity recovery using l1 -constrained quadratic +programming (lasso),” IEEE transactions on information theory, +2009. (document), 2.3, 9.1 + diff --git a/-tAyT4oBgHgl3EQfqfjJ/content/tmp_files/load_file.txt b/-tAyT4oBgHgl3EQfqfjJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ed3edcec2ba65330f954084d9c4e26293783f822 --- /dev/null +++ b/-tAyT4oBgHgl3EQfqfjJ/content/tmp_files/load_file.txt @@ -0,0 +1,2141 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf,len=2140 +page_content='JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 8, AUGUST 2015 1 Knockoffs-SPR: Clean Sample Selection in Learning with Noisy Labels Yikai Wang, Yanwei Fu, and Xinwei Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Abstract—A noisy training set usually leads to the degradation of the generalization and robustness of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' In this paper, we propose a novel theoretically guaranteed clean sample selection framework for learning with noisy labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Specifically, we first present a Scalable Penalized Regression (SPR) method, to model the linear relation between network features and one-hot labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' In SPR, the clean data are identified by the zero mean-shift parameters solved in the regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' We theoretically show that SPR can recover clean data under some conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Under general scenarios, the conditions may be no longer satisfied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' and some noisy data are falsely selected as clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' To solve this problem, we propose a data-adaptive method for Scalable Penalized Regression with Knockoff filters (Knockoffs-SPR), which is provable to control the False-Selection-Rate (FSR) in the selected clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' To improve the efficiency, we further present a split algorithm that divides the whole training set into small pieces that can be solved in parallel to make the framework scalable to large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' While Knockoffs-SPR can be regarded as a sample selection module for a standard supervised training pipeline, we further combine it with a semi-supervised algorithm to exploit the support of noisy data as unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Experimental results on several benchmark datasets and real-world noisy datasets show the effectiveness of our framework and validate the theoretical results of Knockoffs-SPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Our code and pre-trained models will be released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Index Terms—Learning with Noisy Labels, Knockoffs Method, Type-Two Error Control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 1 INTRODUCTION D EEP learning has achieved remarkable success on many supervised learning tasks trained by millions of labeled training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' The performance of deep models heavily relies on the quality of label annotation since neural networks are susceptible to noisy labels and even can easily memorize randomly labeled annotations [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Such noisy labels can lead to the degradation of the generalization and robustness of such models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Critically, it is expensive and difficult to obtain precise labels in many real-world scenarios, thus exposing a realistic challenge for supervised deep models to learn with noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' There are many previous efforts in tackling this challenge by making the models robust to noisy data, such as modifying the network architectures [2]–[5] or loss functions [6]–[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' This paper addresses the challenge by directly selecting clean samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Inspired by the dynamic sample selection methods [9]–[16], we construct a “virtuous” cycle between sample selection and network training: the selected clean samples can improve the network training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' and on the other hand, the improved network has a more powerful ability in picking up clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' As this cycle evolves, the performance can be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' To well establish this cycle, a key question remains: how to effectively differentiate clean data from noisy ones?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Preliminary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Typical principles in existing works [9]–[16] to differentiate clean data from noisy data include large loss [11], inconsistent prediction [17], and irregular feature Yikai Wang and Yanwei Fu contribute equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Xinwei Sun is the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Yikai Wang, Yanwei Fu and Xinwei Sun are with the School of Data Science, Fudan University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' E-mail: {yikaiwang19, yanweifu, sunxinwei}@fudan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content='cn representation [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' The former two principles identify irregular behaviors in the label space, while the last one analyzes the instance representations of the same class in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' In this paper, we propose unifying the label and feature space by making the linear relationship, yi = x⊤ i β + ε, (1) between feature-label pair (xi ∈ Rp: feature vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' yi ∈ Rc: one-hot label vector) of data i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' We also have the fixed (unknown) coefficient matrix β ∈ Rp×c, and random noise ε ∈ Rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Essentially, the linear relationship here is an ideal approximation, as the networks are trained to minimize the divergence between a (soft-max) linear projection of the feature and a one-hot label vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' For a well-trained network, the output prediction of clean data is expected to be as similar to a one-hot vector as possible, while the entropy of the output of noisy data should be large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Thus if the underlying linear relation is well-approximated without soft-max operation, the corresponding data is likely to be clean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' In contrast, the feature-label pair of noisy data may not be approximated well by the linear model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' The simplest way to measure the goodness of the linear model in fitting the feature-label pair is to check the prediction error, or residual, ri = yi − x⊤ i ˆβ, where ˆβ is the estimate of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' The larger ∥r∥ indicates a larger fitting error and thus more possibility for instance i to be outlier/noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Many methods have been proposed to test whether ri is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Particularly, we highlight the classical statistical leave-one-out approach [19] that computes the studentized residual as, ti = yi − x⊤ i ˆβ−i ˆσ−i � 1 + x⊤ i �X⊤ −iX−i �−1 xi �1/2 , (2) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content='00545v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content='LG] 2 Jan 2023 JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 8, AUGUST 2015 2 Images Features Noisy Labels Stage 1: Network Learning Classifier Stage 2: Sample Selection β ∈ Rd×c Noisy Labels Y ∈ Rn×c X ∈ Rn×d Features Noisy Data Indicator = + γ ∈ Rn×c Yi (1, 0, 0) (0, 1, 0) ˜Yi Permute Compare Selected Clean Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Knockoffs-SPR runs a cycle between network learning and sample selection, where clean data are selected via the comparison of the mean-shift parameters between its original label and permuted label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' where ˆσ is the scale estimate and the subscript −i indicates estimates based on the n − 1 observations, leaving out the i-th data for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Equivalently, the linear regression model can be re-formulated into explicitly representing the residual, Y = Xβ + γ + ε, εi,j ∼ N(0, σ2), (3) by introducing a mean-shift parameter γ as in [20] with the feature X ∈ Rn×p, and label Y ∈ Rn×c paired and stacked by rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' For each row of γ ∈ Rn×c, γi represents the predict residual of the corresponding data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' This formulation has been widely studied in different research topics, including economics [21]–[24], robust regression [20], [25], statistical ranking [26], face recognition [27], semi-supervised few-shot learning [28], [29], and Bayesian preference learning [30], to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' This formulation is differently focused on the specific research tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' For example, for the robust regression problem [20], [25], the target is to get a robust estimate ˆβ against the influence of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Here for solving the problem of learning with noisy labels, we are interested in recovering zeros elements of γ, since these elements correspond to clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' SPR [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' To this end, from the statistical perspective, our conference report [31] starts from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' (3) to build up a sample selection framework, dubbed Scalable Penalized Regression (SPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' With a sparse penalty P(γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' λ) on γ, the SPR obtains a regularization solution path of γ(λ) by evolving λ from ∞ to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Then it identifies those samples that are earlier (or at larger λ) selected to be non-zeros as noisy data and those later selected as clean data, with a manually specified ratio of selected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Under the irrepresentable condition [4], [33], the SPR enjoys model selection consistency in the sense that it can recover the set of noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' By feeding only clean data into next-round training, the trained network is less corrupted by the noisy data and hence performs well empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Knockoffs-SPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' However, the irrepresentable condition demands the prior of the ground-truth noisy set, which is not accessible in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' When this condition fails, the trained network with SPR may be still corrupted by a large proportion of noisy data, leading to performance degradation as empirically verified in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' To amend this problem, we provide a data-adaptive sample selection algorithm, in order to well control the expected rate of noisy data in the selected data under the desired level q, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=', q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' As the goal is to identify clean data for the next-round training, we term this rate as the False-Selection- Rate (FSR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' The FSR is the expected rate of the type-II error in sparse regression, as non-zero elements correspond to the noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Our method to achieve the FSR control is inspired by the ideas of Knockoffs in Statistics, which is a recently developed framework for variable selection [1], [2], [34], [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' The Knockoffs framework aims at selecting non-null variables and controlling the False-Discovery-Rate (FDR), by taking as negative controls knockoff features ˜ X, which are constructed as a fake copy for the original features X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Here, the FDR corresponds to the expectation of the type-I error rate in sparse regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Therefore, the vanilla Knockoffs cannot be directly applied to our SPR framework, since FSR is the expected rate of the type-II error and there is no theoretical guarantee in Knockoffs for this control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' To achieve the FSR control, we propose Knockoffs-SPR, which turns to construct the knockoff labels ˜Y via permutation for the original label Y , and incorporates it into a data-partition strategy for FSR control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Formally, we repurpose the knockoffs in Statistics in our SPR method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' and propose a novel data-adaptive sample selection algorithm, dubbed Knockoffs-SPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' It extends SPR in controlling the ratio of noisy data among the selected clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' With this new property, Knockoffs-SPR ensures that the clean pattern is dominant in the data and hence leads to better network training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Specifically, we partition the whole noisy training set into two random subsets and apply the Knockoffs-SPR to two subsets separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' For each time, we use one subset to estimate the intercept β and the other to select the clean data by comparing between the solution paths of γ(λ) and ˜γ(λ) that respectively obtained via regression on noisy labels and the permuted labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' With such a decoupled structure between β and γ, we prove that the FSR can be controlled by any prescribed level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Compared with the original theory of SPR, our new theory enables us to effectively select clean data under general conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Besides, Knockoffs-SPR also enjoys a superior performance over the original SPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Together with network training, the whole framework is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 1 in which the sample selection and the network learning are well incorporated into each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Specifically, we run the network learning process and sample selection process iteratively and repeat this cycle until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' To incorporate Knockoffs-SPR into the end-to-end training pipeline of deep architecture, the simplest way is to directly solve Knockoffs-SPR for each featureA B C DJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 8, AUGUST 2015 3 training mini-batch or training epoch to select clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Solving Knockoffs-SPR for each mini-batch is efficient but suffers from the identifiability issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' The sample size in a mini-batch may be too small to distinguish clean patterns from noisy ones among all classes, especially for large datasets with small batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Solving Knockoffs-SPR for the whole training set is powerful but suffers from the complexity issue, leading to an unacceptable computation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' To resolve these two problems, we strike a balance between complexity and identifiability by proposing a splitting strategy that divides the whole data into small pieces such that each piece is class-balanced with the proper sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' In this regard, the sample size of each piece is small enough to be solved efficiently and large enough to distinguish clean patterns from noisy ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Then Knockoffs- SPR runs on each piece in parallel, making it scalable to large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' As the removed noisy data still contain useful information for network training, we adopt the semi-supervised training pipeline with CutMix [38] where the noisy data are utilized as unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' We conduct extensive experiments to validate the effectiveness of our framework on several benchmark datasets and real-world noisy datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' The results show the efficacy of our Knockoffs-SPR algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Our contributions are as follows: Ideologically, we propose to control the False-Selection- Rate in selecting clean data, under general scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Methodologically, we propose Knockoffs-SPR, a data- adaptive method to control the FSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Theoretically, we prove that the Knockoffs-SPR can control the FSR under any desired level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Algorithmically, we propose a splitting algorithm for better sample selection with balanced identifiability and complexity in large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Experimentally, we demonstrate the effectiveness and efficiency of our method on several benchmark datasets and real-world noisy datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Our conference version of this work, SPR, was published in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Compared with SPR [31], we have the following extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' We identify the limitation of the SPR and consider the FSR control in selecting clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' We propose a new framework: Knockoffs-SPR which is effective in selecting clean data under general scenarios, theoretically and empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' We apply our method on Clothing1M and achieve better results than compared baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Logistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' The rest of this paper is organized as follows: In Section 9, we introduce our SPR algorithm with its noisy set recovery theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' In Section 3, the Knockoffs-SPR algorithm is introduced with its FSR control theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' In Section 4, several training strategies are proposed to well incorporate the Knockoffs-SPR with the network training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' In Section 5, connections are made between our proposed works and several previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' In Section 6, we conduct experiments on several synthetic and real-world noisy datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Section 7 concludes this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 2 CLEAN SAMPLE SELECTION 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content='1 Problem Setup We are given a dataset of image-label pairs {(imgi, yi)}n i=1, where the noisy label yi is corrupted from the ground- truth label y∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' The ground-truth label y∗ i and the corruption process are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Our target is to learn a recognition model f(·) such that it can recognize the true category y∗ i from the image imgi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=', f(imgi) = y∗ i , after training on the noisy label yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' In this paper, we adopt deep neural networks as the recognition model and divide the f(·) into fc(g(·)) where g(·) is the deep model for feature extraction and fc(·) is the final fully-connected layer for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' For each input image imgi, the feature extractor g(·) is used to encode the feature xi := g(imgi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Then the fully-connected layer is used to output the score vector ˆyi = fc(xi) which indicates the chance it belongs to each class and the prediction is provided with ˆyi = argmax(ˆyi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' As the training data contain many noisy labels, simply training from all the data leads to severe degradation of generalization and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Intuitively, if we could identify the clean labels from the noisy training set, and train the network with the clean data, we can reduce the influence of noisy labels and achieve better performance and robustness of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' To achieve this, we thus propose a sample selection algorithm to identify the clean data in the noisy training set with theoretical guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' In this paper, we will use a to represent scalar, a to represent a vector, and A to represent a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' We will annotate a∗ to denote the ground-truth value of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' We use ∥ · ∥F to denote the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content='2 Clean Sample Selection via Penalized Regression Motivated by the leave-one-out approach for outlier detection, we introduce an explicit noisy data indicator γi for each data and assume a linear relation between extracted feature xi and one-hot label yi with noisy data indicator as, yi = x⊤ i β + γi + εi, (4) where yi ∈ Rc is one-hot vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' and xi ∈ Rp, β ∈ Rp×c, γi ∈ Rc, εi ∈ Rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' The noisy data indicator γi can be regarded as the correction of the linear prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' For clean data, yi ∼ N(x⊤ i β∗, σ2Ic) with γ∗ i = 0, and for noisy data y∗ i = yi −γ∗ i ∼ N(x⊤ i β∗, σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' We denote C := {i : γ∗ i = 0} as the ground-truth clean set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' To select clean data for training, we propose Scalable Penalized Regression (SPR), designed as the following sparse learning paradigm, argmin β,γ 1 2 ∥Y − Xβ − γ∥2 F + P(γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' λ), (5) where we have the matrix formulation X ∈ Rn×p, and Y ∈ Rn×c of {xi, yi}n i=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' and P(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' λ) is a row-wise JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 8, AUGUST 2015 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Solution Path of SPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Red lines indicate noisy data while blue lines indicate clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' As λ decreases, the γi gradually solved with non-zero values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' sparse penalty with coefficient parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' So we have P(γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' λ) = �n j=1 P(γi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' λ), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=', group-lasso sparsity with P(γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' λ) = λ � i ∥γi∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' To estimate C, we only need to solve γ with no need to estimate β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Thus to simplify the optimization, we substitute the Ordinary Least Squares (OLS) estimate for β with γ fixed into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' To ensure that ˆβ is identifiable, we apply PCA on X to make p ≪ n so that the X has full-column rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Denote ˜ X = I − X �X⊤X �† X⊤, ˜Y = ˜ XY , the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' (5) is transformed into argmin γ 1 2 ��� ˜Y − ˜ Xγ ��� 2 F + P(γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' λ), (6) which is a standard sparse linear regression for γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Note that in practice we can hardly choose a proper λ that works well in all scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Furthermore, from the equivalence between the penalized regression problem and Huber’s M-estimate, the solution of γ is returned with soft-thresholding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Thus it is not worth finding the precise solution of a single γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Instead, we use a block-wise descent algorithm [39] to solve γ with a list of λs and generate the solution path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' As λ changes from ∞ to 0, the influence of sparse penalty decreases, and γi are gradually solved with non-zero values, in other words, selected by the model, as visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Since earlier selected instance is more possible to be noisy, we rank all samples in the descendent order of their selecting time defined as: Zi = sup {λ : γi (λ) ̸= 0} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' (7) A large Zi means that the γi is earlier selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Then the top samples are identified as noisy data and the other samples are selected as clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' In practice, we select 50% of the data as clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content='3 The Theory of Noisy Set Recovery in SPR The SPR enjoys theoretical guarantees that the noisy data set can be fully recovered with high probability, under the irrepresentable condition [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Formally, consider the vectorized version of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' (6): argmin ⃗γ 1 2 ���⃗y − ˚ X⃗γ ��� 2 2 + λ ∥⃗γ∥1 , (8) where ⃗y, ⃗γ is vectorized from Y , γ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' (6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' ˚ X = Ic ⊗ ˜ X with ⊗ denoting the Kronecker product operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Denote S := supp(⃗γ∗), which is the noisy set Cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' We further denote ˚ XS (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' ˚ XSc) as the column vectors of ˚ X whose indexes are in S (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Sc) and µ ˚ X = maxi∈Sc ∥ ˚ X∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Then we have Theorem 1 (Noisy set recovery).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Assume that: C1, Restricted eigenvalue: λmin( ˚ X⊤ S ˚ XS) = Cmin > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' C2, Irrepresentability: there exists a η ∈ (0, 1], such that ∥ ˚ X⊤ Sc ˚ XS( ˚ X⊤ S ˚ XS)−1∥∞ ≤ 1 − η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' C3, Large error: ⃗γ∗ min := mini∈S |⃗γ∗ i | > h(λ, η, ˚ X, ⃗γ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' where ∥A∥∞ := maxi � j |Ai,j|, and h(λ, η, ˚ X, ⃗γ∗) = λη/�Cminµ ˚ X + λ∥( ˚ X⊤ S ˚ XS)−1sign(⃗γ∗ S)∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Let λ ≥ 2σ√µ ˚ X η √log cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Then with probability greater than 1 − 2(cn)−1, model Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' (8) has a unique solution ˆ⃗γ such that: 1) If C1 and C2 hold, ˆ Cc ⊆ Cc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content='2) If C1, C2 and C3 hold, ˆ Cc = Cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' We present the proof in the appendix, following the treatment in [4], [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' In this theorem, C1 is necessary to get a unique solution, and in our case is mostly satisfied with the natural assumption that the clean data is the majority in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' If C2 holds, the estimated noisy data is the subset of truly noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' This condition is the key to ensuring the success of SPR, which requires divergence between clean and noisy data such that we cannot represent clean data with noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' If C3 further holds, the estimated noisy data is exactly all the truly noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' C3 requires the error measured by γi is large enough to be identified from random noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' If the conditions fail, SPR will fail in a non- vanishing probability, not deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 3 CONTROLLED CLEAN SAMPLE SELECTION In the last section, we stop the solution path at λ such that 50% samples are selected as clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' If this happens to be the rate of clean data, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 1 shows that our SPR can identify the clean data C under the irrepresentable condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' However, the irrepresentable condition and the information of the ground-truth clean set C are practically unknown, making this theory hard to be used in the real life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Particularly, with |Cc| unknown, the algorithm can stop at an improper time such that the noisy rate of the selected clean data ˆC can be still high, making the next-round trained model corrupted a lot by noisy patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' To resolve the problem of false selection in SPR , we in this section propose a data-adaptive early stopping method for the solution path, that targets controlling the expected noisy rate of the selected data dubbed as False-Selection-Rate (FSR) under the desired level q (0 < q < 1): FSR = E � �# � j : j ̸∈ H0 ∩ ˆC � # � j : j ∈ ˆC � ∨ 1 � � , (9) where ˆC = {j : ˆγj = 0} is the recovered clean set of γ, and H0 : γ∗ i = 0 denotes the null hypothesis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=', the sample i belonging to the clean dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Therefore, the FSR in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' (9) targets controlling the false rate among selected null hypotheses, which is also called the expected rate of the type-II error in hypothesis testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 8, AUGUST 2015 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content='1 Knockoffs-SPR To achieve the FSR control, we propose the Knockoffs- SPR for clean sample selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Our method is inspired by knockoff methods [1], [2], [34], [35], [41] with the different focus that we target selecting clean labels via permutation instead of constructing knockoff features to select explanatory variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Specifically, under model (4) we permute the label for each data and construct the permutation ˜y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Then model (4) can be solved for y and ˜y to obtain the solution paths γ(λ) and ˜γ(λ), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' We will show that this construction can pick up clean data from noisy ones, by comparing the selecting time (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' (7)) between γ(λ) and ˜γ(λ) for each data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' On the basis of this construction, we propose to partition the whole dataset into two disjoint parts, with one part for estimating β and the other for learning γ(λ) and ˜γ(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' We will show that the independent structure with such a data partition enables us to construct the comparison statistics whose sign patterns among alternative hypotheses (noisy data) are the independent Bernoulli processes, which is crucial for FSR control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Formally speaking, we split the whole data D into D1 := (X1, Y1) and D2 := (X2, Y2) with ni := |Di|, and implement Knockoffs-SPR on both D1 and D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' In the following, we only introduce the procedure on D2, as the procedure for D1 shares the same spirit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Roughly speaking, the procedure is composed of three steps: i) estimate β on D1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' ii) estimate ˜γ(λ)) on D2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' and iii) construct the comparison statistics and selection filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' We leave detailed discussions for each step in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Step i): Estimating β on D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Our target is to provide an estimate of β that is independent of D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' The simplest strategy is to use the standard OLS estimator to obtain ˆβ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' However, this estimator may not be accurate since it is corrupted by noisy samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' For this consideration, we first run SPR on D1 to get clean data and then solve β via OLS on the estimated clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Step ii): Estimating (γ(λ), ˜γ(λ)) on D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' After obtaining the solution ˆβ1 on D1 , we learn the γ(λ) on D2: 1 2 ���Y2 − X2 ˆβ1 − γ2 ��� 2 F + P(γ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' (10) For each one-hot encoded vector y2,j, we randomly permute the position of 1 and obtain another one-hot vector ˜y2,j ̸= y2,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' For clean data j, the ˜y2,j turns to be a noisy label;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' while for noisy data, the ˜y2,j is switched to another noisy label with probability c−2 c−1 or clean label with probability 1 c−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' After obtaining the permuted matrix as ˜Y2, we learn the solution paths (γ2(λ), ˜γ2(λ)) using the same algorithm as SPR via: � � � � � 1 2 ���Y2 − X2 ˜β1 − γ2 ��� 2 F + � j P(γ2,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' λ), 1 2 ��� ˜Y2 − X2 ˜β1 − ˜γ2 ��� 2 F + � j P(˜γ2,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' (11) Step iii): Comparison statistics and selection filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' After obtaining the solution path (γ2(λ), ˜γ2(λ)), we define sample significance scores with respect to y2,i and ˜y2,i of each i respectively, as the selection time: Zi := sup{λ : Algorithm 1 Knockoffs-SPR Input: subsets D1 and D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Output: clean set of D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 1: Use D1 to fit an linear regression model and get β(D1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 2: Generate permuted label of each sample i in D2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 3: Solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' (26) for D2 and generate {Wi} by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' (12);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 4: Initialize q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content='02 and T = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 5: while q < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content='5 and T = 0 do 6: Compute T by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' (13);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 7: q = q + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content='02;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 8: end while 9: if T is 0 then 10: Construct clean set via half of the samples with largest Wi in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' (14) with T = ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 11: else 12: Construct clean set via samples in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' (14);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 13: end if 14: return clean set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' ∥γ2,i(λ)∥2 ̸= 0} and ˜Zi := sup{λ : ∥˜γ2,i(λ)∥2 ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' With Zi, ˜Zi, we define the Wi as: Wi := Zi · sign(Zi − ˜Zi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' (12) Based on these statistics, we define a data-dependent threshold T as T = max � t > 0 : 1 + # {j : 0 < Wj ≤ t} # {j : −t ≤ Wj < 0} ∨ 1 ≤ q � , (13) or T = 0 if this set is empty, where q is the pre-defined upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Our algorithm will select the clean subset identified by C2 := {j : −T ≤ Wj < 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' (14) Empirically, T may be equal to 0 if the threshold q is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' In this regard, no clean data are selected, which is meaningless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Therefore, we start with a small q and iteratively increase q and calculate T, until an attainable T such that T > 0 to bound the FSR as small as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' In practice, when the FSR cannot be bounded by q = 50%, we will end the selection and simply select half of the most possible clean examples via {Wj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' The whole procedure of Knockoffs-SPR is shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content='2 Statistical Analysis about Knockoffs-SPR In this part, we present the motivations and intuitions of each step in Knockoffs-SPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Data Partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Knockoffs-SPR partitions the dataset D into two subset D1 and D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' This step decomposes the dependency of the estimate of β and γ in that we use D1/D2 to estimate β/γ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Then ˆβ(D1) is independent of ˆγ(D2) if D1 and D2 are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' The independent estimation of β and γ makes it provable for FSR control on D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' As we discussed in step ii, when the original label is clean, its permuted label will be a noisy label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' On the other hand, if the original label is noisy, its permuted label changes to clean with probability 1 c−1 and noisy with JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 8, AUGUST 2015 6 probability c−2 c−1, where c denotes the number of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Note that γ of noisy data is often selected earlier than that of clean data in the solution path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' This implies larger Z values for noisy data than those for clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' As a result, according to the definition of W, a clean sample will ideally have a small negative of W := Z · sign(Z − ˜Z), where Z and ˜Z respectively correspond to the clean label and noisy label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' In contrast for a noisy sample, the W tends to have a large magnitude and has approximately equal probability to be positive or negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Such a different behavior of W between clean and noisy data can help us to identify clean samples from noisy ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Asymmetric comparison statistics W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' The classical way to define comparison statistics is in a symmetric manner, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=', Wi := Zi ∨ ˜Zi · sign(Zi − ˜Zi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' In this way, a clean sample with a noisy permuted label tends to have a large |Wi|, as we expect the noisy label to have a large ˜Zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' However, this is against our target as we only require clean samples to have a small magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' For this purpose, we design asymmetric comparison statistics that only consider the magnitude of the original labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' To see the asymmetric behavior of W for noisy and clean data, we consider the Karush–Kuhn–Tucker (KKT) conditions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' (26) with respect to (γ2,i, ˜γ2,i) γ2,i + ∂P(γ2,i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' λ) ∂γ2,i = x⊤ 2,i(β∗ − ˆβ1) + γ∗ 2,i + ε(2),i, (15a) ˜γ2,i + ∂P(˜γ2,i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' λ) ∂˜γ2,i = x⊤ 2,i(β∗ − ˆβ1) + ˜γ∗ 2,i + ˜ε(2),i, (15b) where ε(2),i ∼i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content='d ˜ε(2),i, |γ∗ 2,i| = |˜γ∗ 2,i| if both y2,i and ˜y2,i are noisy, and P(γ2,i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' λ) := λ|γ2,i| as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' By conditioning on ˆβ1 and denoting ai := x⊤ 2,i(β∗ − ˆβ1), we have that P(Wi > 0) = P(|ai+γ∗ 2,i+ε2,i| > |ai+ ˜γ∗ 2,i+ ˜ε(2),i|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' (16) Then it can be seen that if i is clean, we have γ∗ 2,i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Then Zi tends to be small and besides, it is probable to have Zi < ˜Zi if ˆβ1 can estimate β∗ well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' As a result, Wi tends to be a small negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' On the other hand, if i is noisy, then Zi tends to be large for γi to account for the noisy pattern, and besides, it has equal probability between Zi < ˜Zi and Zi ≥ ˜Zi when ˜y2,i is switched to another noisy label, with probability c−2 c−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' So Wi tends to have a large value and besides, P(Wi > 0) = P(Wi > 0|˜y2,i is noisy)P(˜y2,i is noisy) + P(Wi > 0|˜y2,i is clean)P(˜y2,i is clean) = 1 2 · c − 2 c − 1 + P(Wi > 0|˜y2,i is clean) · 1 c − 1, (17) which falls in the interval of � c−2 c−1 · 1 2, c c−1 · 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' That is to say, P(Wi > 0) ≈ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' In this regard, the clean data corresponds to small negatives of W in the ideal case, which can help to discriminate noisy data with large W with almost equal probability to be positive or negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' For noisy y2,i, we have P(Wi > 0|˜y2,i is noisy) = 1/2 by assuming |γ∗ 2,i| = |˜γ∗ 2,i|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' However, it may not hold in practice when y2,i corresponds to the noisy pattern that has been learned by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' In this regard, it may have |γ∗ 2,i| < |˜γ∗ 2,i| for a randomly permuted label ˜y2,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' To resolve this problem, we instead set the permutation label as the most confident candidate of the model, please refer to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content='1 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Besides, if ˆβ1 can accurately estimate β∗, according to KKT conditions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' (15), we have P(Wi > 0) < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' That is Wi tends to be negative for the clean data, which is beneficial for clean sample selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Data-adaptive threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' The proposed data-adaptive threshold T is directly designed to control the FSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' Specifically, the FSR defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' (9) is equivalent to FSR(t) = E �# {j : γj ̸= 0 and − t ≤ Wj < 0} # {j : −t ≤ Wj < 0} ∨ 1 � , (18) where the denominator denotes the number of selected clean data according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' (14) and the nominator denotes the number of falsely selected noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' This form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' (18) can be further decomposed into, E � # {γj ̸= 0, −t ≤ Wj < 0} 1 + # {γj ̸= 0, 0 < Wj ≤ t} · 1 + # {γj ̸= 0, 0 < Wj ≤ t} # {−t ≤ Wj < 0} ∨ 1 � ≤ E � # {γj ̸= 0, −t ≤ Wj < 0} 1 + # {γj ̸= 0, 0 < Wj ≤ t} 1 + # {0 < Wj ≤ t} # {−t ≤ Wj < 0} ∨ 1 � ≤ E � # {γj ̸= 0, −t ≤ Wj < 0} 1 + # {γj ̸= 0, 0 < Wj ≤ t}q � , (19) where the last inequality comes from the definition of T in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'} +page_content=' To control the FSR, it suffices to bound E � #{γj̸=0, −t≤Wj<0} 1+#{γj̸=0, 0 0 +� . +Throughout all the paper, the words minimum point will be used to denote a local or +global minimum point of a (potential) function. +Definition 1.2. A (global) solution (0, +∞) → Rdst, r �→ u(r), of the differential system +(1.2) (in particular a radially symmetric stationary solution of system (1.1)) is said to be +stable at infinity if u(r) approaches a point of Σmin(V ) as r goes to +∞. If this point of +Σmin(V ) is denoted by u∞, then the solution is said to be stable close to u∞ at infinity. +Notation. For every u∞ in Σmin(V ), let SV, u∞ denote the set of the radially symmetric +stationary solutions of system (1.1) that are stable close to u∞ at infinity. With symbols, +SV, u∞ = +�u : [0, +∞) → Rdst : u satisfies (1.2) and (1.3) and u(r) −−−−→ +r→+∞ u∞ +� . +Let +S0 +V, u∞ = +� +u(0) : u ∈ SV, u∞ +� +, +and let +(1.4) +SV = +� +u∞∈Σmin(V ) +SV, u∞ +and +S0 +V = +� +u∞∈Σmin(V ) +S0 +V, u∞ . +The following statement is an equivalent (simpler) formulation of conclusion 2 of +Corollary 1.1. +Corollary 1.3. For a generic potential V , the subset S0 +V of Rdst is discrete. +1.3 Differential systems governing radially symmetric stationary solutions +The second-order differential system (1.2) is equivalent to the (non-autonomous) 2dst- +dimensional first order differential differential system +(1.5) +� +� +� +� +� +˙u = v +˙v = −dsp − 1 +r +v + ∇V (u) . +Introducing the auxiliary variables τ and c defined as +(1.6) +τ = log(r) +and +c = 1 +r , +4 + +the previous 2dst-dimensional differential system (1.5) is equivalent to each of the following +two 2dst + 1-dimensional autonomous differential systems: +(1.7) +� +� +� +� +� +� +� +uτ = rv +vτ = −(dsp − 1)v + r∇V (u) +rτ = r , +and +(1.8) +� +� +� +� +� +� +� +ur = v +vr = −(dsp − 1)cv + ∇V (u) +cr = −c2 . +Remark. Integrating the third equations of systems (1.7) and (1.8) yields +r = r0eτ−τ0 +and +1 +c − 1 +c0 += r − r0 , +and the parameters τ0 and c0 (which determine in each case the origin of “time”) do +not matter in principle, since those systems are autonomous. However, if the “initial +conditions” r0 and c0 are positive (which is true for the solutions that describe radially +symmetric stationary solutions of system (1.1)), it is natural to choose, in each case, the +origins of time according to equalities (1.6), that is : +τ0 = ln(r0) +and +c0 = 1 +r0 +. +Properties close to origin. +System (1.7) is relevant to provide an insight into the limit +system (1.5) as r goes to 0. The subspace R2dst × {0} (r equal to 0) is invariant by the +flow of this system, and the system reduces on this invariant subspace to +(1.9) +� +uτ = 0 +vτ = −(dsp − 1)v , +see figure 1.1. For every u0 in Rdst, the point (u0, 0Rdst, 0) is an equilibrium of sys- +tem (1.7); let us denote by W u, 0 +V +(u0) the (one-dimensional) unstable manifold of this +equilibrium, for this system, let +(1.10) +W u, 0, + +V +(u0) = W u, 0 +V +(u0) ∩ +�R2dst × (0, +∞) +� , +and let +W u, 0, + +V += +� +u0∈Rdst +W u, 0, + +V +(u0) . +The subspace +(1.11) +Ssym = Rdst × {0Rdst} × {0} +of R2dst+1 can be seen as the higher space dimension analogue of the symmetry (reversibil- +ity) subspace Rdst × {0Rdst} of R2dst (which is relevant for symmetric standing pulses in +space dimension 1, see [1] and subsection 1.7 below); the set W u, 0, + +V +can be seen as the +unstable manifold of this subspace Ssym. +5 + +Figure 1.1: Dynamics of the (equivalent) differential systems (1.7) (for r nonnegative +finite) and (1.8) (for c = 1/r nonnegative finite) in Rdst × Rdst × [0, +∞] (this domain is +three-dimensional if dst is equal to 1, as on the figure). For the limit differential system +(1.9) in the subspace r = 0 (in green), the trajectories are vertical and the solutions +converge towards the horizontal u-axis, defined as Ssym in (1.11), and which is the higher +space dimensional analogue of the symmetry subspace for symmetric standing pulses +in space dimension 1. The point u∞ is a local minimum point of V , so that the point +(u∞, 0Rdst) is a hyperbolic equilibrium for the limit differential system (1.12) in the +subspace c = 0 ⇐⇒ r = +∞ (in blue). Systems (1.7) and (1.8) are autonomous, but the +quantity r (the quantity c) goes monotonously from 0 to +∞ (from +∞ to 0) for all the +solutions in the subspace r > 0 ⇐⇒ c > 0, so that those solutions can be parametrized +with r (with c) as time. The unstable manifold W u, 0, + +V +(u0) is one-dimensional and is a +transverse intersection between the unstable set W u, 0, + +V +of the subspace {r = 0, v = 0Rdst} +and the centre stable manifold W cs, ∞, + +V +(u∞) of the equilibrium (u∞, 0Rdst, c = 0). To +prove the generic transversality of this intersection is the main goal of the paper. The +dotted red curve is the projection onto the (u, r)-subspace of this intersection. The part of +W cs, ∞, + +V +(u∞) which is displayed on the figure can also be seen as the local centre stable +manifold W cs, ∞, + +loc, V, ε1, c1(u∞) defined in (2.10) (with u∞ equal to the point u∞,1 introduced +there). +6 + +wo +u +sym +om +L +0 +(8m) +loc, V, E1, C1 +C1 +L0= +u +8 +E1Properties close to infinity. +System (1.8) is relevant to provide an insight into the limit +system (1.5) as r goes to +∞. The subspace R2dst × {0} of R2dst+1 (c equal to 0, or +in other words r equal to +∞) is invariant by the flow of this system, and the system +reduces on this invariant subspace to +(1.12) +� +ur = v +vr = ∇V (u) . +For every u∞ in Σmin(V ), the point (u∞, 0Rdst, 0) is an equilibrium of system (1.8); let +us consider its global centre-stable manifold in R2dst × (0, +∞), defined as +(1.13) +W cs, ∞, + +V +(u∞) = +� +(u0, v0, c0) ∈ R2dst × (0, +∞) : the solution of system (1.8) +with initial condition (u0, v0, c0) at “time” r0 = 1/c0 is +defined up to +∞ and goes to (u∞, 0, 0) as r goes to +∞ +� +. +This set W cs, ∞, + +V +(u∞) is a dst + 1-dimensional submanifold of R2dst × (0, +∞) (see +subsection 2.4). +Radially symmetric stationary solutions. +Let us consider the involution +ι : R2dst × (0, +∞) → R2dst × (0, +∞) , +(u, v, r) �→ (u, v, 1/r) . +The following lemma, proved in subsection 2.1, formalizes the correspondence between +the radially symmetric stationary solutions stable at infinity for system (1.1) and the +manifolds defined above. +Lemma 1.4. Let u∞ be a point of Σmin(V ). A (global) solution [0, +∞) → Rdst, r �→ u(r) +of system (1.2) belongs to SV, u∞ if and only if its trajectory (in R2dst × (0, +∞)) +(1.14) +��u(r), ˙u(r), r +� : r ∈ (0, +∞) +� +belongs to the intersection +(1.15) +W u, 0, + +V +∩ ι−1�W cs, ∞, + +V +(u∞) +� . +1.4 Transversality of radially symmetric stationary solutions stable at infinity +Definition 1.5. Let u∞ be a point of Σmin(V ). A radially symmetric stationary solution +stable close to u∞ at infinity for system (1.1) (in other words, a function u of SV, u∞) is +said to be transverse if the intersection (1.15) is transverse, in R2dst × (0, +∞), along the +trajectory (1.14). +Remark. The natural analogue of radially symmetric stationary solutions stable at infinity +when space dimension dsp is equal to 1 are symmetric standing pulses stable at infinity +(see Definition 1.5 of [1]), and the natural analogue for such pulses of Definition 1.5 above +7 + +is their elementarity, not their transversality (see Definition 1.4 and Definition 1.6 of [1]). +However, the transversality of a symmetric standing pulse (when the space dimension +dsp equals 1) makes little sense in higher space dimension, because of the singularity at r +equals 0 for the differential systems (1.2) and (1.5), or because of the related fact that +the subspace {r = 0} is invariant for the differential system (1.7). For that reason, the +adjective transverse (not elementary) is chosen to qualify the property considered in +Definition 1.5 above. +1.5 The space of potentials +For the remaining of the paper, let us take and fix an integer k not smaller than 1. Let us +consider the space Ck+1 +b +(Rdst, R) of functions Rd → R of class Ck+1 which are bounded, +as well as their derivatives of order not larger than k + 1, equipped with the norm +∥W∥Ck+1 +b += +max +α multi-index, |α|≤k+1 ∥∂|α| +uαW∥L∞(Rd,R) , +and let us embed the larger space Ck+1(Rdst, R) with the following topology: for V in +this space, a basis of neighbourhoods of V is given by the sets V + O, where O is an +open subset of Ck+1 +b +(Rdst, R) embedded with the topology defined by ∥·∥Ck+1 +b +(which can +be viewed as an extended metric). For comments concerning the choice of this topology, +see subsection 1.4 of [1]. +1.6 Main result +The following generic transversality statement is the main result of this paper. +Theorem 1 (generic transversality of radially symmetric stationary solutions stable at +infinity). There exists a generic subset G of +� +Ck+1(Rdst, R), ∥·∥Ck+1 +b +� +such that, for every +potential function V in G, every radially symmetric stationary solution stable at infinity +of the parabolic system (1.1) is transverse. +Theorem 1 can be viewed as the extension to higher space dimensions (for radially +symmetric solutions) of conclusion 2 of Theorem 1.7 of [1] (which is concerned with +elementary standing pulses stable at infinity in space dimension 1). A short comparison +between these two results and their proofs is provided in the next subsection. For more +comments and a short historical review on transversality results in similar contexts, see +subsection 1.6 of the same reference. +The core of the paper (section 4) is devoted to the proof of the conclusions of Theorem 1 +among potentials which are quadratic past a certain radius (defined in (3.2)), as stated +in Proposition 4.1. The extension to general potentials of Ck+1 +b +(Rdst, R) is carried out in +section 5. +Remark. As in [1] (see Theorem 1.8 of that reference), the same arguments could be +called upon to prove that the following additional conclusions hold, generically with +respect to the potential V : +8 + +1. for every minimum point of V , the smallest eigenvalue of D2V at this minimum +point is simple; +2. every radially symmetric stationary solution stable at infinity of the parabolic system +(1.1) approaches its limit at infinity tangentially to the eigenspace corresponding to +the smallest eigenvalue of D2V at this point. +1.7 Key differences with the generic transversality of standing pulses in +space dimension one +Table 1.1 lists the key differences between the proof of the generic elementarity of +symmetric standing pulses carried out in [1], and the proof of the generic transversality +of radially symmetric stationary solutions carried out in the present paper (implicitly, +the other steps/features of the proofs are similar or identical). The state dimension, +which is simply denoted by d in [1], is here denoted by dst in both cases. Some of the +notation/rigour is lightened. +Symmetric standing pulse +Radially symmetric +stationary solution +Critical point at infinity +critical point e, E = (e, 0Rdst ) +minimum point u∞ +Symmetry subspace Ssym +{(u, v) ∈ R2dst : v = 0}, +dimension dst +{(u, v, r) ∈ R2dst+1 : (v, r) = (0, 0)}, +dimension dst +Differential system +governing the profiles +autonomous, conservative, +regular at Ssym +non-autonomous, dissipative, +singular at reversibility subspace +Direction of the flow +E → Ssym +Ssym → u∞ +Invariant manifold at +infinity +W u(E), dimension dst − m(e) +W cs, ∞, +(u∞), dimension dst + 1 +Invariant manifold at +symmetry subspace +none +W u, 0, +, dimension dst + 1 +Transversality +W u(E) ⋔ Ssym +W cs, ∞, +(u∞) ⋔ W u, 0, + +Transversality of spatially +homogeneous solutions +irrelevant +Proposition 2.2 +Interval Ionce (values +reached only once) +“anywhere” +close to Ssym +M (departure set of Φ) +parametrization of ∂W u +loc, V (E) +and time, dimension dst − m(e) +Ssym and W cs, ∞, + +loc +(u∞) at r = N, +dimension 2dst +N (arrival set of Φ) +R2dst +R2dst × R2dst +W (target manifold) +Ssym +diagonal of N +dim(M) − codim(W) +−m(e) +0 +Condition to be fulfilled +by perturbation W +� +DΦ(W) +�� (0, ψ)� +̸= 0 +� +DΦu(W) +�� (φ, ψ)� +̸= 0 +Perturbation W, case 3 +precluded +W(u0) ̸= 0 +Table 1.1: Formal comparison between the generic elementarity of symmetric standing +pulses (space dimension 1) proved in [1], and the generic transversality of radially +symmetric stationary solutions (higher space dimension dsp) proved in the present paper. +Here are a few additional comments about these differences. +9 + +Concerning the critical point at infinity, u∞ is assumed (here) to be a minimum point, +whereas (in [1]) the Morse index of e is any. Indeed, if the Morse index m(u∞) of u∞ was +positive, then the dimension of the centre-stable manifold W cs, ∞, + +V +(u∞) would be equal +to dst + m(u∞) + 1; as a consequence, proving the transversality of the intersection (1.15) +in that case would require more stringent regularity assumptions on V (see hypothesis +1 of Theorem 4.2 of [1]) while nothing particularly useful could be derived from this +transversality. On the other hand, assuming that u∞ is a minimum point allows to view +its local centre-stable manifold as a graph (u, c) �→ v (see Proposition 2.4), which is +slightly simpler. +Concerning the interval Ionce providing values u reached “only once” by the profile +(Lemma 2.3), the proof of the present paper takes advantage of the dissipation to find a +convenient interval close to the “departure point” u0, as was done in [1] for travelling +fronts (whereas, for standing pulse, the interval is to be found “anywhere”, thanks to the +conservative nature of the differential system governing the profiles, see conclusion 1 of +Proposition 3.3 of [1]). +Concerning the function Φ to which Sard–Smale theorem is applied in the present +paper, both manifolds W u, 0, + and W cs, ∞, +(u∞) depend on the potential V . However, +the transversality of an intersection between these two manifolds can be seen as the +transversality of the image of Φ with the (fixed) diagonal of R2dst × R2dst, for a function +Φ combining the parametrization of these two manifolds. This trick, which is the same +as in [1] for travelling fronts, allows to apply Sard–Smale theorem to a function Φ with a +fixed arrival space N containing a fixed target manifold W (in this case the diagonal of +N). By contrast, for symmetric standing pulses in [1], since the subspace Ssym involved +in the transverse intersection is fixed, the previous trick is unnecessary and the setting is +simpler. +Finally, a technical difference occurs in “case 3” of the proof that the degrees of freedom +provided by perturbing the potential allow to reach enough directions in the arrival state +of Φ (Lemma 4.6, which is the core of the proof). In [1], case 3 is shown to lead to a +contradiction, not only for symmetric standing pulses, but also for asymmetric ones and +for travelling fronts. Here, such a contradiction does not seem to occur (or at least is +more difficult to prove), but this has no harmful consequence: a suitable perturbation of +the potential can still be found in this case. +2 Preliminary properties +2.1 Proof of Lemma 1.4 +Let V denote a potential function in Ck+1(Rdst, R). Let (0, +∞) → Rdst, r �→ u(r) denote +a (global) solution of system (1.2), assumed to be stable close to some point u∞ of +Σmin(V ) at infinity (Definition 1.2). Lemma 1.4 follows from the next lemma. +Lemma 2.1. The derivative ˙u(r) goes to 0 as r goes to +∞. +10 + +Proof. Let us consider the Hamiltonian function +(2.1) +HV : R2dst → R , +(u, v) �→ v2 +2 − V (u) , +and, for every r in (0, +∞), let +h(r) = HV +�u(r), ˙u(r) +� . +It follows from system (1.2) that, for every r in (0, +∞), +(2.2) +˙h(r) = −dsp − 1 +r +˙u(r)2 , +thus the function h(·) decreases, and it follows from the expression (2.1) of the Hamiltonian +that this function converges, as r goes to +∞, towards a finite limit h∞ which is not +smaller than −V (u∞). +Let us proceed by contradiction and assume that h∞ is larger than −V (u∞). Then, it +follows again from the expression (2.1) of the Hamiltonian that the quantity ˙u(r)2 con- +verges towards the positive quantity 2 +�h∞ + V (u∞) +� as r goes to +∞. As a consequence, +it follows from equality (2.2) that h(r) goes to −∞ as r goes to +∞, a contradiction. +Lemma 2.1 is proved. +2.2 Transversality of homogeneous radially symmetric stationary solutions +stable at infinity +Proposition 2.2. For every potential function V in Ck+1(Rdst, R) and for every nonde- +generate minimum point u∞ of V , the constant function +[0, +∞) → Rdst , +r �→ u∞ , +which defines an (homogeneous) radially symmetric stationary solution stable at infinity +for system (1.1) , is transverse (in the sense of Definition 1.5). +Proof. Let V denote a function in Ck+1(Rdst, R) and u∞ denote a nondegenerate minimum +point of V . The function [0, +∞) → Rdst, r �→ u∞ is a (constant) solution of the +differential system (1.5), and the linearization of this differential system around this +solution reads +(2.3) +¨u = −dsp − 1 +r +˙u + D2V (u∞) · u . +Let (0, +∞) → Rdst, r �→ u(r) denote a nonzero solution of this differential system, and, +for every r in (0, +∞), let +v(r) = ˙u(r) +and +U(r) = +�u(r), v(r) +� +and +q(r) = u(r)2 +2 +. +11 + +Then (omitting the dependency on r), +˙q = u · ˙u +and +¨q = ˙u2 + u · ¨u = ˙u2 − dsp − 1 +r +˙q + D2V (u∞) · (u, u) , +so that +d +dr +�rdsp−1 ˙q(r) +� = rdsp−1 +� +¨q + dsp − 1 +r +˙q +� += rdsp−1� ˙u2 + D2V (u∞) · (u, u) +� . +Since r �→ u(r) was assumed to be nonzero, it follows that the quantity rdsp−1 ˙q(r) is +strictly increasing on (0, +∞). To prove the intended conclusion, let us proceed by +contradiction and assume that, for every r in (0, +∞), +�u(r), v(r), r +� belongs: +1. to the tangent space T(u∞,0Rdst ,r)W u, 0, + +V +(u∞), +2. and to the tangent space T(u∞,0Rdst ,r) +� +ι−1�W cs, ∞, + +V +(u∞) +�� +. +As in (1.6), let us introduce the auxiliary variables τ (equal to log(r)) and c (equal to +1/r). With this notation, system (2.3) is equivalent to +(2.4) +� +� +� +� +� +� +� +uτ = rv +vτ = −(dsp − 1)v + rD2V (u∞) · u +rτ = r , +and to +(2.5) +� +� +� +� +� +� +� +ur = v +vr = −(dsp − 1)cv + D2V (u∞) · u +cr = −c2 . +Assumptions 1 and 2 above yield the following conclusions. +1. In view of the limit of system (2.4) as r goes to 0+, it follows from assumption 1 +that there exists δu0 in Rdst such that +�u(r), v(r) +� goes to (δu0, 0Rdst) as r goes to +0+; +2. and in view of the limit of system (2.5) as c goes to 0+, it follows from assumption +2 that +�u(r), v(r) +� goes to (0Rdst, 0Rdst), at an exponential rate, as r goes to +∞. +It follows from these two conclusions that the quantity rdsp−1 ˙q(r) goes to 0 as r goes +to 0+ and as r goes to +∞, a contradiction with the fact (observed above) that this +quantity is strictly increasing with r. Proposition 2.2 is proved. +12 + +2.3 Additional properties close to the origin +Let V denote a potential function in Ck+1(Rdst, R) and let u0 be a point in Rdst. Let +us recall (see subsection 1.3) that the unstable manifold W u, 0 +V +(u0) of the equilibrium +(u0, 0Rdst, 0) for the autonomous differential system (1.7)) is one-dimensional. +As a +consequence there exists a unique solution r �→ u(r) of the differential system (1.2) such +that the image of the map r �→ +�u(r), ˙u(r), r) lies in the intersection W u, 0, + +V +(u0) of this +unstable manifold with the half-space where r is positive (this intersection was defined in +(1.10)); or, in other words, such that +�u(r), ˙u(r) +� goes to (u0, 0) as r goes to 0+. This +solution is defined on some (maximal) interval (0, rmax), where rmax is either a finite +quantity or +∞. The following lemma provides properties of this solution that will be +used in the sequel. To ease its statement, let us assume that rmax is equal to +∞ (only +this case will turn out to be relevant), and let us consider the continuous extension of +u(·) to the interval [0, +∞) (and let us still denote by u(·) this continuous extension). +Lemma 2.3. If u(·) is not identically equal to u0 (in other words, if u0 is not a critical +point of V ), then there exists a positive quantity ronce such that, denoting by Ionce the +interval [0, ronce), the following conclusions hold: +1. the function ˙u(·) does not vanish on Ionce, +2. and, for every r∗ in Ionce and r in [0, +∞), +u(r) = u(r∗) =⇒ r = r∗. +Proof. The linearized system (1.7) at the equilibrium (u0, 0Rdst, 0) reads: +d +dτ +� +� +� +δu +δv +δr +� +� +� = +� +� +� +0 +0 +0 +0 +−(dsp − 1) +∇V (u0) +0 +0 +1 +� +� +� +� +� +� +δu +δv +δr +� +� +� , +thus the tangent space at (u0, 0Rdst, 0) to W u, 0 +V +(u0) (the unstable eigenspace of the matrix +of this system) is spanned by the vector +�0, ∇V (u0)/dsp, 1 +�; it follows that +(2.6) +˙u(r) = r +dsp +∇V (u0) +�1 + or→0+(r) +� . +Thus, if r0 is a sufficiently small positive quantity, then ˙u(·) does not vanish on (0, r0] +(so that conclusion 1 of Lemma 2.3 holds provided that ronce is not larger than r0), and +the map +(2.7) +[0, r0] → Rdst , +r �→ u(r) +is a C1-diffeomorphism onto its image. For r in [0, +∞), let us denote +�u(r), ˙u(r) +� by +U(r). According to the decrease (2.2) of the Hamiltonian, there exists a quantity ronce in +(0, r0) such that, for every r∗ in [0, ronce), +(2.8) +HV +�U(r0) +� < −V +�u(r∗) +� . +13 + +Take r∗ in [0, ronce] and r in [0, +∞), and let us assume that u(r) equals u(r∗). If r was +larger than r0 then it would follow from the expression (2.1) of the Hamiltonian, its +decrease (2.2), and inequality (2.8) that +−V +�u(r) +� ≤ HV +�U(r) +� ≤ HV +�U(r0) +� < −V +�u(r∗) +� , +a contradiction with the equality of u(r) and u(r∗). Thus r is not larger than r0, and +it follows from the one-to-one property of the function (2.7) that r must be equal to +r∗; conclusion 2 of Lemma 2.3 thus holds, and Lemma 2.3 is proved. +2.4 Additional properties close to infinity +Let V1 denote a potential function in Ck+1(Rdst, R) and u1,∞ denote a nondegenerate +minimum point of V1. According to the implicit function theorem, there exists a (small) +neighbourhood νrobust(V1, u1,∞) of Vquad-R and a Ck-function V �→ u∞(V ) defined on +νrobust(V1, u1,∞) and with values in Rdst such that u∞(V1) equals u1,∞ and, for every +V in νrobust(V1, u1,∞), u∞(V ) is a local minimum point of V . The following proposi- +tion is nothing but the local centre-stable manifold theorem applied to the equilibrium +�u∞(V ), 0Rdst, 0 +� of the (autonomous) differential system (1.8), for V close to V1. Addi- +tional comments and references concerning local stable/centre/unstable manifolds are +provided in subsection 2.2 of [1]. +Proposition 2.4 (local centre-stable manifold at infinity). There exist a neighbourhood +ν of V1 in Ck+1(Rdst, R), included in νrobust(V1, u1,∞), such that, if ε1 and c1 denote +sufficiently small positive quantities, then, for every V in ν, there exists a Ck-map +(2.9) +wcs, ∞ +loc, V : BRdst(u1,∞, ε1) × [0, c1] → Rdst , +(u, c) �→ wcs, ∞ +loc, V (u, c) , +such that, for every (u0, v0, c0) in BRdst(u1,∞, ε1) × Rdst × [0, c1], the following two +statements are equivalent: +1. v = wcs, ∞ +loc, V (u, c); +2. the solution r �→ +�u(r), v(r), c(r) +� of the differential system (1.8) with initial condi- +tion (u0, v0, c0) at time r0 = 1/c0 is defined up to +∞, remains in BRdst(u1,∞, ε1)× +Rdst × [0, c1] of all r larger than r0, and goes to +�u∞(V ), 0Rdst, 0 +� as r goes to +∞. +In particular, wcs, ∞ +loc, V +�u∞(V ), 0 +� is equal to 0Rdst. In addition, the map +BRdst(u1,∞, ε1) × [0, c1] × ν → Rdst , +(u, c, V ) �→ wcs, ∞ +loc, V (u, c) +is of class Ck (with respect to u and c and V ), and, for every V in ν, the graph of the +differential at +�u∞(V ), 0) of the map (u, c) �→ wcs, ∞ +loc, V (u, c) is equal to the centre-stable +subspace of the linearization at +�u∞(V ), 0Rdst, 0 +� of the differential system (1.8). +14 + +Let us denote by W cs, ∞, + +loc, V, ε1, c1 +�u∞(V ) +� the graph of the map (2.9) (restricted to positive +values of c), see figure 1.1; with symbols, +(2.10) +W cs, ∞, + +loc, V, ε1, c1 +�u∞(V ) +� = +��u, wcs, ∞ +loc, V (u, c), c +� : (u, c) ∈ BRdst(u1,∞, ε1) × (0, c1] +� +. +This set defines a local centre-manifold (restricted to positive values of c) for the equilib- +rium +�u∞(V ), 0Rdst, 0 +� of the differential system (1.8). Its uniqueness (for positive values +of c) is ensured by the dynamics of the centre component c, which, according to the +third equation of system (1.8), decreases to 0 (see figure 1.1). The global centre-stable +manifold W cs, ∞, + +V +�u∞(V ) +� already defined in (1.13) can be redefined as the points of +R2dst ×(0, +∞) that eventually reach the local centre manifold W cs, ∞, + +loc, V, ε1, c1 +�u∞(V ) +� when +they are transported by the flow of the differential system (1.8). +Remark. If the state dimension dst is equal to 1, then a calculation shows that +wcs, ∞ +loc, V (u, c) = − +�u − u∞(V ) +� �� +V ′′�u∞(V ) +� + dsp − 1 +2 +c + . . . +� +, +where “. . . ” stands for higher order terms in u − u∞(V ) and c. In particular the quantity +∂c∂uwcs, ∞ +loc, V +�u∞(V ), 0 +� is equal to the (negative) quantity −(dsp − 1)/2. The display of +the local centre-stable manifold at infinity on figure 1.1 fits with the sign of this quantity. +3 Tools for genericity +Let +(3.1) +Vfull = Ck+1(Rdst, R) , +and, for a positive quantity R, let +(3.2) +Vquad-R = +� +V ∈ Vfull : for all u in Rd, |u| ≥ R =⇒ V (u) = u2 +2 +� +. +Let us recall the notation SV introduced in (1.4). +Lemma 3.1. For every positive quantity R and for every potential V in Vquad-R, the +following conclusions hold. +1. The flow defined by the differential system (1.2) (governing radially symmetric +stationary solutions of the parabolic system (1.1)) is global (that is, every solution +is defined on (0, +∞)). +2. For every u in SV , the following bound holds: +(3.3) +sup +r∈(0,+∞) +|u(r)| < R . +15 + +Proof. Let V be in Vquad-R. According to the definition (3.2) of Vquad-R, there exists a +positive quantity K such that, for every u in Rdst, +|∇V (u)| ≤ K + |u| . +As a consequence, the following inequalities hold for the right-hand side of the first order +differential system (1.5): +���� +� +v, −dsp − 1 +r +v + ∇V (u) +����� ≤ |v| + dsp − 1 +r +|v| + K + |u| ≤ K + +� +2 + dsp − 1 +r +� +|(u, v)| , +and this bound prevents the solution from blowing up in finite time, which proves +conclusion 1. +Now, take a function u in SV . Let us still denote by u(·) the continuous extension of +this solution to [0, +∞). For every r in [0, +∞), let +q(r) = u(r)2 +2 +and +Q(r) = rdsp−1 ˙q(r) . +Then (omitting the dependency on r), +˙q = u · ˙u +and +¨q = ˙u2 + u · ¨u = ˙u2 − dsp − 1 +r +˙q + u · ∇V (u) , +so that +˙Q = rdsp−1 +� +¨q + dsp − 1 +r +˙q +� += rdsp−1� ˙u2 + u · ∇V (u) +� . +According to the definition (3.2) of Vquad-R, there exists a positive quantity δ (sufficiently +small) so that, for every w in Rdst, +(3.4) +|w| ≥ R − δ =⇒ w · ∇V (w) ≥ w2 +2 . +Let us proceed by contradiction and assume that supr∈(0,+∞) |u(r)| is not smaller than +R. Since u(·) is stable at infinity and since the critical points of V belong to the open +ball BRdst(0, R − δ), it follows that the set +�r ∈ [0, +∞) : |u(r)| ≥ R +� +is nonempty; let rout denote the minimum of this set. For the same reason, the set +�r ∈ (rout, +∞) : |u(r)| < R − δ +� +is also nonempty. Let rback denote the infimum of this last set. It follows from these +definitions that rback is larger than rout and that, for every r in (rout, rback), according to +inequality (3.4), +(3.5) +˙Q(r) ≥ rdsp−1 +� +˙u2(r) + u2(r) +2 +� +> 0 . +16 + +If on the one hand rout equals 0 then |u(0)| is not smaller than R and, since Q(0) equals +0, it follows from inequality (3.5) that Q(·) is positive on (0, rback), so that the same is +true for ˙q(·). Thus q(·) is strictly increasing on [0, rback] and |u(rback)| must be larger +than |u(rout)|, a contradiction with the definition of rback. If on the other hand rout is +positive, then |u(rout)| is equal to R and ˙q(rout) is nonnegative so that the same is true +for Q(rout), and it again follows from inequality (3.5) that Q(·) is positive on (0, rback), +yielding the same contradiction. Conclusion 2 of Lemma 3.1 is proved. +Notation. For every positive quantity R and every potential V in Vquad-R, let +(3.6) +SV : (0, +∞)2 × R2dst → R2dst , +�(rinit, r), (uinit, vinit) +� �→ SV +�(rinit, r), (uinit, vinit) +� +denote the (globally defined) flow of the (non-autonomous) differential system (1.5) for +this potential V . In other words, for every rinit in (0, +∞) and (uinit, vinit) in R2dst, the +function +(0, +∞) → R2dst , +r �→ SV +�(rinit, r1), (uinit, vinit) +� +is the solution of the differential system (1.5) for the initial condition (uinit, vinit) at r +equals rinit. According to subsection 1.3, the flow SV may be extended to the larger set +(0, +∞)2 × R2dst ∪ [0, +∞)2 × Rdst × {0Rdst} ; +according to this extension, for every u0 in Rdst, the solution taking its values in the +(one-dimensional) unstable manifold W u, 0, + +V +(u0) reads: +(3.7) +[0, +∞) → Rdst , +r �→ SV +�(0, r), (u0, 0Rdst) +� . +4 Generic transversality among potentials that are quadratic +past a given radius +4.1 Notation and statement +Let us recall the notation SV and SV, u∞ introduced in (1.4). +Proposition 4.1. There exists a generic subset of Vquad-R such that, for every potential +V in this subset, every radially symmetric stationary solution stable at infinity of the +parabolic system (1.1) (in other words, every u in SV ) is transverse. +4.2 Reduction to a local statement +Let V1 denote a potential function in Vquad-R and u1,∞ denote a nondegenerate minimum +point of V1. According to the implicit function theorem, there exists a (small) neighbour- +hood νrobust(V1, u1,∞) of Vquad-R and a Ck-function u∞(·) defined on νrobust(V1, u1,∞) and +with values in Rdst such that u∞(V1) equals u1,∞ and, for every V in νrobust(V1, u1,∞), +u∞(V ) is a local minimum point of V . The following local generic transversality statement +yields Proposition 4.1 (as shown below). +17 + +Proposition 4.2. There exists a neighbourhood νV1, u1,∞ of V1 in νrobust(V1, u1,∞) and +a generic subset νV1, u1,∞, gen of νV1, u1,∞ such that, for every V in νV1, u1,∞, gen, every +radially symmetric stationary solution stable close to u∞(V ) at infinity of the parabolic +system (1.1) (in other words, every u in SV, u∞(V )) is transverse. +Proof that Proposition 4.2 yields Proposition 4.1. Let us denote by Vquad-R-Morse the +dense open subset of Vquad-R defined by the Morse property: +(4.1) +Vquad-R-Morse = {V ∈ Vquad-R : all critical points of V are nondegenerate} . +Let V1 denote a potential function in Vquad-R-Morse. According to the Morse property +its minimum points are isolated and since V1 is in Vquad-R they belong to the open ball +BRd(0, R), so that those minimum points are in finite number. Assume that Proposi- +tion 4.2 holds. With the notation of this proposition, let us consider the following two +intersections, at each time over all minimum points u1,∞ of V1: +(4.2) +νV1 = +� +νV1, u1,∞ +and +νV1, gen = +� +νV1, u1,∞, gen . +Since those are finite intersections, νV1 is still a neighbourhood of V1 in Vquad-R and the +set νV1, gen is still a generic subset of νV1. This shows that the set +{V ∈ Vquad-R-Morse : +every u in SV, u∞(V ) is transverse} +is locally generic. Applying Lemma 4.3 of [1] as in Subsection 5.2 of this reference shows +that this local genericity implies the global genericity stated in Proposition 4.1, which is +therefore proved. +4.3 Proof of the local statement (Proposition 4.2) +4.3.1 Setting +For the remaining part of this section, let us fix a potential function V1 in Vquad-R and a +nondegenerate minimum point u1,∞ of V1. Let ν be a neighbourhood of V1 in Vquad-R, +included in νrobust(V1, u1,∞), and let ε1 and c1 be positive quantities, with ν and ε1 and +c1 small enough so that the conclusions of Proposition 2.4 hold. Let +r1 = 1/c1 +and +M = Rdst × BRdst(u1,∞, ε1) +and +Λ = ν , +and +N = (R2dst)2 +and +W = {(A, B) ∈ N : A = B} +, +thus W is the diagonal of N. Let N denote an integer not smaller than r1, and let us +consider the functions +Φu : Rdst × Λ → R2dst , +(u0, V ) �→ SV +�(0, N), (u0, 0Rdst) +� , +and +Φcs : BRdst(u1,∞, ε1) × Λ → R2dst , +(uN, V ) �→ +�uN, wcs, ∞ +loc, V (uN, 1/N) +� , +and the function +(4.3) +Φ : M × Λ → N , +(m, V ) = (u0, uN, V ) �→ +�Φu(u0, V ), Φcs(uN, V ) +� . +18 + +4.3.2 Equivalent characterizations of transversality +Let us consider the set +SΛ,u1,∞,N = +�(V, u) : V ∈ Λ and u ∈ SV, u∞(V ) and u(N) ∈ BRdst(u1,∞, ε1) +� . +Proposition 4.3. The map +(4.4) +Φ−1(W) → SΛ,u1,∞,N , +(u0, u, V ) �→ +� +V, r �→ SV +�(0, r), (u0, 0Rdst +�� +is well defined and one-to-one. +Proof. The image by Φ of a point (u0, uN, V ) of M × Λ belongs to the diagonal W of +N if and only if Φu(u0, V ) equals Φcs(uN, V ), and in this case the function u : r �→ +SV +�(0, r), (u0, 0Rdst +� belongs to SV, u∞(V ) and u(N) (which is equal to uN) belongs to +BRdst(u1,∞, ε1), so that (V, u) belongs to SΛ,u1,∞,N. The map (4.4) above is thus well +defined. +Now, for every (V, u) in SΛ,u1,∞,N, if we denote by u0 the limit limr→0+ u(r) and by +uN the vector u(N), then (u0, uN, V ) is the only possible antecedent of (V, u) by the map +(4.4). In addition, +SV +�(0, N), (u0, 0Rdst) +� = +�uN, ˙u(N) +� , +and since u(r) goes to u∞(V ) as r goes to +∞, the vector +�u(N), ˙u(N), 1/N +� must +belong to the centre-stable manifold W cs, ∞, + +V +�u∞(V ) +� of u∞(V ), so that, according to +the definition of wcs, ∞ +loc, V , +˙u(N) = wcs, ∞ +loc, V +�u(N), 1/N +� , +and this yields the equality between Φu(u0, V ) and Φcs(uN, V ). Thus Φ(V, u) belongs to +W and (u0, uN, V ) belongs to Φ−1(W). Proposition 4.3 is proved. +Proposition 4.4. For every potential function V in Λ, the following two statements are +equivalent. +1. The image of the function M → N, m �→ Φ(m, V ) is transverse to W. +2. Every u in SV, u∞(V ) such that u(N) is in BRdst(u1,∞, ε1) is transverse. +Remark. According to Proposition 2.2, for every V in Λ, the constant function r �→ u∞(V ), +which belongs to SV , is already (a priori) known to be transverse, therefore only +nonconstant solutions matter in statement 2 of this proposition. +Proof. Let us consider (m2, V2) in M × Λ such that Φ(m2, V2) is in W, let (u2,0, u2,N) +denote the components of m2, and let r �→ u2(r) and r �→ U2(r) denote the functions +satisfying, for all r in [0, +∞), +U2(r) = +�u2(r), ˙u2(r) +� = SV +�(0, r), (u2,0, 0Rdst +� . +Let us consider the map +∆Φ : M → R2dst , +(u0, uN) �→ Φu(u0, V2) − Φcs(uN, V2) , +19 + +and let us write, only for this proof, DΦ and DΦu and DΦcs and D(∆Φ) for the +differentials of Φ and Φu and Φcs and ∆Φ at (m2, V2) and with respect to all variables in +M (but not with respect to V ). According to Definition 1.5, the transversality of u2 is +defined as the transversality of the intersection W u, 0, + +V2 +∩ ι−1� +W cs, ∞, + +V2 +�u∞(V2) +�� +along +the trajectory of U2. This transversality can be considered at a single point, no matter +which, of the trajectory U2 +�(0, +∞) +�, in particular at the point Φu(u2,0, V2) which is +equal to Φcs�u2(N), V 2 +�, and is equivalent to the transversality of the dst-dimensional +manifolds +W u, 0, + +V2 +∩ +�R2dst × {N} +� +and +ι−1� +W cs, ∞, + +V2 +�u∞(V2) +�� +∩ +�R2dst × {N} +� +in R2dst ×{N}. It is therefore equivalent to the surjectivity of the map D(∆Φ) (statement +(B) in Lemma 4.5 below). On the other hand, the image of the function M → N, +m �→ Φ(m, V2) is transverse at Φ(m, V2) to the diagonal W of N if and only if the image +of DΦ contains a complementary space of this diagonal (statement (A) in Lemma 4.5 +below)). Thus Proposition 4.4 is a consequence of the next lemma. +Lemma 4.5. The following two statements are equivalent. +(A) The image of DΦ contains a complementary subspace of the diagonal W of N. +(B) The map D(∆Φ) is surjective. +Proof. If statement (A) holds, then, for every (α, β) in N, there exist γ in R2dst and δm +in Tm2M such that +(4.5) +(γ, γ) + DΦ · δm = (α, β) , +so that +(4.6) +D(∆Φ) · δm = α − β , +and statement (B) holds. Conversely, if statement (B) holds, then, for every (α, β) in +N, there exists δm in Tm2M such that (4.6) holds, and as a consequence, if (δu0, δuN) +denote the components of δm, then α − DΦu(δu0) is equal to β − DΦcs(δuN), and if +this vector is denoted by γ, then equality (4.5) holds, and this shows that statement (A) +holds. +As explained above, Proposition 4.4 follows from Lemma 4.5, and is therefore proved. +4.3.3 Checking hypothesis 1 of Theorem 4.2 of [1] +The function Φ is as regular as the flow SV , thus of class Ck. It follows from the definitions +of M and N and W that +dim(M) − codim(W) = (dst + dst) − 2dst = 0 , +so that hypothesis 1 of Theorem 4.2 of [1] is fulfilled. +20 + +4.3.4 Checking hypothesis 2 of Theorem 4.2 of [1] +For every V in Vquad-R, let us recall the notation SV introduced in (3.6) and (3.7) for the +flow of the differential system (1.5). Take (m2, V2) in the set Φ−1(W). Let (u2,0, u2,N) +denote the components of m2, and, for every r in (0, +∞), let us write +U2(r) = +�u2(r), v2(r) +� = SV2 +�(0, r), (u2,0, 0Rdst) +� . +Let us write +DΦ +and +DΦu +and +DΦcs +for the full differentials (with respect to arguments m in M and V in Λ) of the three +functions Φ and Φu and Φcs respectively at the points +�u2,0, u2,N, V2 +�, +�u2,0, V2 +� and +�u2,N, V2 +�. Checking hypothesis 2 of Theorem 4.2 of [1] amounts to prove that +(4.7) +im(DΦ) + W = N . +If u2(·) is constant (that is, identically equal to u∞(V2)), then equality (4.7) follows from +Proposition 2.2. Thus, let us assume that u2(·) is nonconstant. In this case, equality +(4.7) is a consequence of the following lemma. +Lemma 4.6. For every nonzero vector (φ2, ψ2) in R2dst, there exists a function W in +Ck+1 +b +(Rdst, R) such that +supp(W) ⊂ BRd(0, R) , +(4.8) +and +�DΦu · (0, 0, W) +�� (φ2, ψ2) +� ̸= 0 , +(4.9) +and +DΦcs · (0, 0, W) = 0R2dst . +(4.10) +Proof that Lemma 4.6 yields equality (4.7). Inequality (4.9) shows that the orthogonal +complement, in R2dst, of the directions that can be reached by DΦu·(0, 0, W) for potentials +W satisfying (4.8) and (4.10) is reduced to 0R2dst; in other words, all directions of R2dst +can be reached by that means. This shows that +im(DΦ) ⊃ R2dst × {0R2dst} , +and since the subspace at the right-hand side of this inclusion is transverse to W in +R4dst, this proves equality (4.7) (and shows that hypothesis 2 of Theorem 4.2 of [1] is +fulfilled). +Proof of Lemma 4.6. Let (φ2, ψ2) denote a nonzero vector in R2dst, let W be a function +in Ck+1 +b +(Rdst, R) satisfying the inclusion +(4.11) +supp(W) ⊂ BRd(0, R) \ BRdst(u1,∞, ε1) , +and observe that inclusion (4.8) and equality (4.10) follow from this inclusion (4.11). Let +us consider the linearization of the differential system (1.2), for the potential V2, around +the solution r �→ U2(r): +(4.12) +d +dr +� +δu(r) +δv(r) +� += +� +0 +id +D2V2 +�u2(r) +� +−dsp−1 +r +� � +δu(r) +δv(r) +� +, +21 + +and let T(r, r′) denote the family of evolution operators obtained by integrating this +linearized differential system between r and r′. It follows from the variation of constants +formula that +(4.13) +DΦu · (0, 0, W) = +� N +−∞ +T(r, N) +� +0, ∇W +�u2(r) +�� +dr . +For every r in (0, +∞), let T ∗(r, N) denote the adjoint operator of T(r, N), and let +(4.14) +�φ(r), ψ(r) +� = T ∗(r, N) · (φ2, ψ2) . +According to expression (4.13), inequality (4.9) reads +� N +−∞ +�� +0, ∇W +�u2(r) +�� ��� T ∗(r, N) · (φ2, ψ2) +� +dr ̸= 0 , +or equivalently +(4.15) +� N +−∞ +∇W +�u2(r) +� · ψ(r) dr ̸= 0 . +Due to the expression of the linearized differential system (4.12), (φ, ψ) is a solution of +the adjoint linearized system +(4.16) +� ˙φ(r) +˙ψ(r) +� += − +� +0 +D2V2 +�u2(r) +� +id +−dsp−1 +r +� � +φ(r) +ψ(r) +� +. +According to Lemma 2.3 (and since u2(·) was assumed to be nonconstant), there exists +positive quantity ronce such that, if we denote by Ionce the interval (0, ronce], then ˙u2(·) +does not vanish on Ionce, and, for all r∗ in Ionce and r in R, +(4.17) +u2(r) = u2(r∗) =⇒ r = r∗ . +In addition, up to replacing ronce by a smaller positive quantity, it may be assumed that +the following conclusions hold: +u2(Ionce) ∩ BRdst(u1,∞, ε1) = ∅ . +To complete the proof three cases have to be considered. +Case 1. +There exists r∗ in Ionce such that ψ(r∗) is not collinear to ˙u2(r∗). +In this case, the construction of a potential function W satisfying inclusion (4.11) and +inequality (4.9) (and thus the conclusions of Lemma 4.6) is the same as in the proof of +Lemma 5.7 of [1]. +If case 1 does not occur, then ψ(r) is collinear to ˙u2(r), and since ˙u2(·) does not vanish +on Ionce, there exists a C1-function α : Ionce → R such that, for every r in Ionce, +(4.18) +ψ(r) = α(r) ˙u2(r) . +The next cases 2 and 3 differ according to whether the function α(·) is constant or not. +22 + +Case 2. +For every r in Ionce, equality (4.18) holds for some nonconstant function α(·). +In this case there exists r∗ in Ionce such that ˙α(r∗) is nonzero, and again the construction +of a potential function W satisfying inclusion (4.11) and inequality (4.9) (and thus the +conclusions of Lemma 4.6) is the same as in the proof of Lemma 5.7 of [1]. +Case 3. +For every r in Ionce, ψ(r) = α ˙u2(r) for some real (constant) quantity α. +In this case the quantity α cannot be 0 or else, due to (4.16) and (4.18), both φ(·) +and ψ(·) would identically vanish on Ionce and thus on (0, +∞), a contradiction with the +assumptions of Lemma 4.6. Thus, without loss of generality, we may assume that α is +equal to 1. If supp(W) is included in a sufficiently small neighbourhood of u2,0, then +W(·) vanishes on u2 +�[ronce, N] +� and the integral on the left-hand side of inequality (4.15) +reads +� ronce +0 +∇W +�u2(r) +� · ˙u2(r) dr = W +�u2(ronce) +� − W(u2,0) = −W(u2,0) , +so that inequality (4.15) holds as soon as W(u2,0) is nonzero. Lemma 4.6 is proved. +Remark. By contrast with the proof of the generic elementarity of standing pulses in +[1], case 3 above cannot be easily precluded. Indeed, let us assume that, for every r in +Ionce, ψ(r) is equal to α ˙u2(r) for some nonzero (constant) quantity α. Without loss of +generality, we may assume that α is equal to 1. Then, it follows from the second equation +of (4.16) that, still for every r in Ionce (omitting the dependency on r), +φ = dsp − 1 +r +ψ − ˙ψ = dsp − 1 +r +˙u2 − ¨u2 = 2(dsp − 1) +r +˙u2 − ∇V2(u2) , +and it follows from the first equation of (4.16) that +−D2V2(u2) ˙u2 = ˙φ = −2(dsp − 1) +r2 +˙u2 + 2(dsp − 1) +r +¨u2 − D2V2(u2) ˙u2 , +and thus, after simplification, +¨u2 = 1 +r ˙u2 , +or equivalently +˙u2 = r +dsp +∇V (u2) . +As illustrated by equality (2.6), this last equality indeed holds if ∇V2 is constant on the +set u2(Ionce). Case 3 can therefore not be a priori precluded, and if it may be argued +that this case is “unlikely” (non generic), the direct argument provided above in this +case is simpler. By contrast, in [1] for standing pulses in space dimension one (dsp equal +to 1), this case could not occur because ψ was assumed to be nonzero on the symmetry +subspace, defined here as {(v, r) = (0Rdst, 0)}, see (1.11). +4.3.5 Conclusion +As seen in sub-subsection 4.3.3, hypothesis 1 of Theorem 4.2 of [1] is fulfilled for the +function Φ defined in (4.3), and since Lemma 4.6 yields equality (4.7), hypothesis 2 of this +23 + +theorem is also fulfilled. The conclusion of this theorem ensures that there exists a generic +subset Λgen, N of Λ such that, for every V in Λgen, N, the image of the function M → N, +m �→ Φ(m, V ) is transverse to the diagonal W of N. According to Proposition 4.4, it +follows that every u in SV, u∞(V ) such that u(N) is in BRdst(u1,∞, ε1) is transverse. The +set +Λgen = +� +N∈N, N≥r0 +Λgen, N +is still a generic subset of Λ. For every V in Λgen and every u in SV, u∞(V ), since u(r) +goes to u∞(V ) as r goes to +∞, there exists N such that u(N) is in BRdst(u1,∞, ε1), and +according to the previous statements u is transverse. In other words, the conclusions of +Proposition 4.2 hold with: +νV1, u1,∞ = ν = Λ +and +νV1, u1,∞, gen = Λgen . +5 Proof of the main results +Proposition 4.1 shows the genericity of the property considered in Theorem 1, but only +inside the space Vquad-R of the potentials that are quadratic past some radius R. In this +section, the arguments will be adapted to obtain the genericity of the same property +in the space Vfull (that is Ck+1(Rdst, R)) of all potentials, endowed with the extended +topology (see subsection 1.5). They are identical to those of section 9 of [1]. Let us recall +the notation SV introduced in (1.4), and, for every positive quantity R, let us consider +the set +SV,R = +� +u ∈ SV : +sup +r∈[0,+∞) +|u(r)| ≤ R +� +. +Exactly as shown in subsection 9.1 of [1], Theorem 1 follows from the next proposition. +Proposition 5.1. For every positive quantity R, there exists a generic subset Vfull-⋔-S-R +of Vfull such that, for every potential V in this subset, every radially symmetric stationary +solution stable at infinity in SV,R is transverse. +Proof. Let R denote a positive quantity, let V1 denote a potential function in Vquad-(R+1), +and let u1,∞ denote a nondegenerate minimum point of V1. Let us consider the neigh- +bourhood νV1, u1,∞ of V1 in Vquad-(R+1) provided by Proposition 4.2 for these objects, +together with the quantities ε1, c1, and r1 introduced in sub-subsection 4.3.1. Up to +replacing νV1, u1,∞ by its interior, we may assume that it is open in Vquad-(R+1). As in +sub-subsection 4.3.1, let us consider an integer N not smaller than r1, and the same +function Φ : M × Λ → N as in (4.3). +Here is the sole difference with the setting of sub-subsection 4.3.1: by contrast with the +non-compact set M defining the departure set of Φ, let us consider the compact subset +MN defined as: +MN = BRdst(0Rdst, N) × BRdst(u1,∞, ε1) . +Thus the integer N now serves two purposes: the “time” (radius) at which the intersection +between unstable and centre-stable manifolds is considered, and the radius of the ball +24 + +containing the departure points of the unstable manifolds that are considered. These +purposes are independent (two different integers instead of the single integer N may as +well be introduced). Let us consider the set: +OV1,u1,∞,N = +� +V ∈ νV1, u1,∞ : Φ(MN, V ) is transverse to W in N +� +. +As shown in Proposition 4.4, this set OV1,u1,∞,N is made of the potential functions V in +νV1, u1,∞ such that every u in SV, u∞(V ) such that u(N) is in BRdst(u1,∞, ε1) and u(0) is in +BRdst(0Rdst, N), is transverse. This set contains the generic subset νV1, u1,∞, gen = Λgen of +νV1, u1,∞ and is therefore generic (thus, in particular, dense) in νV1, u1,∞. By comparison +with νV1, u1,∞, gen, the additional feature of this set OV1,u1,∞,N is that it is open: exactly +as in the proof of Lemma 9.2 of [1], this openness follows from the intrinsic openness of a +transversality property and the compactness of MN. +Let us make the additional assumption that the potential V1 is a Morse function. Then, +the set of minimum points of V1 is finite and depends smoothly on V in a neighbourhood +νrobust(V1) of V1. Intersecting the sets νV1, u1,∞ and OV1,u1,∞,N above over all the minimum +points u1,∞ of V1 provides an open neighbourhood νV1 of V1 and an open dense subset +OV1,N of νV1 such that, for all V in νV1, every radially symmetric stationary solution +stable close to a minimum point of V at infinity, and equal at origin to some point of +BRdst(0Rdst, N), is transverse. +Denoting by int(A) the interior of a set A and using the notation of subsection 4.4 of +[1], let us introduce the sets +˜νV1 = res−1 +R,∞ ◦ resR,(R+1)(νV1) , +and +˜OV1,N = res−1 +R,∞ ◦ resR,(R+1)(OV1,N) , +and +˜Oext +V1,N = ˜OV1,N ⊔ int +�Vfull \ ˜νV1 +� . +It follows from these definitions that ˜Oext +V1,N is a dense open subset of Vfull (for more +details, see Lemma 9.3 of [1]). +Since Vquad-(R+1) is a separable space, it is second-countable, and can be covered by a +countable number of sets of the form νV1. With symbols, there exists a countable family +(V1,i)i∈N of potentials of Vquad-(R+1)-Morse so that +Vquad-(R+1)-Morse = +� +i∈N +νV1,i . +Let us consider the set +Vfull-⋔-S-R = Vfull-Morse ∩ +� +� +� +(i,N)∈N2 +˜Oext +V1,i,N +� +� , +where Vfull-Morse is the set of potentials in Vfull which are Morse functions. This set is a +countable intersection of dense open subsets of Vfull, and is therefore a generic subset of +Vfull. And, for every potential V in this set Vfull-⋔-S-R, every radially symmetric stationary +solution stable at infinity in SV,R is transverse (for more details, see Lemma 9.4 of [1]). +Proposition 5.1 is proved. +25 + +As already mentioned at the beginning of this section, Theorem 1 follows from Proposi- +tion 5.1. Finally, Corollary 1.1 follows from Theorem 1 (for more details, see subsection 9.4 +of [1]). +Acknowledgements +This paper owes a lot to numerous fruitful discussions with Romain +Joly, about both its content and the content of the companion paper [1] written in +collaboration with him. +References +[1] +R. Joly and E. Risler. “Generic transversality of travelling fronts, standing fronts, +and standing pulses for parabolic gradient systems”. In: arXiv (2023), pp. 1–69. +arXiv: 2301.02095 (cit. on pp. 3, 5, 7–10, 14, 18, 20–26). +[2] +E. Risler. “Global behaviour of bistable solutions for gradient systems in one +unbounded spatial dimension”. In: arXiv (2022), pp. 1–91. arXiv: 1604.02002 +(cit. on p. 3). +[3] +E. Risler. “Global behaviour of bistable solutions for hyperbolic gradient systems +in one unbounded spatial dimension”. In: arXiv (2022), pp. 1–75. arXiv: 1703.01221 +(cit. on p. 3). +[4] +E. Risler. “Global behaviour of radially symmetric solutions stable at infinity for +gradient systems”. In: arXiv (2022), pp. 1–52. arXiv: 1703.02134 (cit. on p. 3). +[5] +E. Risler. “Global relaxation of bistable solutions for gradient systems in one +unbounded spatial dimension”. In: arXiv (2022), pp. 1–69. arXiv: 1604.00804 +(cit. on p. 3). +Emmanuel Risler +Université de Lyon, INSA de Lyon, CNRS UMR 5208, Institut Camille Jordan, +F-69621 Villeurbanne, France. +emmanuel.risler@insa-lyon.fr +26 + diff --git a/19E0T4oBgHgl3EQfugE5/content/tmp_files/load_file.txt b/19E0T4oBgHgl3EQfugE5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b0df6e4668bfa6aad0ae2340b679f5ac5147d5fa --- /dev/null +++ b/19E0T4oBgHgl3EQfugE5/content/tmp_files/load_file.txt @@ -0,0 +1,1199 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf,len=1198 +page_content='Generic transversality of radially symmetric stationary solutions stable at infinity for parabolic gradient systems Emmanuel Risler January 9, 2023 This paper is devoted to the generic transversality of radially symmetric stationary solutions of nonlinear parabolic systems of the form ∂tw(x, t) = −∇V �w((x, t)) � + ∆xw(x, t) , where the space variable x is multidimensional and unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' It is proved that, generically with respect to the potential V , radially symmetric stationary solutions that are stable at infinity (in other words, that approach a minimum point of V at infinity in space) are transverse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' as a consequence, the set of such solutions is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' This result can be viewed as the extension to higher space dimensions of the generic elementarity of symmetric standing pulses, proved in a companion paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' It justifies the generic character of the discreteness hypothesis concerning this set of stationary solutions, made in another companion paper devoted to the global behaviour of (time dependent) radially symmetric solutions stable at infinity for such systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 2020 Mathematics Subject Classification: 35K57, 37C20, 37C29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Key words and phrases: parabolic gradient systems, radially symmetric stationary solutions, generic transversality, Morse–Smale theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='02605v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='AP] 6 Jan 2023 Contents 1 Introduction 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 An insight into the main result .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 Radially symmetric stationary solutions stable at infinity .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 Differential systems governing radially symmetric stationary solutions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 Transversality of radially symmetric stationary solutions stable at infinity 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5 The space of potentials .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6 Main result .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7 Key differences with the generic transversality of standing pulses in space dimension one .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 9 2 Preliminary properties 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 Proof of Lemma 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 Reduction to a local statement .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 Proof of the local statement (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 Setting .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 Equivalent characterizations of transversality .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 Checking hypothesis 1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 of [1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 Checking hypothesis 2 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 of [1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5 Conclusion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 23 5 Proof of the main results 24 2 1 Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 An insight into the main result The purpose of this paper is to prove the generic transversality of radially symmetric stationary solutions stable at infinity for gradient systems of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) ∂tw(x, t) = −∇V �w((x, t)) � + ∆xw(x, t) , where time variable t is real, space variable x lies in the spatial domain Rdsp with dsp an integer not smaller than 2, the state function (x, t) �→ w(x, t) takes its values in Rdst with dst a positive integer, and the nonlinearity is the gradient of a scalar potential function V : Rdst → R, which is assumed to be regular (of class at least C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' An insight into the main result of this paper (Theorem 1 on page 8) is provided by the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For a generic potential V , the following conclusions hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' every radially symmetric stationary solution stable at infinity of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) is robust with respect to small perturbations of V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' the set of all such solutions is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The discreteness stated in conclusion 2 of this corollary is a required assumption for the main result of [4], which describes the global behaviour of radially symmetric (time dependent) solutions stable at infinity for the parabolic system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 provides a rigorous proof that this assumption holds generically with respect to V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' This paper can be viewed as a supplement of the article [1], which is devoted to the generic transversality of bistable travelling fronts and standing pulses stable at infinity for parabolic systems of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) in (unbounded) space dimension one, and which provides a rigorous proof of the genericity of similar assumptions made in [2, 3, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The ideas, the nature of the results, and the scheme of the proof are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 Radially symmetric stationary solutions stable at infinity A function u : [0, +∞) → Rdst, r �→ u(r) defines a radially symmetric stationary solution of the parabolic system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) if and only if it satisfies, on (0, +∞), the (non-autonomous) differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) ¨u(r) = −dsp − 1 r ˙u(r) + ∇V �u(r) � , where ˙u and ¨u stand for the first and second derivatives of r �→ u(r), together with the limit (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3) ˙u(r) → 0 as r → 0+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Observe that, in this case, u(·) is actually the restriction to [0, +∞) of an even function in C3(R, Rd st) which is a solution (on R) of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) (the limit (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3) ensures 3 that equality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) still makes sense and holds at r equals 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In other words, provided that condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3) holds, it is equivalent to assume that system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) holds on (0, +∞) or on [0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' By abuse of language, the terminology radially symmetric stationary solution of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) will refer, all along the paper, to functions u : [0, +∞) → Rdst satisfying these conditions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3) (even if, formally, it is rather the function Rd sp → Rd st, x �→ u �|x| � that fits with this terminology).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us denote by Σmin(V ) the set of nondegenerate (local or global) minimum points of V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' with symbols, Σmin(V ) = �u ∈ Rdst : ∇V (u) = 0 and D2V (u) > 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Throughout all the paper, the words minimum point will be used to denote a local or global minimum point of a (potential) function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' A (global) solution (0, +∞) → Rdst, r �→ u(r), of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) (in particular a radially symmetric stationary solution of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1)) is said to be stable at infinity if u(r) approaches a point of Σmin(V ) as r goes to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' If this point of Σmin(V ) is denoted by u∞, then the solution is said to be stable close to u∞ at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For every u∞ in Σmin(V ), let SV, u∞ denote the set of the radially symmetric stationary solutions of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) that are stable close to u∞ at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' With symbols, SV, u∞ = �u : [0, +∞) → Rdst : u satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3) and u(r) −−−−→ r→+∞ u∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let S0 V, u∞ = � u(0) : u ∈ SV, u∞ � , and let (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4) SV = � u∞∈Σmin(V ) SV, u∞ and S0 V = � u∞∈Σmin(V ) S0 V, u∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The following statement is an equivalent (simpler) formulation of conclusion 2 of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For a generic potential V , the subset S0 V of Rdst is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 Differential systems governing radially symmetric stationary solutions The second-order differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) is equivalent to the (non-autonomous) 2dst- dimensional first order differential differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5) � � � � � ˙u = v ˙v = −dsp − 1 r v + ∇V (u) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Introducing the auxiliary variables τ and c defined as (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6) τ = log(r) and c = 1 r , 4 the previous 2dst-dimensional differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5) is equivalent to each of the following two 2dst + 1-dimensional autonomous differential systems: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7) � � � � � � � uτ = rv vτ = −(dsp − 1)v + r∇V (u) rτ = r , and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8) � � � � � � � ur = v vr = −(dsp − 1)cv + ∇V (u) cr = −c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Integrating the third equations of systems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8) yields r = r0eτ−τ0 and 1 c − 1 c0 = r − r0 , and the parameters τ0 and c0 (which determine in each case the origin of “time”) do not matter in principle, since those systems are autonomous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' However, if the “initial conditions” r0 and c0 are positive (which is true for the solutions that describe radially symmetric stationary solutions of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1)), it is natural to choose, in each case, the origins of time according to equalities (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6), that is : τ0 = ln(r0) and c0 = 1 r0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Properties close to origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7) is relevant to provide an insight into the limit system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5) as r goes to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The subspace R2dst × {0} (r equal to 0) is invariant by the flow of this system, and the system reduces on this invariant subspace to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='9) � uτ = 0 vτ = −(dsp − 1)v , see figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For every u0 in Rdst, the point (u0, 0Rdst, 0) is an equilibrium of sys- tem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' let us denote by W u, 0 V (u0) the (one-dimensional) unstable manifold of this equilibrium, for this system, let (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='10) W u, 0, + V (u0) = W u, 0 V (u0) ∩ �R2dst × (0, +∞) � , and let W u, 0, + V = � u0∈Rdst W u, 0, + V (u0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The subspace (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='11) Ssym = Rdst × {0Rdst} × {0} of R2dst+1 can be seen as the higher space dimension analogue of the symmetry (reversibil- ity) subspace Rdst × {0Rdst} of R2dst (which is relevant for symmetric standing pulses in space dimension 1, see [1] and subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7 below);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' the set W u, 0, + V can be seen as the unstable manifold of this subspace Ssym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 5 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1: Dynamics of the (equivalent) differential systems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7) (for r nonnegative finite) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8) (for c = 1/r nonnegative finite) in Rdst × Rdst × [0, +∞] (this domain is three-dimensional if dst is equal to 1, as on the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For the limit differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='9) in the subspace r = 0 (in green), the trajectories are vertical and the solutions converge towards the horizontal u-axis, defined as Ssym in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='11), and which is the higher space dimensional analogue of the symmetry subspace for symmetric standing pulses in space dimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The point u∞ is a local minimum point of V , so that the point (u∞, 0Rdst) is a hyperbolic equilibrium for the limit differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='12) in the subspace c = 0 ⇐⇒ r = +∞ (in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Systems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8) are autonomous, but the quantity r (the quantity c) goes monotonously from 0 to +∞ (from +∞ to 0) for all the solutions in the subspace r > 0 ⇐⇒ c > 0, so that those solutions can be parametrized with r (with c) as time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The unstable manifold W u, 0, + V (u0) is one-dimensional and is a transverse intersection between the unstable set W u, 0, + V of the subspace {r = 0, v = 0Rdst} and the centre stable manifold W cs, ∞, + V (u∞) of the equilibrium (u∞, 0Rdst, c = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' To prove the generic transversality of this intersection is the main goal of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The dotted red curve is the projection onto the (u, r)-subspace of this intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The part of W cs, ∞, + V (u∞) which is displayed on the figure can also be seen as the local centre stable manifold W cs, ∞, + loc, V, ε1, c1(u∞) defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='10) (with u∞ equal to the point u∞,1 introduced there).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 6 wo u sym om L 0 (8m) loc, V, E1, C1 C1 L0= u 8 E1Properties close to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8) is relevant to provide an insight into the limit system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5) as r goes to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The subspace R2dst × {0} of R2dst+1 (c equal to 0, or in other words r equal to +∞) is invariant by the flow of this system, and the system reduces on this invariant subspace to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='12) � ur = v vr = ∇V (u) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For every u∞ in Σmin(V ), the point (u∞, 0Rdst, 0) is an equilibrium of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' let us consider its global centre-stable manifold in R2dst × (0, +∞), defined as (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='13) W cs, ∞, + V (u∞) = � (u0, v0, c0) ∈ R2dst × (0, +∞) : the solution of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8) with initial condition (u0, v0, c0) at “time” r0 = 1/c0 is defined up to +∞ and goes to (u∞, 0, 0) as r goes to +∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' This set W cs, ∞, + V (u∞) is a dst + 1-dimensional submanifold of R2dst × (0, +∞) (see subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Radially symmetric stationary solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us consider the involution ι : R2dst × (0, +∞) → R2dst × (0, +∞) , (u, v, r) �→ (u, v, 1/r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The following lemma, proved in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1, formalizes the correspondence between the radially symmetric stationary solutions stable at infinity for system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) and the manifolds defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let u∞ be a point of Σmin(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' A (global) solution [0, +∞) → Rdst, r �→ u(r) of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) belongs to SV, u∞ if and only if its trajectory (in R2dst × (0, +∞)) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='14) ��u(r), ˙u(r), r � : r ∈ (0, +∞) � belongs to the intersection (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='15) W u, 0, + V ∩ ι−1�W cs, ∞, + V (u∞) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 Transversality of radially symmetric stationary solutions stable at infinity Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let u∞ be a point of Σmin(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' A radially symmetric stationary solution stable close to u∞ at infinity for system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) (in other words, a function u of SV, u∞) is said to be transverse if the intersection (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='15) is transverse, in R2dst × (0, +∞), along the trajectory (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The natural analogue of radially symmetric stationary solutions stable at infinity when space dimension dsp is equal to 1 are symmetric standing pulses stable at infinity (see Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5 of [1]), and the natural analogue for such pulses of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5 above 7 is their elementarity, not their transversality (see Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 and Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6 of [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' However, the transversality of a symmetric standing pulse (when the space dimension dsp equals 1) makes little sense in higher space dimension, because of the singularity at r equals 0 for the differential systems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5), or because of the related fact that the subspace {r = 0} is invariant for the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For that reason, the adjective transverse (not elementary) is chosen to qualify the property considered in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5 The space of potentials For the remaining of the paper, let us take and fix an integer k not smaller than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us consider the space Ck+1 b (Rdst,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' R) of functions Rd → R of class Ck+1 which are bounded,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' as well as their derivatives of order not larger than k + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' equipped with the norm ∥W∥Ck+1 b = max α multi-index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' |α|≤k+1 ∥∂|α| uαW∥L∞(Rd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='R) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' and let us embed the larger space Ck+1(Rdst,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' R) with the following topology: for V in this space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' a basis of neighbourhoods of V is given by the sets V + O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' where O is an open subset of Ck+1 b (Rdst,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' R) embedded with the topology defined by ∥·∥Ck+1 b (which can be viewed as an extended metric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For comments concerning the choice of this topology, see subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6 Main result The following generic transversality statement is the main result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Theorem 1 (generic transversality of radially symmetric stationary solutions stable at infinity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' There exists a generic subset G of � Ck+1(Rdst, R), ∥·∥Ck+1 b � such that, for every potential function V in G, every radially symmetric stationary solution stable at infinity of the parabolic system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Theorem 1 can be viewed as the extension to higher space dimensions (for radially symmetric solutions) of conclusion 2 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7 of [1] (which is concerned with elementary standing pulses stable at infinity in space dimension 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' A short comparison between these two results and their proofs is provided in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For more comments and a short historical review on transversality results in similar contexts, see subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6 of the same reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The core of the paper (section 4) is devoted to the proof of the conclusions of Theorem 1 among potentials which are quadratic past a certain radius (defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2)), as stated in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The extension to general potentials of Ck+1 b (Rdst, R) is carried out in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' As in [1] (see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8 of that reference), the same arguments could be called upon to prove that the following additional conclusions hold, generically with respect to the potential V : 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' for every minimum point of V , the smallest eigenvalue of D2V at this minimum point is simple;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' every radially symmetric stationary solution stable at infinity of the parabolic system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) approaches its limit at infinity tangentially to the eigenspace corresponding to the smallest eigenvalue of D2V at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7 Key differences with the generic transversality of standing pulses in space dimension one Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 lists the key differences between the proof of the generic elementarity of symmetric standing pulses carried out in [1], and the proof of the generic transversality of radially symmetric stationary solutions carried out in the present paper (implicitly, the other steps/features of the proofs are similar or identical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The state dimension, which is simply denoted by d in [1], is here denoted by dst in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Some of the notation/rigour is lightened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Symmetric standing pulse Radially symmetric stationary solution Critical point at infinity critical point e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' E = (e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 0Rdst ) minimum point u∞ Symmetry subspace Ssym {(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' v) ∈ R2dst : v = 0},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' dimension dst {(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' r) ∈ R2dst+1 : (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' r) = (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 0)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' dimension dst Differential system governing the profiles autonomous,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' conservative,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' regular at Ssym non-autonomous,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' dissipative,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' singular at reversibility subspace Direction of the flow E → Ssym Ssym → u∞ Invariant manifold at infinity W u(E),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' dimension dst − m(e) W cs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' +(u∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' dimension dst + 1 Invariant manifold at symmetry subspace none W u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' dimension dst + 1 Transversality W u(E) ⋔ Ssym W cs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' +(u∞) ⋔ W u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' + Transversality of spatially homogeneous solutions irrelevant Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 Interval Ionce (values reached only once) “anywhere” close to Ssym M (departure set of Φ) parametrization of ∂W u loc, V (E) and time, dimension dst − m(e) Ssym and W cs, ∞, + loc (u∞) at r = N, dimension 2dst N (arrival set of Φ) R2dst R2dst × R2dst W (target manifold) Ssym diagonal of N dim(M) − codim(W) −m(e) 0 Condition to be fulfilled by perturbation W � DΦ(W) �� (0, ψ)� ̸= 0 � DΦu(W) �� (φ, ψ)� ̸= 0 Perturbation W, case 3 precluded W(u0) ̸= 0 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1: Formal comparison between the generic elementarity of symmetric standing pulses (space dimension 1) proved in [1], and the generic transversality of radially symmetric stationary solutions (higher space dimension dsp) proved in the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Here are a few additional comments about these differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 9 Concerning the critical point at infinity, u∞ is assumed (here) to be a minimum point, whereas (in [1]) the Morse index of e is any.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Indeed, if the Morse index m(u∞) of u∞ was positive, then the dimension of the centre-stable manifold W cs, ∞, + V (u∞) would be equal to dst + m(u∞) + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' as a consequence, proving the transversality of the intersection (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='15) in that case would require more stringent regularity assumptions on V (see hypothesis 1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 of [1]) while nothing particularly useful could be derived from this transversality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' On the other hand, assuming that u∞ is a minimum point allows to view its local centre-stable manifold as a graph (u, c) �→ v (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4), which is slightly simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Concerning the interval Ionce providing values u reached “only once” by the profile (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3), the proof of the present paper takes advantage of the dissipation to find a convenient interval close to the “departure point” u0, as was done in [1] for travelling fronts (whereas, for standing pulse, the interval is to be found “anywhere”, thanks to the conservative nature of the differential system governing the profiles, see conclusion 1 of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 of [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Concerning the function Φ to which Sard–Smale theorem is applied in the present paper, both manifolds W u, 0, + and W cs, ∞, +(u∞) depend on the potential V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' However, the transversality of an intersection between these two manifolds can be seen as the transversality of the image of Φ with the (fixed) diagonal of R2dst × R2dst, for a function Φ combining the parametrization of these two manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' This trick, which is the same as in [1] for travelling fronts, allows to apply Sard–Smale theorem to a function Φ with a fixed arrival space N containing a fixed target manifold W (in this case the diagonal of N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' By contrast, for symmetric standing pulses in [1], since the subspace Ssym involved in the transverse intersection is fixed, the previous trick is unnecessary and the setting is simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Finally, a technical difference occurs in “case 3” of the proof that the degrees of freedom provided by perturbing the potential allow to reach enough directions in the arrival state of Φ (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6, which is the core of the proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In [1], case 3 is shown to lead to a contradiction, not only for symmetric standing pulses, but also for asymmetric ones and for travelling fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Here, such a contradiction does not seem to occur (or at least is more difficult to prove), but this has no harmful consequence: a suitable perturbation of the potential can still be found in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 2 Preliminary properties 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 Proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 Let V denote a potential function in Ck+1(Rdst, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let (0, +∞) → Rdst, r �→ u(r) denote a (global) solution of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2), assumed to be stable close to some point u∞ of Σmin(V ) at infinity (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 follows from the next lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The derivative ˙u(r) goes to 0 as r goes to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 10 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us consider the Hamiltonian function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) HV : R2dst → R , (u, v) �→ v2 2 − V (u) , and, for every r in (0, +∞), let h(r) = HV �u(r), ˙u(r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' It follows from system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) that, for every r in (0, +∞), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) ˙h(r) = −dsp − 1 r ˙u(r)2 , thus the function h(·) decreases, and it follows from the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) of the Hamiltonian that this function converges, as r goes to +∞, towards a finite limit h∞ which is not smaller than −V (u∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us proceed by contradiction and assume that h∞ is larger than −V (u∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Then, it follows again from the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) of the Hamiltonian that the quantity ˙u(r)2 con- verges towards the positive quantity 2 �h∞ + V (u∞) � as r goes to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' As a consequence, it follows from equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) that h(r) goes to −∞ as r goes to +∞, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 Transversality of homogeneous radially symmetric stationary solutions stable at infinity Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For every potential function V in Ck+1(Rdst, R) and for every nonde- generate minimum point u∞ of V , the constant function [0, +∞) → Rdst , r �→ u∞ , which defines an (homogeneous) radially symmetric stationary solution stable at infinity for system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) , is transverse (in the sense of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let V denote a function in Ck+1(Rdst, R) and u∞ denote a nondegenerate minimum point of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The function [0, +∞) → Rdst, r �→ u∞ is a (constant) solution of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5), and the linearization of this differential system around this solution reads (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3) ¨u = −dsp − 1 r ˙u + D2V (u∞) · u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let (0, +∞) → Rdst, r �→ u(r) denote a nonzero solution of this differential system, and, for every r in (0, +∞), let v(r) = ˙u(r) and U(r) = �u(r), v(r) � and q(r) = u(r)2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 11 Then (omitting the dependency on r), ˙q = u · ˙u and ¨q = ˙u2 + u · ¨u = ˙u2 − dsp − 1 r ˙q + D2V (u∞) · (u, u) , so that d dr �rdsp−1 ˙q(r) � = rdsp−1 � ¨q + dsp − 1 r ˙q � = rdsp−1� ˙u2 + D2V (u∞) · (u, u) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Since r �→ u(r) was assumed to be nonzero, it follows that the quantity rdsp−1 ˙q(r) is strictly increasing on (0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' To prove the intended conclusion, let us proceed by contradiction and assume that, for every r in (0, +∞), �u(r), v(r), r � belongs: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' to the tangent space T(u∞,0Rdst ,r)W u, 0, + V (u∞), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' and to the tangent space T(u∞,0Rdst ,r) � ι−1�W cs, ∞, + V (u∞) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' As in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6), let us introduce the auxiliary variables τ (equal to log(r)) and c (equal to 1/r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' With this notation, system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3) is equivalent to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4) � � � � � � � uτ = rv vτ = −(dsp − 1)v + rD2V (u∞) · u rτ = r , and to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5) � � � � � � � ur = v vr = −(dsp − 1)cv + D2V (u∞) · u cr = −c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Assumptions 1 and 2 above yield the following conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In view of the limit of system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4) as r goes to 0+, it follows from assumption 1 that there exists δu0 in Rdst such that �u(r), v(r) � goes to (δu0, 0Rdst) as r goes to 0+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' and in view of the limit of system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5) as c goes to 0+, it follows from assumption 2 that �u(r), v(r) � goes to (0Rdst, 0Rdst), at an exponential rate, as r goes to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' It follows from these two conclusions that the quantity rdsp−1 ˙q(r) goes to 0 as r goes to 0+ and as r goes to +∞, a contradiction with the fact (observed above) that this quantity is strictly increasing with r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 Additional properties close to the origin Let V denote a potential function in Ck+1(Rdst, R) and let u0 be a point in Rdst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us recall (see subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3) that the unstable manifold W u, 0 V (u0) of the equilibrium (u0, 0Rdst, 0) for the autonomous differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7)) is one-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' As a consequence there exists a unique solution r �→ u(r) of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) such that the image of the map r �→ �u(r), ˙u(r), r) lies in the intersection W u, 0, + V (u0) of this unstable manifold with the half-space where r is positive (this intersection was defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='10));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' or, in other words, such that �u(r), ˙u(r) � goes to (u0, 0) as r goes to 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' This solution is defined on some (maximal) interval (0, rmax), where rmax is either a finite quantity or +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The following lemma provides properties of this solution that will be used in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' To ease its statement, let us assume that rmax is equal to +∞ (only this case will turn out to be relevant), and let us consider the continuous extension of u(·) to the interval [0, +∞) (and let us still denote by u(·) this continuous extension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' If u(·) is not identically equal to u0 (in other words, if u0 is not a critical point of V ), then there exists a positive quantity ronce such that, denoting by Ionce the interval [0, ronce), the following conclusions hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' the function ˙u(·) does not vanish on Ionce, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' and, for every r∗ in Ionce and r in [0, +∞), u(r) = u(r∗) =⇒ r = r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The linearized system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7) at the equilibrium (u0, 0Rdst, 0) reads: d dτ � � � δu δv δr � � � = � � � 0 0 0 0 −(dsp − 1) ∇V (u0) 0 0 1 � � � � � � δu δv δr � � � , thus the tangent space at (u0, 0Rdst, 0) to W u, 0 V (u0) (the unstable eigenspace of the matrix of this system) is spanned by the vector �0, ∇V (u0)/dsp, 1 �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' it follows that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6) ˙u(r) = r dsp ∇V (u0) �1 + or→0+(r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Thus, if r0 is a sufficiently small positive quantity, then ˙u(·) does not vanish on (0, r0] (so that conclusion 1 of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 holds provided that ronce is not larger than r0), and the map (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7) [0, r0] → Rdst , r �→ u(r) is a C1-diffeomorphism onto its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For r in [0, +∞), let us denote �u(r), ˙u(r) � by U(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' According to the decrease (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) of the Hamiltonian, there exists a quantity ronce in (0, r0) such that, for every r∗ in [0, ronce), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8) HV �U(r0) � < −V �u(r∗) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 13 Take r∗ in [0, ronce] and r in [0, +∞), and let us assume that u(r) equals u(r∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' If r was larger than r0 then it would follow from the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) of the Hamiltonian, its decrease (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2), and inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8) that −V �u(r) � ≤ HV �U(r) � ≤ HV �U(r0) � < −V �u(r∗) � , a contradiction with the equality of u(r) and u(r∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Thus r is not larger than r0, and it follows from the one-to-one property of the function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7) that r must be equal to r∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' conclusion 2 of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 thus holds, and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 Additional properties close to infinity Let V1 denote a potential function in Ck+1(Rdst, R) and u1,∞ denote a nondegenerate minimum point of V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' According to the implicit function theorem, there exists a (small) neighbourhood νrobust(V1, u1,∞) of Vquad-R and a Ck-function V �→ u∞(V ) defined on νrobust(V1, u1,∞) and with values in Rdst such that u∞(V1) equals u1,∞ and, for every V in νrobust(V1, u1,∞), u∞(V ) is a local minimum point of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The following proposi- tion is nothing but the local centre-stable manifold theorem applied to the equilibrium �u∞(V ), 0Rdst, 0 � of the (autonomous) differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8), for V close to V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Addi- tional comments and references concerning local stable/centre/unstable manifolds are provided in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 (local centre-stable manifold at infinity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' There exist a neighbourhood ν of V1 in Ck+1(Rdst, R), included in νrobust(V1, u1,∞), such that, if ε1 and c1 denote sufficiently small positive quantities, then, for every V in ν, there exists a Ck-map (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='9) wcs, ∞ loc, V : BRdst(u1,∞, ε1) × [0, c1] → Rdst , (u, c) �→ wcs, ∞ loc, V (u, c) , such that, for every (u0, v0, c0) in BRdst(u1,∞, ε1) × Rdst × [0, c1], the following two statements are equivalent: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' v = wcs, ∞ loc, V (u, c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' the solution r �→ �u(r), v(r), c(r) � of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8) with initial condi- tion (u0, v0, c0) at time r0 = 1/c0 is defined up to +∞, remains in BRdst(u1,∞, ε1)× Rdst × [0, c1] of all r larger than r0, and goes to �u∞(V ), 0Rdst, 0 � as r goes to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In particular, wcs, ∞ loc, V �u∞(V ), 0 � is equal to 0Rdst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In addition, the map BRdst(u1,∞, ε1) × [0, c1] × ν → Rdst , (u, c, V ) �→ wcs, ∞ loc, V (u, c) is of class Ck (with respect to u and c and V ), and, for every V in ν, the graph of the differential at �u∞(V ), 0) of the map (u, c) �→ wcs, ∞ loc, V (u, c) is equal to the centre-stable subspace of the linearization at �u∞(V ), 0Rdst, 0 � of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 14 Let us denote by W cs, ∞, + loc, V, ε1, c1 �u∞(V ) � the graph of the map (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='9) (restricted to positive values of c), see figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' with symbols, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='10) W cs, ∞, + loc, V, ε1, c1 �u∞(V ) � = ��u, wcs, ∞ loc, V (u, c), c � : (u, c) ∈ BRdst(u1,∞, ε1) × (0, c1] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' This set defines a local centre-manifold (restricted to positive values of c) for the equilib- rium �u∞(V ), 0Rdst, 0 � of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Its uniqueness (for positive values of c) is ensured by the dynamics of the centre component c, which, according to the third equation of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8), decreases to 0 (see figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The global centre-stable manifold W cs, ∞, + V �u∞(V ) � already defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='13) can be redefined as the points of R2dst ×(0, +∞) that eventually reach the local centre manifold W cs, ∞, + loc, V, ε1, c1 �u∞(V ) � when they are transported by the flow of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' If the state dimension dst is equal to 1, then a calculation shows that wcs, ∞ loc, V (u, c) = − �u − u∞(V ) � �� V ′′�u∞(V ) � + dsp − 1 2 c + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' � , where “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' ” stands for higher order terms in u − u∞(V ) and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In particular the quantity ∂c∂uwcs, ∞ loc, V �u∞(V ), 0 � is equal to the (negative) quantity −(dsp − 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The display of the local centre-stable manifold at infinity on figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 fits with the sign of this quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 3 Tools for genericity Let (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) Vfull = Ck+1(Rdst, R) , and, for a positive quantity R, let (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) Vquad-R = � V ∈ Vfull : for all u in Rd, |u| ≥ R =⇒ V (u) = u2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us recall the notation SV introduced in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For every positive quantity R and for every potential V in Vquad-R, the following conclusions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The flow defined by the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) (governing radially symmetric stationary solutions of the parabolic system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1)) is global (that is, every solution is defined on (0, +∞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For every u in SV , the following bound holds: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3) sup r∈(0,+∞) |u(r)| < R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let V be in Vquad-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' According to the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) of Vquad-R, there exists a positive quantity K such that, for every u in Rdst, |∇V (u)| ≤ K + |u| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' As a consequence, the following inequalities hold for the right-hand side of the first order differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5): ���� � v, −dsp − 1 r v + ∇V (u) ����� ≤ |v| + dsp − 1 r |v| + K + |u| ≤ K + � 2 + dsp − 1 r � |(u, v)| , and this bound prevents the solution from blowing up in finite time, which proves conclusion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Now, take a function u in SV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us still denote by u(·) the continuous extension of this solution to [0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For every r in [0, +∞), let q(r) = u(r)2 2 and Q(r) = rdsp−1 ˙q(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Then (omitting the dependency on r), ˙q = u · ˙u and ¨q = ˙u2 + u · ¨u = ˙u2 − dsp − 1 r ˙q + u · ∇V (u) , so that ˙Q = rdsp−1 � ¨q + dsp − 1 r ˙q � = rdsp−1� ˙u2 + u · ∇V (u) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' According to the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) of Vquad-R, there exists a positive quantity δ (sufficiently small) so that, for every w in Rdst, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4) |w| ≥ R − δ =⇒ w · ∇V (w) ≥ w2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us proceed by contradiction and assume that supr∈(0,+∞) |u(r)| is not smaller than R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Since u(·) is stable at infinity and since the critical points of V belong to the open ball BRdst(0, R − δ), it follows that the set �r ∈ [0, +∞) : |u(r)| ≥ R � is nonempty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' let rout denote the minimum of this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For the same reason, the set �r ∈ (rout, +∞) : |u(r)| < R − δ � is also nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let rback denote the infimum of this last set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' It follows from these definitions that rback is larger than rout and that, for every r in (rout, rback), according to inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5) ˙Q(r) ≥ rdsp−1 � ˙u2(r) + u2(r) 2 � > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 16 If on the one hand rout equals 0 then |u(0)| is not smaller than R and, since Q(0) equals 0, it follows from inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5) that Q(·) is positive on (0, rback), so that the same is true for ˙q(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Thus q(·) is strictly increasing on [0, rback] and |u(rback)| must be larger than |u(rout)|, a contradiction with the definition of rback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' If on the other hand rout is positive, then |u(rout)| is equal to R and ˙q(rout) is nonnegative so that the same is true for Q(rout), and it again follows from inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5) that Q(·) is positive on (0, rback), yielding the same contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Conclusion 2 of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For every positive quantity R and every potential V in Vquad-R, let (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6) SV : (0, +∞)2 × R2dst → R2dst , �(rinit, r), (uinit, vinit) � �→ SV �(rinit, r), (uinit, vinit) � denote the (globally defined) flow of the (non-autonomous) differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5) for this potential V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In other words, for every rinit in (0, +∞) and (uinit, vinit) in R2dst, the function (0, +∞) → R2dst , r �→ SV �(rinit, r1), (uinit, vinit) � is the solution of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5) for the initial condition (uinit, vinit) at r equals rinit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' According to subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3, the flow SV may be extended to the larger set (0, +∞)2 × R2dst ∪ [0, +∞)2 × Rdst × {0Rdst} ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' according to this extension, for every u0 in Rdst, the solution taking its values in the (one-dimensional) unstable manifold W u, 0, + V (u0) reads: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7) [0, +∞) → Rdst , r �→ SV �(0, r), (u0, 0Rdst) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 4 Generic transversality among potentials that are quadratic past a given radius 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 Notation and statement Let us recall the notation SV and SV, u∞ introduced in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' There exists a generic subset of Vquad-R such that, for every potential V in this subset, every radially symmetric stationary solution stable at infinity of the parabolic system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) (in other words, every u in SV ) is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 Reduction to a local statement Let V1 denote a potential function in Vquad-R and u1,∞ denote a nondegenerate minimum point of V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' According to the implicit function theorem, there exists a (small) neighbour- hood νrobust(V1, u1,∞) of Vquad-R and a Ck-function u∞(·) defined on νrobust(V1, u1,∞) and with values in Rdst such that u∞(V1) equals u1,∞ and, for every V in νrobust(V1, u1,∞), u∞(V ) is a local minimum point of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The following local generic transversality statement yields Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 (as shown below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 17 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' There exists a neighbourhood νV1, u1,∞ of V1 in νrobust(V1, u1,∞) and a generic subset νV1, u1,∞, gen of νV1, u1,∞ such that, for every V in νV1, u1,∞, gen, every radially symmetric stationary solution stable close to u∞(V ) at infinity of the parabolic system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) (in other words, every u in SV, u∞(V )) is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proof that Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 yields Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us denote by Vquad-R-Morse the dense open subset of Vquad-R defined by the Morse property: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) Vquad-R-Morse = {V ∈ Vquad-R : all critical points of V are nondegenerate} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let V1 denote a potential function in Vquad-R-Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' According to the Morse property its minimum points are isolated and since V1 is in Vquad-R they belong to the open ball BRd(0, R), so that those minimum points are in finite number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Assume that Proposi- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' With the notation of this proposition, let us consider the following two intersections, at each time over all minimum points u1,∞ of V1: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) νV1 = � νV1, u1,∞ and νV1, gen = � νV1, u1,∞, gen .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Since those are finite intersections, νV1 is still a neighbourhood of V1 in Vquad-R and the set νV1, gen is still a generic subset of νV1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' This shows that the set {V ∈ Vquad-R-Morse : every u in SV, u∞(V ) is transverse} is locally generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 of [1] as in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 of this reference shows that this local genericity implies the global genericity stated in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1, which is therefore proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 Proof of the local statement (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 Setting For the remaining part of this section, let us fix a potential function V1 in Vquad-R and a nondegenerate minimum point u1,∞ of V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let ν be a neighbourhood of V1 in Vquad-R, included in νrobust(V1, u1,∞), and let ε1 and c1 be positive quantities, with ν and ε1 and c1 small enough so that the conclusions of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let r1 = 1/c1 and M = Rdst × BRdst(u1,∞, ε1) and Λ = ν , and N = (R2dst)2 and W = {(A, B) ∈ N : A = B} , thus W is the diagonal of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let N denote an integer not smaller than r1, and let us consider the functions Φu : Rdst × Λ → R2dst , (u0, V ) �→ SV �(0, N), (u0, 0Rdst) � , and Φcs : BRdst(u1,∞, ε1) × Λ → R2dst , (uN, V ) �→ �uN, wcs, ∞ loc, V (uN, 1/N) � , and the function (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3) Φ : M × Λ → N , (m, V ) = (u0, uN, V ) �→ �Φu(u0, V ), Φcs(uN, V ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 Equivalent characterizations of transversality Let us consider the set SΛ,u1,∞,N = �(V, u) : V ∈ Λ and u ∈ SV, u∞(V ) and u(N) ∈ BRdst(u1,∞, ε1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The map (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4) Φ−1(W) → SΛ,u1,∞,N , (u0, u, V ) �→ � V, r �→ SV �(0, r), (u0, 0Rdst �� is well defined and one-to-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The image by Φ of a point (u0, uN, V ) of M × Λ belongs to the diagonal W of N if and only if Φu(u0, V ) equals Φcs(uN, V ), and in this case the function u : r �→ SV �(0, r), (u0, 0Rdst � belongs to SV, u∞(V ) and u(N) (which is equal to uN) belongs to BRdst(u1,∞, ε1), so that (V, u) belongs to SΛ,u1,∞,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The map (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4) above is thus well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Now, for every (V, u) in SΛ,u1,∞,N, if we denote by u0 the limit limr→0+ u(r) and by uN the vector u(N), then (u0, uN, V ) is the only possible antecedent of (V, u) by the map (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In addition, SV �(0, N), (u0, 0Rdst) � = �uN, ˙u(N) � , and since u(r) goes to u∞(V ) as r goes to +∞, the vector �u(N), ˙u(N), 1/N � must belong to the centre-stable manifold W cs, ∞, + V �u∞(V ) � of u∞(V ), so that, according to the definition of wcs, ∞ loc, V , ˙u(N) = wcs, ∞ loc, V �u(N), 1/N � , and this yields the equality between Φu(u0, V ) and Φcs(uN, V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Thus Φ(V, u) belongs to W and (u0, uN, V ) belongs to Φ−1(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For every potential function V in Λ, the following two statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The image of the function M → N, m �→ Φ(m, V ) is transverse to W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Every u in SV, u∞(V ) such that u(N) is in BRdst(u1,∞, ε1) is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' According to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2, for every V in Λ, the constant function r �→ u∞(V ), which belongs to SV , is already (a priori) known to be transverse, therefore only nonconstant solutions matter in statement 2 of this proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us consider (m2, V2) in M × Λ such that Φ(m2, V2) is in W, let (u2,0, u2,N) denote the components of m2, and let r �→ u2(r) and r �→ U2(r) denote the functions satisfying, for all r in [0, +∞), U2(r) = �u2(r), ˙u2(r) � = SV �(0, r), (u2,0, 0Rdst � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us consider the map ∆Φ : M → R2dst , (u0, uN) �→ Φu(u0, V2) − Φcs(uN, V2) , 19 and let us write, only for this proof, DΦ and DΦu and DΦcs and D(∆Φ) for the differentials of Φ and Φu and Φcs and ∆Φ at (m2, V2) and with respect to all variables in M (but not with respect to V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' According to Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5, the transversality of u2 is defined as the transversality of the intersection W u, 0, + V2 ∩ ι−1� W cs, ∞, + V2 �u∞(V2) �� along the trajectory of U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' This transversality can be considered at a single point, no matter which, of the trajectory U2 �(0, +∞) �, in particular at the point Φu(u2,0, V2) which is equal to Φcs�u2(N), V 2 �, and is equivalent to the transversality of the dst-dimensional manifolds W u, 0, + V2 ∩ �R2dst × {N} � and ι−1� W cs, ∞, + V2 �u∞(V2) �� ∩ �R2dst × {N} � in R2dst ×{N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' It is therefore equivalent to the surjectivity of the map D(∆Φ) (statement (B) in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' On the other hand, the image of the function M → N, m �→ Φ(m, V2) is transverse at Φ(m, V2) to the diagonal W of N if and only if the image of DΦ contains a complementary space of this diagonal (statement (A) in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5 below)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Thus Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 is a consequence of the next lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The following two statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' (A) The image of DΦ contains a complementary subspace of the diagonal W of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' (B) The map D(∆Φ) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' If statement (A) holds, then, for every (α, β) in N, there exist γ in R2dst and δm in Tm2M such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5) (γ, γ) + DΦ · δm = (α, β) , so that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6) D(∆Φ) · δm = α − β , and statement (B) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Conversely, if statement (B) holds, then, for every (α, β) in N, there exists δm in Tm2M such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6) holds, and as a consequence, if (δu0, δuN) denote the components of δm, then α − DΦu(δu0) is equal to β − DΦcs(δuN), and if this vector is denoted by γ, then equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5) holds, and this shows that statement (A) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' As explained above, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5, and is therefore proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 Checking hypothesis 1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 of [1] The function Φ is as regular as the flow SV , thus of class Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' It follows from the definitions of M and N and W that dim(M) − codim(W) = (dst + dst) − 2dst = 0 , so that hypothesis 1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 of [1] is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 Checking hypothesis 2 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 of [1] For every V in Vquad-R, let us recall the notation SV introduced in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7) for the flow of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Take (m2, V2) in the set Φ−1(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let (u2,0, u2,N) denote the components of m2, and, for every r in (0, +∞), let us write U2(r) = �u2(r), v2(r) � = SV2 �(0, r), (u2,0, 0Rdst) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us write DΦ and DΦu and DΦcs for the full differentials (with respect to arguments m in M and V in Λ) of the three functions Φ and Φu and Φcs respectively at the points �u2,0, u2,N, V2 �, �u2,0, V2 � and �u2,N, V2 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Checking hypothesis 2 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 of [1] amounts to prove that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7) im(DΦ) + W = N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' If u2(·) is constant (that is, identically equal to u∞(V2)), then equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7) follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Thus, let us assume that u2(·) is nonconstant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In this case, equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7) is a consequence of the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For every nonzero vector (φ2, ψ2) in R2dst, there exists a function W in Ck+1 b (Rdst, R) such that supp(W) ⊂ BRd(0, R) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8) and �DΦu · (0, 0, W) �� (φ2, ψ2) � ̸= 0 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='9) and DΦcs · (0, 0, W) = 0R2dst .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='10) Proof that Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6 yields equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='9) shows that the orthogonal complement, in R2dst, of the directions that can be reached by DΦu·(0, 0, W) for potentials W satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='10) is reduced to 0R2dst;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' in other words, all directions of R2dst can be reached by that means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' This shows that im(DΦ) ⊃ R2dst × {0R2dst} , and since the subspace at the right-hand side of this inclusion is transverse to W in R4dst, this proves equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7) (and shows that hypothesis 2 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 of [1] is fulfilled).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let (φ2, ψ2) denote a nonzero vector in R2dst, let W be a function in Ck+1 b (Rdst, R) satisfying the inclusion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='11) supp(W) ⊂ BRd(0, R) \\ BRdst(u1,∞, ε1) , and observe that inclusion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8) and equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='10) follow from this inclusion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us consider the linearization of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2), for the potential V2, around the solution r �→ U2(r): (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='12) d dr � δu(r) δv(r) � = � 0 id D2V2 �u2(r) � −dsp−1 r � � δu(r) δv(r) � , 21 and let T(r, r′) denote the family of evolution operators obtained by integrating this linearized differential system between r and r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' It follows from the variation of constants formula that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='13) DΦu · (0, 0, W) = � N −∞ T(r, N) � 0, ∇W �u2(r) �� dr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For every r in (0, +∞), let T ∗(r, N) denote the adjoint operator of T(r, N), and let (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='14) �φ(r), ψ(r) � = T ∗(r, N) · (φ2, ψ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' According to expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='13), inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='9) reads � N −∞ �� 0, ∇W �u2(r) �� ��� T ∗(r, N) · (φ2, ψ2) � dr ̸= 0 , or equivalently (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='15) � N −∞ ∇W �u2(r) � · ψ(r) dr ̸= 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Due to the expression of the linearized differential system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='12), (φ, ψ) is a solution of the adjoint linearized system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='16) � ˙φ(r) ˙ψ(r) � = − � 0 D2V2 �u2(r) � id −dsp−1 r � � φ(r) ψ(r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' According to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 (and since u2(·) was assumed to be nonconstant), there exists positive quantity ronce such that, if we denote by Ionce the interval (0, ronce], then ˙u2(·) does not vanish on Ionce, and, for all r∗ in Ionce and r in R, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='17) u2(r) = u2(r∗) =⇒ r = r∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In addition, up to replacing ronce by a smaller positive quantity, it may be assumed that the following conclusions hold: u2(Ionce) ∩ BRdst(u1,∞, ε1) = ∅ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' To complete the proof three cases have to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' There exists r∗ in Ionce such that ψ(r∗) is not collinear to ˙u2(r∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In this case, the construction of a potential function W satisfying inclusion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='11) and inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='9) (and thus the conclusions of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6) is the same as in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7 of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' If case 1 does not occur, then ψ(r) is collinear to ˙u2(r), and since ˙u2(·) does not vanish on Ionce, there exists a C1-function α : Ionce → R such that, for every r in Ionce, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='18) ψ(r) = α(r) ˙u2(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The next cases 2 and 3 differ according to whether the function α(·) is constant or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 22 Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For every r in Ionce, equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='18) holds for some nonconstant function α(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In this case there exists r∗ in Ionce such that ˙α(r∗) is nonzero, and again the construction of a potential function W satisfying inclusion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='11) and inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='9) (and thus the conclusions of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6) is the same as in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7 of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For every r in Ionce, ψ(r) = α ˙u2(r) for some real (constant) quantity α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In this case the quantity α cannot be 0 or else, due to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='16) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='18), both φ(·) and ψ(·) would identically vanish on Ionce and thus on (0, +∞), a contradiction with the assumptions of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Thus, without loss of generality, we may assume that α is equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' If supp(W) is included in a sufficiently small neighbourhood of u2,0, then W(·) vanishes on u2 �[ronce, N] � and the integral on the left-hand side of inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='15) reads � ronce 0 ∇W �u2(r) � · ˙u2(r) dr = W �u2(ronce) � − W(u2,0) = −W(u2,0) , so that inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='15) holds as soon as W(u2,0) is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' By contrast with the proof of the generic elementarity of standing pulses in [1], case 3 above cannot be easily precluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Indeed, let us assume that, for every r in Ionce, ψ(r) is equal to α ˙u2(r) for some nonzero (constant) quantity α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Without loss of generality, we may assume that α is equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Then, it follows from the second equation of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='16) that, still for every r in Ionce (omitting the dependency on r), φ = dsp − 1 r ψ − ˙ψ = dsp − 1 r ˙u2 − ¨u2 = 2(dsp − 1) r ˙u2 − ∇V2(u2) , and it follows from the first equation of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='16) that −D2V2(u2) ˙u2 = ˙φ = −2(dsp − 1) r2 ˙u2 + 2(dsp − 1) r ¨u2 − D2V2(u2) ˙u2 , and thus, after simplification, ¨u2 = 1 r ˙u2 , or equivalently ˙u2 = r dsp ∇V (u2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' As illustrated by equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6), this last equality indeed holds if ∇V2 is constant on the set u2(Ionce).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Case 3 can therefore not be a priori precluded, and if it may be argued that this case is “unlikely” (non generic), the direct argument provided above in this case is simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' By contrast, in [1] for standing pulses in space dimension one (dsp equal to 1), this case could not occur because ψ was assumed to be nonzero on the symmetry subspace, defined here as {(v, r) = (0Rdst, 0)}, see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5 Conclusion As seen in sub-subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3, hypothesis 1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 of [1] is fulfilled for the function Φ defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3), and since Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6 yields equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7), hypothesis 2 of this 23 theorem is also fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The conclusion of this theorem ensures that there exists a generic subset Λgen, N of Λ such that, for every V in Λgen, N, the image of the function M → N, m �→ Φ(m, V ) is transverse to the diagonal W of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' According to Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4, it follows that every u in SV, u∞(V ) such that u(N) is in BRdst(u1,∞, ε1) is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The set Λgen = � N∈N, N≥r0 Λgen, N is still a generic subset of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For every V in Λgen and every u in SV, u∞(V ), since u(r) goes to u∞(V ) as r goes to +∞, there exists N such that u(N) is in BRdst(u1,∞, ε1), and according to the previous statements u is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In other words, the conclusions of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 hold with: νV1, u1,∞ = ν = Λ and νV1, u1,∞, gen = Λgen .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 5 Proof of the main results Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 shows the genericity of the property considered in Theorem 1, but only inside the space Vquad-R of the potentials that are quadratic past some radius R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In this section, the arguments will be adapted to obtain the genericity of the same property in the space Vfull (that is Ck+1(Rdst, R)) of all potentials, endowed with the extended topology (see subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' They are identical to those of section 9 of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us recall the notation SV introduced in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4), and, for every positive quantity R, let us consider the set SV,R = � u ∈ SV : sup r∈[0,+∞) |u(r)| ≤ R � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Exactly as shown in subsection 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 of [1], Theorem 1 follows from the next proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For every positive quantity R, there exists a generic subset Vfull-⋔-S-R of Vfull such that, for every potential V in this subset, every radially symmetric stationary solution stable at infinity in SV,R is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let R denote a positive quantity, let V1 denote a potential function in Vquad-(R+1), and let u1,∞ denote a nondegenerate minimum point of V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us consider the neigh- bourhood νV1, u1,∞ of V1 in Vquad-(R+1) provided by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 for these objects, together with the quantities ε1, c1, and r1 introduced in sub-subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Up to replacing νV1, u1,∞ by its interior, we may assume that it is open in Vquad-(R+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' As in sub-subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1, let us consider an integer N not smaller than r1, and the same function Φ : M × Λ → N as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Here is the sole difference with the setting of sub-subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1: by contrast with the non-compact set M defining the departure set of Φ, let us consider the compact subset MN defined as: MN = BRdst(0Rdst, N) × BRdst(u1,∞, ε1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Thus the integer N now serves two purposes: the “time” (radius) at which the intersection between unstable and centre-stable manifolds is considered, and the radius of the ball 24 containing the departure points of the unstable manifolds that are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' These purposes are independent (two different integers instead of the single integer N may as well be introduced).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us consider the set: OV1,u1,∞,N = � V ∈ νV1, u1,∞ : Φ(MN, V ) is transverse to W in N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' As shown in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4, this set OV1,u1,∞,N is made of the potential functions V in νV1, u1,∞ such that every u in SV, u∞(V ) such that u(N) is in BRdst(u1,∞, ε1) and u(0) is in BRdst(0Rdst, N), is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' This set contains the generic subset νV1, u1,∞, gen = Λgen of νV1, u1,∞ and is therefore generic (thus, in particular, dense) in νV1, u1,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' By comparison with νV1, u1,∞, gen, the additional feature of this set OV1,u1,∞,N is that it is open: exactly as in the proof of Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 of [1], this openness follows from the intrinsic openness of a transversality property and the compactness of MN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us make the additional assumption that the potential V1 is a Morse function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Then, the set of minimum points of V1 is finite and depends smoothly on V in a neighbourhood νrobust(V1) of V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Intersecting the sets νV1, u1,∞ and OV1,u1,∞,N above over all the minimum points u1,∞ of V1 provides an open neighbourhood νV1 of V1 and an open dense subset OV1,N of νV1 such that, for all V in νV1, every radially symmetric stationary solution stable close to a minimum point of V at infinity, and equal at origin to some point of BRdst(0Rdst, N), is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Denoting by int(A) the interior of a set A and using the notation of subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 of [1], let us introduce the sets ˜νV1 = res−1 R,∞ ◦ resR,(R+1)(νV1) , and ˜OV1,N = res−1 R,∞ ◦ resR,(R+1)(OV1,N) , and ˜Oext V1,N = ˜OV1,N ⊔ int �Vfull \\ ˜νV1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' It follows from these definitions that ˜Oext V1,N is a dense open subset of Vfull (for more details, see Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 of [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Since Vquad-(R+1) is a separable space, it is second-countable, and can be covered by a countable number of sets of the form νV1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' With symbols, there exists a countable family (V1,i)i∈N of potentials of Vquad-(R+1)-Morse so that Vquad-(R+1)-Morse = � i∈N νV1,i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us consider the set Vfull-⋔-S-R = Vfull-Morse ∩ � � � (i,N)∈N2 ˜Oext V1,i,N � � , where Vfull-Morse is the set of potentials in Vfull which are Morse functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' This set is a countable intersection of dense open subsets of Vfull, and is therefore a generic subset of Vfull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' And, for every potential V in this set Vfull-⋔-S-R, every radially symmetric stationary solution stable at infinity in SV,R is transverse (for more details, see Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 of [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 25 As already mentioned at the beginning of this section, Theorem 1 follows from Proposi- tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Finally, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 follows from Theorem 1 (for more details, see subsection 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 of [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Acknowledgements This paper owes a lot to numerous fruitful discussions with Romain Joly, about both its content and the content of the companion paper [1] written in collaboration with him.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' References [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Joly and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Risler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' “Generic transversality of travelling fronts, standing fronts, and standing pulses for parabolic gradient systems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In: arXiv (2023), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 1–69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' arXiv: 2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='02095 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' on pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 3, 5, 7–10, 14, 18, 20–26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' [2] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Risler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' [3] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Risler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' “Global behaviour of bistable solutions for hyperbolic gradient systems in one unbounded spatial dimension”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In: arXiv (2022), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 1–75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' arXiv: 1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='01221 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' [4] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Risler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' “Global behaviour of radially symmetric solutions stable at infinity for gradient systems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In: arXiv (2022), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 1–52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' arXiv: 1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='02134 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' [5] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Risler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' “Global relaxation of bistable solutions for gradient systems in one unbounded spatial dimension”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In: arXiv (2022), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 1–69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' arXiv: 1604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='00804 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Emmanuel Risler Université de Lyon, INSA de Lyon, CNRS UMR 5208, Institut Camille Jordan, F-69621 Villeurbanne, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' emmanuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='risler@insa-lyon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='fr 26' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} diff --git a/2NFAT4oBgHgl3EQfDRyP/content/tmp_files/2301.08415v1.pdf.txt b/2NFAT4oBgHgl3EQfDRyP/content/tmp_files/2301.08415v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1c295bc1c93bf12d54f2f2aa49aeb47b2d25f422 --- /dev/null +++ b/2NFAT4oBgHgl3EQfDRyP/content/tmp_files/2301.08415v1.pdf.txt @@ -0,0 +1,1220 @@ +Low-energy quasi-circular electron correlations with charge order +wavelength in Bi2Sr2CaCu2O8+δ +K. Scott,1, 2 E. Kisiel,3 T. J. Boyle,1, 2, 4 R. Basak,3 G. Jargot,5 S. Das,3 S. Agrestini,6 +M. Garcia-Fernandez,6 J. Choi,6 J. Pelliciari,7 J. Li,7 Y. D. Chuang,8 R. D. Zhong,9 +J. A. Schneeloch,9 G. D. Gu,9 F. L´egar´e,5 A. F. Kemper,10 Ke-Jin Zhou,6 V. Bisogni,7 +S. Blanco-Canosa,11, 12 A. Frano,3, 13 F. Boschini,5, 14 and E. H. da Silva Neto1, 2, ∗ +1Department of Physics, Yale University, New Haven, Connecticut 06520, USA +2Energy Sciences Institute, Yale University, West Haven, Connecticut 06516, USA +3Department of Physics, University of California San Diego, La Jolla, California 92093, USA +4Department of Physics and Astronomy, University of California, Davis, California 95616, USA +5Centre ´Energie Mat´eriaux T´el´ecommunications, +Institut National de la Recherche Scientifique, Varennes, Qu´ebec J3X 1S2, Canada +6Diamond Light Source, Harwell Campus, Didcot OX11 0DE, United Kingdom +7National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, NY 11973, USA +8Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA +9Condensed Matter Physics and Materials Science, Brookhaven National Laboratory, Upton, NY, USA +10Department of Physics, North Carolina State University, Raleigh, NC 27695, U.S.A. +11Donostia International Physics Center, DIPC, 20018 Donostia-San Sebastian, Basque Country, Spain +12IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Spain +13Canadian Institute for Advanced Research, Toronto, ON, M5G 1M1, Canada +14Quantum Matter Institute, University of British Columbia, Vancouver, BC V6T 1Z4, Canada +∗ Corresponding Author: eduardo.dasilvaneto@yale.edu +arXiv:2301.08415v1 [cond-mat.str-el] 20 Jan 2023 + +2 +ABSTRACT +In the study of dynamic charge order correlations in the cuprates, most high energy- +resolution resonant inelastic x-ray scattering (RIXS) measurements have focused on mo- +menta along the high-symmetry directions of the copper oxide plane. However, electron +scattering along other in-plane directions should not be neglected as they may contain in- +formation relevant, for example, to the origin of charge order correlations or to our un- +derstanding of the isotropic scattering responsible for strange metal behavior in cuprates. +We report high-resolution resonant inelastic x-ray scattering (RIXS) experiments that re- +veal the presence of dynamic electron correlations over the qx-qy scattering plane in under- +doped Bi2Sr2CaCu2O8+δ with Tc = 54 K. We use the softening of the RIXS-measured bond +stretching phonon line as a marker for the presence of charge-order-related dynamic electron +correlations. The experiments show that these dynamic correlations exist at energies below +approximately 70 meV and are centered around a quasi-circular manifold in the qx-qy scat- +tering plane with radius equal to the magnitude of the charge order wave vector, qCO. We +also demonstrate how this phonon-tracking procedure provides the necessary experimental +precision to rule out fluctuations of short-range directional charge order (i.e. centered around +[qx = ±qCO, qy = 0] and [qx = 0, qy = ±qCO]) as the origin of the observed correlations. +INTRODUCTION +Dynamic fluctuations from periodic charge order (CO) pervade the phase diagram of +cuprate superconductors, perhaps even more than superconductivity itself [1]. The detec- +tion of these fluctuations over energy and momentum was enabled by several recent advances +in the energy resolution of resonant inelastic x-ray scattering (RIXS) instruments operat- +ing in the soft x-ray regime. In the case of YBa2Cu3O6+δ, Cu-L3 RIXS detects dynamic +correlations at the charge order wavevector, qCO, with a characteristic energy scale of ap- +proximately 20 meV [2]. It has been proposed that these low-energy short-range dynamic +charge order correlations are a key ingredient to the strange metal behavior [3, 4] charac- +terized by linear-in-temperature resistivity [5, 6]. On one hand, this temperature behav- +ior is often associated with an isotropic scattering rate that depends only on temperature +in units of energy and Planck’s constant (i.e. ∝ kBT/ℏ, sometimes called the Planckian +regime) [7–11], as supported by recent angle-dependent magnetoresistance measurements of +La1.6−xNd0.4SrxCuO4 [12]. On the other hand, combined transport and RIXS studies have + +3 +recently shown an unexpected link between linear-in-temperature resistivity and charge or- +der in YBa2Cu3O6+δ [13, 14]. Combined, these latest results suggest that fluctuations of the +charge order should somehow result in an effective isotropic scattering. Still, high-resolution +RIXS experiments have largely focused on the fluctuations along the high-symmetry crystal- +lographic directions only, leaving the full structure of electron correlations within the copper +oxide plane unknown. +Recently, in Bi2Sr2CaCu2O8+δ (Bi-2212), RIXS measurements found the existence of a +quasi-circular pattern in the qx-qy plane at finite energies and with the same wave vector +magnitude as that of the observed static charge order peak at q = [qx = ±qCO, qy = +0] and [qx = 0, qy = ±qCO] – i.e. +dynamic correlations with charge order wavelength +along all direction in the CuO2 plane [15]. Although the medium energy-resolution of those +measurements (∆E ≈ 0.8 eV) precluded a more precise determination of their energy profile, +the results suggested that these quasi-circular dynamic correlations (QCDCs) appear broad +over the mid-infrared ranges (defined approximately as 100 to 900 meV). This scattering +manifold, which may result from combined short- and long-range Coulomb interactions [15– +17], would provide a large variety of wave vectors for connecting all points of the Fermi +surface (i.e. an effective isotropic scattering). However, it is not yet experimentally known +if this manifold extends to electron scattering at lower energies, in the quasi-elastic regime. +To experimentally investigate this scenario we used high energy-resolution (≈ 37 meV) Cu-L3 +RIXS qx-qy mapping of the electronic correlations in Bi-2212. Using the softening of the bond +stretching (BS) phonon in RIXS as a marker of charge order correlations, our measurements +reveal the presence of low-energy quasi-circular dynamic electronic correlations with |q| ≈ +qCO. +RESULTS +High-resolution RIXS mapping of dynamic correlations in the qx-qy plane +We performed measurements at φ = 0◦, 25◦, 30◦, 35◦, 45◦, where φ is defined as the az- +imuthal angle from the qx axis. For each φ, we acquired RIXS spectra at different values of +in-plane momentum-transfer q = |q| by varying the incident angle on the sample. Through- +out the paper, values of q are reported in reciprocal lattice units (r.l.u.), where one r.l.u. is +defined as 2π/a and a = 3.82 ˚A (the lattice constant along φ = 0◦). In Fig. 1 (A and B), we + +4 +show representative spectra obtained at q near qCO for φ = 0◦ and 30◦, and energies below +1.1 eV. In these two cases, the minimal model that fits the data includes five contributions: +a quasi-elastic peak, a bond-stretching phonon peak at ≈ 70 meV, a peak at ≈ 135 meV +(likely from a two-phonon process), a broad paramagnon and a broad background feature +of unknown origin. A similar assessment can be made regarding all other high-resolution +spectra acquired in this work. In this type of fitting analysis, the QCDCs are not explicitly +accounted for and it is generally difficult to disentangle overlapping contributions to the +RIXS spectra using a fitting model with so many parameters, thus precluding the extraction +of the exact spectral profile of the QCDCs with any reasonable confidence. Still, we note +that this high-resolution data is consistent with the previously reported medium-resolution +data [15], which can be verified by integration of the high-resolution data (see supplementary +materials, Fig. S7). +It is likely that the spectral intensity of QCDCs in Bi-2212 is so dilute over energy +as to preclude the extraction of their spectral structure amidst stronger paramagnon and +phonon signals. Still, here we develop a different method to detect QCDCs at lower energies, +by tracking the evolution of the bond-stretching (BS) phonon over the qx-qy plane. This +method is based on the phenomenology revealed by several recent RIXS measurements of +the cuprates along qx and qy, which indicate an apparent softening of the BS phonon peak +at the momentum location of the static CO peak [18–23]. In the case of Bi-2212, it has +been proposed that the apparent softening of the BS phonon in RIXS is due to an interplay +between low-energy fluctuations of the charge order and BS phonons that results in a Fano- +like interference [18, 21, 23, 24]. Another possibility is that the apparent softening is simply +the result of the phonon peak and a low-energy charge order peak overlapping, as recently +suggested by measurements of both YBa2Cu3O6+δ and Bi-2212 [25]. In either interpretation +the location of the phonon softening can be used as a marker for low-energy charge order +correlations. +Figure 1 (C and D) shows the spectra acquired as a function of q for φ = 0◦ and 30◦, +respectively, focusing on the region of the BS phonon. +At φ = 0◦, it is clear that the +phonon peak position softens to its lowest energy value at q = qCO ≈ 0.29 r.l.u. (Fig. 1C). +Careful observation of the spectra taken along φ = 30◦ shows a similar softening effect with +the lowest phonon energy position occuring for q ≈ qCO (Fig. 1D). Figure 2A shows the +mapping of the BS phonon mode at φ = 0◦ and 30◦ obtained after subtraction of the fitted + +5 +3 +2 +1 +0 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +ϕ=0° +q=0.27 r.l.u. +1 +0 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +ϕ=30° +q=0.27 r.l.u. +IRIXS (arb. units) +IRIXS (arb. units) +Intensity (arb. units) +Energy Loss (meV) +A +B +C +Energy Loss (eV) +10 +8 +6 +4 +2 +120 +80 +40 +Energy Loss (eV) +ϕ=0° +D +ϕ=30° +120 +80 +40 +q (r.l.u.) +0.1 +0.12 +0.14 +0.16 +0.17 +0.18 +0.19 +0.2 +0.21 +0.22 +0.23 +0.24 +0.25 +0.26 +0.27 +0.28 +0.29 +0.30 +0.31 +0.32 +0.33 +0.34 +0.35 +0.36 +0.37 +0.39 +0.41 +0.43 +0.45 +0.47 +FIG. 1. RIXS spectra and fitting. (A and B) Examples of spectra at q = 0.27 r.l.u. for φ = 0◦ +and 30◦, respectively (open circles). The red lines are fits to the spectra, composed of a quasi- +elastic peak (pink), a BS phonon peak at ≈ 70 meV (blue), a peak at ≈ 135 meV (likely from +a two-phonon process) (purple), a broad paramagnon (orange) and a broad background feature +of unknown origin (brown). (C and D) RIXS measured BS phonon peak for various values of q +measured for φ = 0◦ and 30◦, respectively (black circles). The blue lines are the fits to the spectra. +The vertical orange dashed lines, indicating the lowest phonon peak position at each φ, are shown +to help the reader observe the phonon dispersions in the raw data. +elastic line, once again showing the softening of the RIXS phonon even at φ = 30◦. To +precisely determine the locations of the softening in the qx-qy plane, we fit the spectra to +extract the dispersion of the BS phonon for each φ (Fig. 2B). We observe a softening of the +RIXS measured phonon line for all φ, except for φ = 45◦. Remarkably all observed softening +occurs at a value of q ≈ qCO, precisely as expected for QCDCs at low energies. +Discriminating QCDCs from short-range directional order +Dynamic correlations emanating from short range order are bound to be broad in q. It +is therefore reasonable to ask whether the measured qx-qy profile of the BS phonon could + +6 +0.4 +0.3 +0.2 +0.1 +100 +60 +20 +100 +60 +20 +70 +60 +50 +0.4 +0.3 +0.2 +0.1 +ϕ: +q (r.l.u.) +q (r.l.u.) +Energy Loss (meV) +Phonon Energy (meV) +1 +2 +1 +2 +3 +A +B +ϕ=0° + 0° + 25° + 30° + 35° + 45° +ϕ=30° +FIG. 2. Location of low-energy dynamic correlations extracted from the phonon dis- +persion. (A) Energy-momentum structure of the excitations at φ = 0◦ and 30◦ after subtraction +of the elastic line. The image is constructed from RIXS spectra deconvoluted from the energy res- +olution. (B) Location of the phonon peak obtained by fitting the RIXS spectra deconvoluted from +energy resolution for different φ (see Materials and Methods and also Supplementary Materials, +Fig. S3). The solid lines are obtained by fitting the q-dependence of the phonon peak (circles) with +a negative Lorentzian function plus a linear background. The shaded regions around the solid lines +are generated from the 95% confidence interval obtained for the various fits to the spectra (see +Materials and Methods for details). The solid lines for φ = 0◦ and 30◦ in (B) appear as dashed +white lines in (A). +simply be the result of diffuse scattering from short-range directional order. The fundamental +difference between QCDCs and short-range directional order is that the former forms a +manifold of dynamic correlations centered at q = qCO (similar to Brazovskii-type fluctuations +[15, 26]), while the the latter results in dynamic correlations around q = [qx = ±qCO, qy = 0] +and q = [qx = 0, qy = ±qCO] (more details on M1 and M2 are provided in the Materials and +Methods section). To contrast these scenarios we consider two simple toy models. In both +cases we start with a flat |q|-independent phonon mode at 72 meV, which is a reasonable +approximation given the small dispersion of the BS phonon in the absence of charge order +[25, 27]. In the first model (M1) we construct the QCDCs scenario, where the q-cuts for + +7 +-0.5 +-0.25 +0 +0.25 +0.5 +-0.5 +-0.25 +0 +0.25 +0.5 +qx (r.l.u.) +qy (r.l.u.) +-0.5 +-0.25 +0 +0.25 +0.5 +40 +45 +50 +55 +60 +65 +70 +Energy (meV) +qx (r.l.u.) +0.1 +0.2 +0.3 +0.4 +0.5 +40 +45 +50 +55 +60 +65 +70 +75 +Energy (meV) +0.1 +0.2 +0.3 +0.4 +0.5 +0.1 +0.25 +0.4 +0° +90° +180° +270° +φ +q (r.l.u.) +A +B +C +D +E +T=Tc +TTc +FIG. S5. Comparison of models to experiments at 2-ID and I21. Polar plot contrasting +M1, M2 models (orange and green solid lines) and the experimental data (blue and red symbols). +The error bars are obtained from the fits to the phonon dispersion as described in the Materials and +Methods section. On the left side, the model was adjusted for a higher value of qCO for comparison +with the data obtained at 2-ID. The data at 2-ID is consistent with M1 and not with M2. In the +main text only the data from I21 is shown because for those experiments the sample crystal axes +could be aligned in situ from structural diffraction peaks by using the photodiode detector in that +chamber. See Materials and Methods section for details. + +18 +Medium resolution RIXS +In Fig. S6 we show the results of medium resolution RIXS done at the qRIXS endstation +at the Advanced Light Source in the Lawrence Berkeley National Laboratory. The data +were obtained by integrating the RIXS spectra over the −0.5 to 0.7 eV energy window and +normalizing them by spectra integrated over all energies, which allows a comparison between +the three different dopings. The data was also symmetrized about the high-symmetry φ = +45◦ direction. In Fig. S6(D-I) the solid lines are fits of a Gaussian function plus a linear +background to the data. The maps in Fig. S6(A-C) were generated from the fits in Fig. S6(G- +I), respectively. +The gray bars in Fig. S6(D-F) are centered at the average radii of the +correlations, obtained from averaging over φ the peak positions obtained from the fits in +Fig. S6(G-I). The widths of the grey bars in Fig. S6(D and E) are obtained from the 95% +confidence intervals obtained from the fits in Fig. S6(G and H), summing them in quadrature +and taking their square root. The same procedure underestimates the uncertainty for the +Tc = 54 K. Instead the width of the grey bar in Fig. S6(F) is calculated by taking the lowest +and largest peak positions over all φ, taking into account the 95% confidence intervals from +the fits in Fig. S6(I). The width of the grey bar in Fig. S6(F) The data used to generate +Fig. S6(C, F and I) were used in a previous publication [Boschini et al. Nat. Comm. 12, 1- +8 2021]. The new data follows the same experimental procedures as the previously published +data, so we direct the reader to [Boschini et al. Nat. Comm. 12, 1-8 2021] for further details +of the experimental procedure. + +19 +0 +0.2 0.4 +q (r.l.u.) +20 +30 +40 +50 +60 +70 +80 +90 +100 +0 +0.1 0.2 0.3 0.4 +q (r.l.u.) +18 +20 +22 +24 +26 +28 +30 +32 +34 +36 +38 +40 +Intensity (a.u.) +0 +0.2 0.4 +q (r.l.u.) +20 +30 +40 +50 +60 +70 +80 +90 +100 +0 +0.1 0.2 0.3 0.4 +q (r.l.u.) +18 +20 +22 +24 +26 +28 +30 +32 +34 +36 +38 +40 +Intensity (a.u.) +0 +0.2 0.4 +q (r.l.u.) +20 +30 +40 +50 +60 +70 +80 +90 +100 +0 +0.1 0.2 0.3 0.4 +q (r.l.u.) +18 +20 +22 +24 +26 +28 +30 +32 +34 +36 +38 +40 +Intensity (a.u.) +100° +45° +-10° +0.1 +0.2 +0.3 +0.4 +22 +23 +24 +25 +26 +27 +q (r.l.u.) +Intensity (a.u.) +100° +45° +-10° +0.1 +0.2 +0.3 +0.4 +22 +23 +24 +25 +26 +27 +q (r.l.u.) +Intensity (a.u.) +95° +45° +-10° +0.1 +0.2 +0.3 +0.4 +22 +23 +24 +25 +26 +27 +q (r.l.u.) +Intensity (a.u.) +[-10°, 5°] +[15°, 25°] +[35°, 55°] +[-10°, 10°] +[20°, 30°] +[40°, 50°] +[-10°, 10°] +[15°, 30°] +[45°] +-10° +0° +5° +15° +25° +35° +45° +55° +65° +75° +85° +90° +95° +-10° +0° +10° +20° +30° +40° +50° +60° +70° +80° +90° +100° +-10° +-5° +0° +5° +10° +15° +30° +45° +60° +75° +80° +90° +95° +85° +100° +Overdoped +Tc = 60K +Optimally doped +Tc = 91K +Underdoped +Tc = 54K +A +B +C +D +E +F +G +H +I +FIG. S6. Doping dependence from medium resolution RIXS (A-C) Normalized energy- +integrated RIXS mapping showing high energy quasi-circular electron correlations in overdoped, +optimally doped and underdoped samples, respectively. (D-F) q-cuts integrated over different φ +ranges, as specified in the legends. (G-I) The normalized energy-integrated RIXS data used to +used to generate (A and D), (B and E) and (C and F). + +20 +1.0 +0.8 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +1.0 +0.5 +0.0 +0.8 +0.4 +0.0 +q (r.l.u.) +Intensity (arb. units) +∫100 meVIRIXS / ∫IRIXS +700 meV +Intensity (arb. units) + ϕ=0° + 25° + 30° + 35° + 45° +q (r.l.u.) +ϕ = 30° +A +B +FIG. S7. Energy-integrated RIXS maps from high resolution RIXS on Bi2212 under- +doped Tc=54 K at I21. A Energy-Loss RIXS spectrum for (q,φ)=(0.28 rlu, 30o). The orange +shadow highlights the energy integration window [0.1,0.7] eV. 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URL https://www.nature.com/articles/nature07400. + diff --git a/2NFAT4oBgHgl3EQfDRyP/content/tmp_files/load_file.txt b/2NFAT4oBgHgl3EQfDRyP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c09fd060ff551c82c3a568599c5b81ce86f8855c --- /dev/null +++ b/2NFAT4oBgHgl3EQfDRyP/content/tmp_files/load_file.txt @@ -0,0 +1,1360 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf,len=1359 +page_content='Low-energy quasi-circular electron correlations with charge order wavelength in Bi2Sr2CaCu2O8+δ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' Scott,1, 2 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' Kisiel,3 T.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' United Kingdom 7National Synchrotron Light Source II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' Brookhaven National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' Upton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' NY 11973,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' USA 8Advanced Light Source,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' Lawrence Berkeley National Laboratory,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' USA 10Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' North Carolina State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' Raleigh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' NC 27695,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' 11Donostia International Physics Center, DIPC, 20018 Donostia-San Sebastian, Basque Country, Spain 12IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Spain 13Canadian Institute for Advanced Research, Toronto, ON, M5G 1M1, Canada 14Quantum Matter Institute, University of British Columbia, Vancouver, BC V6T 1Z4, Canada ∗ Corresponding Author: eduardo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='dasilvaneto@yale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='edu arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='08415v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='str-el] 20 Jan 2023 2 ABSTRACT In the study of dynamic charge order correlations in the cuprates, most high energy- resolution resonant inelastic x-ray scattering (RIXS) measurements have focused on mo- menta along the high-symmetry directions of the copper oxide plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' However, electron scattering along other in-plane directions should not be neglected as they may contain in- formation relevant, for example, to the origin of charge order correlations or to our un- derstanding of the isotropic scattering responsible for strange metal behavior in cuprates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' We report high-resolution resonant inelastic x-ray scattering (RIXS) experiments that re- veal the presence of dynamic electron correlations over the qx-qy scattering plane in under- doped Bi2Sr2CaCu2O8+δ with Tc = 54 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' We use the softening of the RIXS-measured bond stretching phonon line as a marker for the presence of charge-order-related dynamic electron correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' The experiments show that these dynamic correlations exist at energies below approximately 70 meV and are centered around a quasi-circular manifold in the qx-qy scat- tering plane with radius equal to the magnitude of the charge order wave vector, qCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' We also demonstrate how this phonon-tracking procedure provides the necessary experimental precision to rule out fluctuations of short-range directional charge order (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' centered around [qx = ±qCO, qy = 0] and [qx = 0, qy = ±qCO]) as the origin of the observed correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' INTRODUCTION Dynamic fluctuations from periodic charge order (CO) pervade the phase diagram of cuprate superconductors, perhaps even more than superconductivity itself [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' The detec- tion of these fluctuations over energy and momentum was enabled by several recent advances in the energy resolution of resonant inelastic x-ray scattering (RIXS) instruments operat- ing in the soft x-ray regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' In the case of YBa2Cu3O6+δ, Cu-L3 RIXS detects dynamic correlations at the charge order wavevector, qCO, with a characteristic energy scale of ap- proximately 20 meV [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' It has been proposed that these low-energy short-range dynamic charge order correlations are a key ingredient to the strange metal behavior [3, 4] charac- terized by linear-in-temperature resistivity [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' On one hand, this temperature behav- ior is often associated with an isotropic scattering rate that depends only on temperature in units of energy and Planck’s constant (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' ∝ kBT/ℏ, sometimes called the Planckian regime) [7–11], as supported by recent angle-dependent magnetoresistance measurements of La1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='6−xNd0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='4SrxCuO4 [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' On the other hand, combined transport and RIXS studies have 3 recently shown an unexpected link between linear-in-temperature resistivity and charge or- der in YBa2Cu3O6+δ [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' Combined, these latest results suggest that fluctuations of the charge order should somehow result in an effective isotropic scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' Still, high-resolution RIXS experiments have largely focused on the fluctuations along the high-symmetry crystal- lographic directions only, leaving the full structure of electron correlations within the copper oxide plane unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' Recently, in Bi2Sr2CaCu2O8+δ (Bi-2212), RIXS measurements found the existence of a quasi-circular pattern in the qx-qy plane at finite energies and with the same wave vector magnitude as that of the observed static charge order peak at q = [qx = ±qCO, qy = 0] and [qx = 0, qy = ±qCO] – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' dynamic correlations with charge order wavelength along all direction in the CuO2 plane [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' Although the medium energy-resolution of those measurements (∆E ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='8 eV) precluded a more precise determination of their energy profile, the results suggested that these quasi-circular dynamic correlations (QCDCs) appear broad over the mid-infrared ranges (defined approximately as 100 to 900 meV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' This scattering manifold, which may result from combined short- and long-range Coulomb interactions [15– 17], would provide a large variety of wave vectors for connecting all points of the Fermi surface (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' an effective isotropic scattering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' However, it is not yet experimentally known if this manifold extends to electron scattering at lower energies, in the quasi-elastic regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' To experimentally investigate this scenario we used high energy-resolution (≈ 37 meV) Cu-L3 RIXS qx-qy mapping of the electronic correlations in Bi-2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' Using the softening of the bond stretching (BS) phonon in RIXS as a marker of charge order correlations, our measurements reveal the presence of low-energy quasi-circular dynamic electronic correlations with |q| ≈ qCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' RESULTS High-resolution RIXS mapping of dynamic correlations in the qx-qy plane We performed measurements at φ = 0◦, 25◦, 30◦, 35◦, 45◦, where φ is defined as the az- imuthal angle from the qx axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' For each φ, we acquired RIXS spectra at different values of in-plane momentum-transfer q = |q| by varying the incident angle on the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' Through- out the paper, values of q are reported in reciprocal lattice units (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' ), where one r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' is defined as 2π/a and a = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='82 ˚A (the lattice constant along φ = 0◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' 1 (A and B), we 4 show representative spectra obtained at q near qCO for φ = 0◦ and 30◦, and energies below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='1 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' In these two cases, the minimal model that fits the data includes five contributions: a quasi-elastic peak, a bond-stretching phonon peak at ≈ 70 meV, a peak at ≈ 135 meV (likely from a two-phonon process), a broad paramagnon and a broad background feature of unknown origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' A similar assessment can be made regarding all other high-resolution spectra acquired in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' In this type of fitting analysis, the QCDCs are not explicitly accounted for and it is generally difficult to disentangle overlapping contributions to the RIXS spectra using a fitting model with so many parameters, thus precluding the extraction of the exact spectral profile of the QCDCs with any reasonable confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' Still, we note that this high-resolution data is consistent with the previously reported medium-resolution data [15], which can be verified by integration of the high-resolution data (see supplementary materials, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' It is likely that the spectral intensity of QCDCs in Bi-2212 is so dilute over energy as to preclude the extraction of their spectral structure amidst stronger paramagnon and phonon signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' Still, here we develop a different method to detect QCDCs at lower energies, by tracking the evolution of the bond-stretching (BS) phonon over the qx-qy plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' This method is based on the phenomenology revealed by several recent RIXS measurements of the cuprates along qx and qy, which indicate an apparent softening of the BS phonon peak at the momentum location of the static CO peak [18–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' In the case of Bi-2212, it has been proposed that the apparent softening of the BS phonon in RIXS is due to an interplay between low-energy fluctuations of the charge order and BS phonons that results in a Fano- like interference [18, 21, 23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' Another possibility is that the apparent softening is simply the result of the phonon peak and a low-energy charge order peak overlapping, as recently suggested by measurements of both YBa2Cu3O6+δ and Bi-2212 [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' In either interpretation the location of the phonon softening can be used as a marker for low-energy charge order correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' Figure 1 (C and D) shows the spectra acquired as a function of q for φ = 0◦ and 30◦, respectively, focusing on the region of the BS phonon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' At φ = 0◦, it is clear that the phonon peak position softens to its lowest energy value at q = qCO ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='29 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' 1C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' Careful observation of the spectra taken along φ = 30◦ shows a similar softening effect with the lowest phonon energy position occuring for q ≈ qCO (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' 1D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' Figure 2A shows the mapping of the BS phonon mode at φ = 0◦ and 30◦ obtained after subtraction of the fitted 5 3 2 1 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='0 ϕ=0° q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='27 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' 1 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='0 ϕ=30° q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='27 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' IRIXS (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' units) IRIXS (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' units) Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' units) Energy Loss (meV) A B C Energy Loss (eV) 10 8 6 4 2 120 80 40 Energy Loss (eV) ϕ=0° D ϕ=30° 120 80 40 q (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='14 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='47 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' RIXS spectra and fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' (A and B) Examples of spectra at q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='27 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' for φ = 0◦ and 30◦, respectively (open circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' The red lines are fits to the spectra, composed of a quasi- elastic peak (pink), a BS phonon peak at ≈ 70 meV (blue), a peak at ≈ 135 meV (likely from a two-phonon process) (purple), a broad paramagnon (orange) and a broad background feature of unknown origin (brown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' (C and D) RIXS measured BS phonon peak for various values of q measured for φ = 0◦ and 30◦, respectively (black circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' The blue lines are the fits to the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' The vertical orange dashed lines, indicating the lowest phonon peak position at each φ, are shown to help the reader observe the phonon dispersions in the raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' elastic line, once again showing the softening of the RIXS phonon even at φ = 30◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' To precisely determine the locations of the softening in the qx-qy plane, we fit the spectra to extract the dispersion of the BS phonon for each φ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' 2B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' We observe a softening of the RIXS measured phonon line for all φ, except for φ = 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' Remarkably all observed softening occurs at a value of q ≈ qCO, precisely as expected for QCDCs at low energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' Discriminating QCDCs from short-range directional order Dynamic correlations emanating from short range order are bound to be broad in q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' It is therefore reasonable to ask whether the measured qx-qy profile of the BS phonon could 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='1 100 60 20 100 60 20 70 60 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='1 ϕ: q (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=') q (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=') Energy Loss (meV) Phonon Energy (meV) 1 2 1 2 3 A B ϕ=0° 0° 25° 30° 35° 45° ϕ=30° FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' Location of low-energy dynamic correlations extracted from the phonon dis- persion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' (A) Energy-momentum structure of the excitations at φ = 0◦ and 30◦ after subtraction of the elastic line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' The image is constructed from RIXS spectra deconvoluted from the energy res- olution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' (B) Location of the phonon peak obtained by fitting the RIXS spectra deconvoluted from energy resolution for different φ (see Materials and Methods and also Supplementary Materials, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' The solid lines are obtained by fitting the q-dependence of the phonon peak (circles) with a negative Lorentzian function plus a linear background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' The shaded regions around the solid lines are generated from the 95% confidence interval obtained for the various fits to the spectra (see Materials and Methods for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' The solid lines for φ = 0◦ and 30◦ in (B) appear as dashed white lines in (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' simply be the result of diffuse scattering from short-range directional order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' The fundamental difference between QCDCs and short-range directional order is that the former forms a manifold of dynamic correlations centered at q = qCO (similar to Brazovskii-type fluctuations [15, 26]), while the the latter results in dynamic correlations around q = [qx = ±qCO, qy = 0] and q = [qx = 0, qy = ±qCO] (more details on M1 and M2 are provided in the Materials and Methods section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' To contrast these scenarios we consider two simple toy models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' In both cases we start with a flat |q|-independent phonon mode at 72 meV, which is a reasonable approximation given the small dispersion of the BS phonon in the absence of charge order [25, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=' In the first model (M1) we construct the QCDCs scenario, where the q-cuts for 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='25 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='25 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='5 qx (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=') qy (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='25 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='5 40 45 50 55 60 65 70 Energy (meV) qx (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='5 40 45 50 55 60 65 70 75 Energy (meV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='4 0° 90° 180° 270° φ q (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'} +page_content=') A B C D E T=Tc T 20kHz, which is non- +realistic, since spectrograms usually filter out those frequen- +cies. We consider two attacks against the Kinetics-Sounds +dataset; the first is to add a low-amplitude sine wave com- +ponent with f = 800Hz to the audio signal, and the second +is to add band-limited noise 5kHz < f < 6kHz. The spec- +trograms and the absolute difference between the attacked +spectrograms and the clean spectrogram are shown in Fig- +ure 7. Since no clear artifacts are observed in the spectro- +grams, human inspection fails to label the spectrograms as +attacked. The CDA and ASR rates of the backdoor-attacked +models for both attacks are shown in Table 6. The attacks +achieve a relatively high ASR. +4.4. Audiovisual Backdoor Attacks +Now, we combine video and audio attacks to build a +multi-modal audiovisual backdoor attack. The way we do + +BadNet +FTrojan +1500 +1500 +Poisoned +Clean +1000 +1000 +500 +500 +0 +0 +2 +4 +0 +2 +4 +Frame Lag +Motion Blur +600 +600 +400 +400 +200 +200 +0 +0 +0 +2 +4 +0 +2 +4 +EntropyBadNet +Frame Lag +100 +ASR +ASR +CDA +CDA +Accuracy(%) +75 +50 +25 +0 +0 +25 +50 +75 +100 +0 +25 +50 +75 +100 +Percentage Pruned (%Late Fusion +Early Fusion +Clean Audio +Sine Attack +High Freq. Attack +Clean Audio +Sine Attack +High Freq. Attack +Clean Video +80.25 / - +81.74 / 70.98 +80.96 / 77.91 +84.72 / - +83.48 / 92.23 +83.94 / 93.72 +BadNet +77.33 / 66.97 +78.63 / 99.74 +77.33 / 99.87 +87.50 / 99.29 +85.10 / 99.87 +85.75 / 100.00 +Blend +79.60 / 75.06 +80.76 / 99.68 +79.08 / 99.61 +86.08 / 98.19 +83.55 / 99.81 +85.43 / 99.87 +SIG +78.50 / 68.33 +80.12 / 99.87 +79.02 / 100.00 +86.92 / 99.81 +84.97 / 100.00 +85.95 / 100.00 +WaNet +77.66 / 68.39 +79.79 / 99.94 +79.02 / 99.94 +86.46 / 98.96 +84.97 / 100.00 +85.88 / 100.00 +FTrojan +79.66 / 67.16 +80.76 / 99.48 +79.99 / 99.29 +86.08 / 98.58 +84.65 / 99.94 +85.49 / 100.00 +Frame Lag +79.08 / 63.41 +80.57 / 99.74 +79.47 / 99.87 +86.08 / 98.19 +84.59 / 99.94 +84.65 / 100.00 +Video Corruption +78.11 / 64.57 +78.24 / 99.68 +77.66 / 99.94 +86.59 / 99.29 +84.59 / 100.00 +85.43 / 100.00 +Motion Blur +79.79 / 69.24 +80.70 / 99.68 +79.86 / 99.94 +86.40 / 98.58 +84.65 / 100.00 +85.62 / 100.00 +Table 7. Audiovisual Backdoor Attacks (Kinetics-Sounds). The entries in the table report the CDA(%)/ASR(%) of attacking late and +early fused audiovisual networks. When a single modality is attacked, late fusion has a low ASR compared to early fusion. When both +modalities are attacked, the ASR of both late and early fusion are high. +Figure 7. Clean and Attacked Audio Spectrograms. The uti- +lized audio backdoor attacks are not only audibly imperceptible +but also leave no perceptible artifacts in the Mel spectrogram. The +spectrogram of each attack is followed by the absolute difference +of the attacked spectrogram with the clean one. +it is by taking our attacked models from Sections 4.2 and +4.3 and applying early or late fusion. For early fusion, we +extract video and audio features using our trained audio and +video backbones, and we then train a classifier on the con- +catenation of the features. In late fusion, the video and au- +dio networks predict independently on the input, and then +the individual logits are aggregated to produce the final pre- +diction. To answer the three questions posed in Section 3.3, +we run experiments in which both modalities are attacked +and others in which only a single modality is attacked for +both early and late fusion setups (Table 7). We summarize +the results as follows. (1) Attacking two modalities con- +sistently improves ASR and even CDA in some cases. (2) +Attacking a single modality is good enough to achieve a +high ASR in the case of early fusion but not late fusion. +(3) Early fusion enables the best of both worlds for the at- +tacker, namely, a high CDA and an almost perfect ASR. On +the other hand, late fusion experiences some serious drops +in ASR in the unimodal attack setup. An interesting find- +ing in these experiments is the following: if the outsourcer +has the option to outsource the most expensive modality, +training wise, while training other modalities in-house, ap- +plying late fusion could be used as a defense mechanism, +especially in the presence of more clean modalities. +5. Conclusion +Backdoor attacks present a serious and exploitable vul- +nerability against both unimodal and multi-modal video +action recognition models. We showed how existing im- +age backdoor attacks could be extended either statically +or dynamically to develop powerful backdoor attacks that +achieve both a high clean data accuracy and a high attack +success rate. Besides existing image backdoor attacks, there +exists a set of natural video backdoor attacks, such as mo- +tion blur and frame lag, that are resilient to existing image +backdoor defenses. Given that videos are usually accom- +panied by audio, we showed two ways in which one could +attack audio classifiers in a human inaudible manner. The + +Clean Spectrogram +80 +40 +0 +Sine Attack +80 +40 +0 +Mel Frequency +80 +40 +0 +High Frequency Attack +80 +40 +0 +80 +40 +0 +0 +50 +100 +150 +200 +250 +300 +Frameattacked video and audio models are then used to train an +audiovisual action recognition model by applying both early +and late fusion. Different combinations of poisoned modal- +ities are tested, concluding that: (1) poisoning two modal- +ities could achieve extremely high attack success rates in +both late and early fusion settings, and (2) if a single modal- +ity is poisoned, unlike early fusion, late fusion could reduce +the effectiveness of the backdoor. 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In IJCAI, 2022. 2 + diff --git a/39AzT4oBgHgl3EQfD_qi/content/tmp_files/load_file.txt b/39AzT4oBgHgl3EQfD_qi/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ec9da3286f4759d61f2b8255a45aab27620d5cf0 --- /dev/null +++ b/39AzT4oBgHgl3EQfD_qi/content/tmp_files/load_file.txt @@ -0,0 +1,890 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf,len=889 +page_content='Look, Listen, and Attack: Backdoor Attacks Against Video Action Recognition Hasan Abed Al Kader Hammoud1 Shuming Liu1 Mohammad Alkhrasi2 Fahad AlBalawi2 Bernard Ghanem1 1 King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia 2 Saudi Data and Artificial Intelligence Authority (SDAIA), Riyadh, Saudi Arabia {hasanabedalkader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='hammoud,shuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='liu,bernard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='ghanem} @kaust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='sa {mkhrashi,falbalawi} @sdaia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='sa Abstract Deep neural networks (DNNs) are vulnerable to a class of attacks called “backdoor attacks”, which create an as- sociation between a backdoor trigger and a target label the attacker is interested in exploiting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' A backdoored DNN per- forms well on clean test images, yet persistently predicts an attacker-defined label for any sample in the presence of the backdoor trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Although backdoor attacks have been ex- tensively studied in the image domain, there are very few works that explore such attacks in the video domain, and they tend to conclude that image backdoor attacks are less effective in the video domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' In this work, we revisit the traditional backdoor threat model and incorporate addi- tional video-related aspects to that model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' We show that poisoned-label image backdoor attacks could be extended temporally in two ways, statically and dynamically, leading to highly effective attacks in the video domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' In addition, we explore natural video backdoors to highlight the seri- ousness of this vulnerability in the video domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' And, for the first time, we study multi-modal (audiovisual) backdoor attacks against video action recognition models, where we show that attacking a single modality is enough for achiev- ing a high attack success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Introduction A fundamental requirement for the deployment of deep neural networks (DNNs) in real-world tasks is their safety and robustness against possible vulnerabilities and security breaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' This requirement is, in essence, the motivation behind exploring adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' One particularly in- teresting adversarial attack is “backdoor attacks”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Backdoor attacks or neural trojan attacks explore the scenario in which a user with limited computational capabilities downloads pretrained DNNs from an untrusted party or outsources the training procedure to such a party that we refer to as the ad- versary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The adversary provides the user with a model that performs well on an unseen validation set, but produces a pre-defined class label in the presence of an attacker-defined trigger called the backdoor trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The association between the backdoor trigger and the attacker-specified label is cre- ated by training the DNN on poisoned training samples, which are samples polluted by the attacker’s trigger [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' In poisoned-label attacks, unlike clean-label attacks, the at- tacker also switches the label of the poisoned samples to the intended target label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Considerable attention has been paid to explore back- door attacks and defenses for 2D image classification mod- els [5,22,25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' However, little attention has been paid to ex- ploring backdoor attacks and defenses against video action recognition models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The disappointing conclusion uncov- ered by [87] regarding the limited effectiveness of image backdoor attacks on videos stunted further development of video backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Unfortunately, the attacks consid- ered in [87] were limited to only visible patch-based clean- label attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Moreover, [87] directly adopted the 2D back- door attack threat model without incorporating important video-specific considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' To this end, and as opposed to [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' we first revisit and revise the commonly adopted 2D poisoned-label backdoor threat model by incorporating additional constraints that are inherently imposed by video systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' These constraints arise due to the presence of the temporal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' We then explore two ways to extend image backdoor attacks to incorporate the temporal dimension into the attack to enable more video-specific backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' In particular, image backdoor attacks could be either extended statically by ap- plying the same attack to each frame of the video or dynam- ically by adjusting the attack parameters differently for each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Then, three novel natural video backdoor attacks are presented to highlight the seriousness of the risks associ- ated with backdoor attacks in the video domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' We then test the attacked models against three 2D backdoor defenses and discuss the reason behind the failure of those methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='00986v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='CV] 3 Jan 2023 We also study, for the first time, audiovisual backdoor at- tacks, where we ablate the importance and contribution of each modality on the performance of the attack for both late and early fusion settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' We show that attacking a single modality is enough to achieve a high attack success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Our contributions are twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' (1) We re- visit the traditional backdoor attack threat model and incor- porate video-related aspects, such as video subsampling and spatial cropping, into the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' We also extend existing image backdoor attacks to the video domain in two differ- ent ways, statically and dynamically, after which we pro- pose three novel natural video backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Through extensive experiments, we provide evidence that the previ- ous perception of image backdoor attacks in the video do- main is not necessarily true, especially in the poisoned-label attack setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' (2) To the best of our knowledge, this work is the first to investigate audiovisual backdoor attacks against video action recognition models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Related Work Backdoor Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Backdoor attacks were first introduced in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The attack, called BadNet, was based on adding a patch to the corner of a subset of training images to create a backdoor that could be triggered by the attacker at will.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Fol- lowing BadNet, [44] proposed optimizing for the values of the patch to obtain a more effective backdoor attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Shortly after the development of patch-based backdoor attacks, the community realized the importance of adding an invisibility constraint to the design of backdoor triggers to bypass any human inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Works such as [9] proposed blending the backdoor trigger with the image rather than stamping it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' [37] generated backdoor attacks using the least signifi- cant bit algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' [52] generated warping fields to warp the image content as a backdoor trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' [14] went one step fur- ther and designed learnable transformations to generate op- timal backdoor triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' After many attacks were proposed in the spatial domain [10, 37, 40, 45, 56, 57, 67, 72, 75], and others in the latent representation domain [13,53,80,88,91], [19, 25, 71, 82, 84] proposed to switch attention to the fre- quency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' [25] utilized frequency heatmaps proposed in [81] to create backdoor attacks that target the most sen- sitive frequency components of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' [19] proposed blending low frequency content from a trigger image with training images as a poisoning technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' In our work, we extend the 2D backdoor threat model to the video domain by incorporating video-related aspects into it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' We also ex- tend five image backdoor attacks into the video domain and propose three natural video backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Backdoor Defenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Backdoor attack literature was im- mediately opposed by various defenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Backdoor de- fenses are generally of five types: preprocessing-based [12,47,55], model reconstruction-based [38,42,74,83,89], trigger synthesis-based [23,24,29,43,54,58,62,68], model diagonsis-based [15, 34, 46, 76, 90], and sample-filtering based [8, 20, 26, 30, 61, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Early backdoor defenses such as [68] hypothesized that backdoor attacks create a short- cut between all samples and the poisoned class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Based on that, they solved an optimization problem to find whether a trigger of an abnormally small norm exists that would flip all samples to one label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Later, multiple improved itera- tions of this method were proposed, such as [23, 43, 83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Fine pruning [42] suggested that the backdoor is triggered by particular neurons that are dormant in the absence of the trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Therefore, the authors proposed pruning the least active neurons on clean samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' STRIP [20] showed that blending clean samples with other clean samples would yield a higher entropy compared to when clean images are blended with poisoned samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Activation clustering [8] uses KMeans to cluster the activations of an inspection, a potentially poisoned data set, into two clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' A large sil- houette distance between the two clusters would uncover the poisoned samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' In our work, we show that current image backdoor attacks have limited effectiveness in de- fending against backdoor attacks in the video domain, es- pecially against the proposed natural video attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Video Action Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Video action recognition mod- els, which only leverage the raw frames of a video, can be categorized into two categories, CNN-based networks and transformer-based networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' 2D CNN-based methods are built on top of pretrained image recognition networks with well-designed modules to capture the temporal relationship between multiple frames [41,49,69,70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Those methods are computationally efficient as they use 2D convolutional ker- nels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' To learn stronger spatial-temporal representations, 3D CNN-based methods were proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' These methods utilize 3D kernels to jointly leverage the spatio-temporal context within a video clip [17, 18, 64, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' To better initialize the network, I3D [7] inflated the weights of 2D pretrained im- age recognition models to adapt them to 3D CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Real- izing the importance of computational efficiency, S3D [78] and R(2+1)D [66] proposed to disentangle spatial and tem- poral convolutions to reduce computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Recently, transformer-based action recognition models were able to achieve better performance in large training data sets com- pared to CNN-based models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' [4,6,16,48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' In this work, we test backdoor attacks against three action recognition architectures, namely I3D, SlowFast, and TSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Audiovisual Action Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' In addition to frames, a line of action recognition models [1,27,28,51] has used the accompanying audio to better understand activities such as “playing music” or “washing dishes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' To take advantage of existing CNN and transformer-based models, the Log-Mel spectrogram was introduced to convert audio data from a non-structured signal into a 2D representation in time and frequency usable by these models [2,3,35,77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Current au- diovisual action recognition methods are divided into two Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Traditional Backdoor Attack Pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' After selecting a backdoor trigger and a target label, the attacker poisons a subset of the training data referred to as the poisoned dataset (Dp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The label of the poisoned dataset is fixed to a target poisoning label specified by the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The attacker trains jointly on clean (non-poisoned) samples (Dc) and poisoned samples leading to a backdoored model, which outputs the target label in the presence of the backdoor trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' categories based on when the audio and visual signals are merged in the recognition pipeline: early fusion and late fu- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Early fusion combines features before classification, which can better capture features [32,77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The disadvantage of early fusion is that there is a higher risk of overfitting to the training data [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Late fusion, on the other hand, treats the video and audio networks separately, and the predictions of each network are carried out independently, after which the logits are aggregated to make a final prediction [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' For the first time, we test backdoor attacks against audiovisual action recognition networks in both late and early fusion setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Video Backdoor Attacks 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The Traditional Threat Model The commonly adopted threat model for backdoor at- tacks dates back to the works that studied those attacks against 2D image classification models [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The victim outsources the training process to a trainer who is given ac- cess to both the victim’s training data and the network ar- chitecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The victim only accepts the model provided by the trainer if it performs well on the victim’s private val- idation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The attacker aims to maximize the effective- ness of the embedded backdoor attack [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' We refer to the model’s performance on the validation set as clean data accuracy (CDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The effectiveness of the backdoor attack is measured by the attack success rate (ASR), which is de- fined as the percentage of test examples not labeled as the target class that are classified as the target class when the backdoor pattern is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' To achieve this goal, the at- tacker applies a backdoor trigger to a subset of the train- ing images and then, in the poisoned-label setup, switches the labels of those images to a target class of choice before training begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' A more powerful backdoor attack is one that is visually imperceptible (usually measured in terms of ℓ2/ℓ∞-norm, PSNR, SSIM, or LPIPS) but achieves both a high CDA and a high ASR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' This is summarized in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' More formally, we denote the classifier which is param- eterized by θ as fθ : X → Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' It maps the input x ∈ X, such as images or videos, to class labels y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Let Gη : X → X indicate an attacker-specific poisoned im- age generator that is parameterized by some trigger-specific parameters η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The generator may be image-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Fi- nally, let S : Y → Y be an attacker-specified label shifting function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' In our case, we consider the scenario in which the attacker is trying to flip all the labels into one particular la- bel, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' S : Y → t, where t ∈ Y is an attacker-specified label that will be activated in the presence of the backdoor trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Let D = {(xi, yi)}N i=1 indicate the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The attacker splits D into two subsets, a clean subset Dc and a poisoned subset Dp, whose images are poisoned by Gη and labels are poisoned by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The poisoning rate is the ra- tio α = |Dp| |D| , generally a lower poisoning rate is associated with a higher clean data accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The attacker typically trains the network by minimizing the cross-entropy loss on Dc∪Dp, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' minimizes E(x,y)∼Dc∪Dp[LCE(fθ(x), y)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The attacker aims to achieve high accuracy on the user’s valida- tion set Dval while being able to trigger the poisoned-label, t, in the presence of the backdoor trigger, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' fθ(Gη(x)) = t, ∀x ∈ X (ideally).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' From Images to Videos Unlike images, videos have an additional dimension, the temporal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' This dimension introduces new rules to the game between the attacker and the victim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' More Settings Training Stage Inference Stage "Eat" "Jump\' fe fe Poisoned Network Target Label = "Eat" Backdoor Trigger Clean Samples (Dc) Poisoned Video (Gn(α)) (α) Label = "Eat" Clean Video Ground Truth LabelFigure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Static vs Dynamic Backdoor Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Static backdoor attacks apply the same trigger across all frames along the temporal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' On the other hand, dynamic attacks apply a different trigger per frame along the temporal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' precisely, the attacker now has an additional dimension to hide the backdoor trigger, leading to a higher level of im- perceptibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The backdoor attack could be applied to all the frames or a subset of the frames statically, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' the same trigger is applied to each frame, or dynamically, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' a different trigger is applied to each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' On the other hand, the testing pipeline now imposes harsher conditions against the backdoor attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Video recognition models tend to test the model on multiple sub-sampled clips with vari- ous crops [7,18,41] which might, in turn, destroy the back- door trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' For example, if the trigger is applied to a single frame, it might not be sampled, and if the trigger is applied to the corner of the image, it might be cropped out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The threat model presented in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='1 was di- rectly adopted in [87], which to the best of our knowledge, is the only previous work that considered backdoor attacks for video action recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Our work sheds light on the aforementioned video- related aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='2, we show the effect of the number of frames poisoned on CDA and ASR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' We also show how existing 2D methods could be extended both stat- ically and dynamically to suit the video domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' For exam- ple, BadNet [22] applies a fixed patch as a backdoor trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The patch could be applied statically using the same pixel values and the same position along the temporal dimension or applied dynamically by changing the position and possi- bly the pixel values of the patch for each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Figure 2 shows a BadNet attack when applied in a static and dynamic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Additionally, we show how simple yet natural video “artifacts” could be used as backdoor triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' More specif- ically, lag in a video, motion blur, and compression glitches could all be used as naturally occurring backdoor triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Audiovisual Backdoor Attacks Videos are naturally accompanied by audio signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Sim- ilarly to how the video modality could be attacked, the audio signal could also be attacked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The interesting question that arises is how backdoor attacks would perform in a multi- modal setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' In the experiments of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='4, we answer the following questions: (1) What is the effect of having two attacked modalities on CDA and ASR?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' (2) What happens if only one modality is attacked and the other is left clean?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' (3) What is the difference in performance between late and early fusion in terms of CDA and ASR?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Experimental Settings Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' We consider three standard benchmark datasets used in video action recognition: UCF-101 [60], HMDB- 51 [36], and Kinetics-Sounds [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Kinetics-Sounds is a subset of Kinetics400 that contains classes that can be clas- sified from the audio signal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' classes where audio is use- ful for action recognition [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Kinetics-Sounds is particu- larly interesting for Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='4, where we explore backdoor attacks against audio and audiovisual classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Network Architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Following common practice, for the visual modality, we use a dense sampling strategy to sub-sample 32 frames per video to fine-tune a pretrained I3D network on the target dataset [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='2, we also show results using TSM [41] and SlowFast [18] net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' All three models adopt ResNet-50 as the backbone and are pretrained on Kinetics-400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Similarly to [2], for the audio modality, a ResNet-18 is trained from scratch on Mel-Spectrograms composed of 80 Mel bands sub-sampled temporally to a fixed length of 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Attack Setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' For the video modality, we study and extend the following image-based backdoor attacks to the video domain: BadNet [22], Blend [9], SIG [5], WaNet [52], and FTrojan [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' We also explore three additional natural video backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' For the audio modality, we consider two attacks: sine attack and high-frequency noise attack, both of which we explain later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Following [22,25,52], the target class is arbitrarily set to the first class t=0 t=1 t=12 t=13 t=14 t=N-1 t=N Static DynamicFigure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Visualization of 2D Backdoor Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Image backdoor attacks mainly differ according to the backdoor trigger used to poison the training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' They could be extended either statically or dynamically based on how the attack is applied across the frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' of each data set (class 0), and the poisoning rate is set to 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Unless otherwise stated, the considered image back- door attacks poison all frames of the sampled clips during training and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Evaluation Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' As is commonly done in the back- door literature, we evaluate the performance of the model using clean data accuracy (CDA) and attack success rate (ASR) explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' CDA represents the usual validation/test accuracy on an unseen dataset hence mea- suring the generalizability of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' On the other hand, ASR measures the effectiveness of the attack when the poi- son is applied to the validation/test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' In addition, we test the attacked models against some of the early 2D back- door defenses, more precisely against activation clustering (AC) [8], STRIP [20], and pruning [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Implementation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Our method is built on MMAc- tion2 library [11], and follows their default training config- urations and testing protocols, except for the learning rate and the number of training epochs (check Supplementary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' All experiments were run using 4 NVIDIA A100 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Video Backdoor Attacks Extending Image Backdoor Attacks to the Video Do- main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' As mentioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='2, image backdoor attacks could be extended either statically by applying an attack in the same way across all frames or dynamically by adjusting the attack parameters for different frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' We consider five attacks that differ according to the applied backdoor trig- ger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' BadNet applies a patch as a trigger, Blend blends a trigger image to the original image, SIG superimposes a si- nusoidal trigger to the image, WaNet warps the content of the image, and FTrojan poisons a high- and mid- frequency component in the discrete cosine transform (DCT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Figure 3 visualizes all five attacks on the same video frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Each of the considered methods could be extended dynamically as follows: BadNet: change the patch location for each frame;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Blend: blend a uniform noise that is different per frame;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' SIG: change the frequency of the sine component superim- posed with each frame;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' WaNet: generate a different warp- ing field for each frame;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' FTrojan: select a different DCT basis to perturb at each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Note that Blend and FTro- jan are generally imperceptible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Visualizations and saliency UCF101 HMDB51 KineticsSound CDA(%) ASR(%) CDA(%) ASR(%) CDA(%) ASR(%) Baseline 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='95 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='59 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='41 BadNet 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='95 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='63 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='35 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='89 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='97 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='09 Blend 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='29 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='26 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='37 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='73 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='12 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='54 SIG 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='97 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='97 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='50 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='80 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='84 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='87 WaNet 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='05 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='84 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='95 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='61 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='38 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='09 FTrojan 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='16 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='34 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='10 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='52 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='45 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='86 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Statically Extended 2D Backdoor Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Statically extending 2D backdoor attacks to the video domain leads to high CDA and ASR across all three considered datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' UCF101 HMDB51 KineticsSound CDA(%) ASR(%) CDA(%) ASR(%) CDA(%) ASR(%) Baseline 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='95 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='59 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='41 BadNet 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='11 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='97 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='08 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='54 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='25 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='74 Blend 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='21 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='44 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='03 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='95 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='67 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='79 SIG 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='24 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='00 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='63 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='00 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='84 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='00 WaNet 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='29 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='79 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='22 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='80 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='25 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='61 FTrojan 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='16 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='34 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='19 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='69 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='25 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='27 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Dynamically Extended 2D Backdoor Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Dynam- ically extending 2D backdoor attacks to the video domain leads to high CDA and ASR across all three considered datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' maps for each attack are found in the Supplementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Tables 1 and 2 show the CDA and ASR of the I3D mod- els attacked using various backdoor attacks on UCF-101, HMDB-51, and Kinetics-Sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Contrary to the conclu- sion presented in [87], we find that backdoor attacks are actually highly effective in the video domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The CDA of the attacked models is very similar to that of the clean unattacked model (baseline), surpassing it in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Extending attacks dynamically, almost always, improves CDA and ASR compared to extending them statically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Natural Video Backdoors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' A more interesting attack is one that seems natural and could bypass human inspec- tion [50, 73, 79, 86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' There are several natural “glitches” that occur in the video domain and that one could exploit to design a natural backdoor attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' For example, videos might contain some frame lag, motion blur, video compres- sion corruptions, camera focus/defocus, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' In Table 3, we report the CDA and ASR of three natural backdoor attacks: Clean BadNet Blend SIG WaNet FTrojanUCF101 HMDB51 KineticsSound CDA(%) ASR(%) CDA(%) ASR(%) CDA(%) ASR(%) Baseline 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='95 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='59 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='41 Frame Lag 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='94 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='20 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='04 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='76 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='51 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='19 Video Corrupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='26 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='87 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='22 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='22 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='74 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='51 Motion Blur 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='97 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='92 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='17 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='52 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='19 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='22 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Natural Video Backdoor Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Natural attacks against video action recognition models could achieve high CDA and ASR while looking completely natural to human inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' SlowFast TSM CDA(%) ASR(%) CDA(%) ASR(%) Baseline 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='72 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='77 BadNet 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='64 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='47 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='69 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='78 SIG 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='70 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='97 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='77 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='47 FTrojan 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='25 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='52 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='21 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='00 Frame Lag 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='43 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='97 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='63 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='96 Video Corruption 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='54 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='76 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='08 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='97 Motion Blur 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='46 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='55 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='50 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='39 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Video Backdoor Attacks Against Different Archi- tectures (UCF-101).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' When tested against network architectures other than I3D such as TSM and SlowFast, both image and natural backdoor attacks can still achieve high CDA and high ASR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' frame lag (lagging video), video compression glitch (which we refer to as Video Corruption), and motion blur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Interest- ingly, these attacks could achieve both high clean data ac- curacy and high attack success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' It is worth noting that for frame lag, a two-frame lag is used for UCF-101 and a three-frame lag is used for HMDB-51 and Kinetics-Sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' More details are provided in the Supplementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Attacks Against Different Architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' So far, all at- tacks have been experimented with against an I3D network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' To further explore the behavior of backdoor attacks against other video recognition models, we test a subset of the con- sidered attacks against a 2D based model, TSM, and another 3D based model, SlowFast, on UCF-101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Table 4 shows that all the aforementioned backdoor attacks perform sig- nificantly well in terms of CDA and ASR against both TSM and SlowFast architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Note that even though TSM is a 2D based model, our proposed natural video backdoor at- tacks still succeed in attacking it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Recommendations for Video Backdoor Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' As men- tioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='2, the attacker must select a number of frames to poison per video, keeping in mind that the video will be sub-sampled and randomly cropped during evalua- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Since the attacker is the one who trained the network in the first place, he/she has access to the processing pipeline and could exploit this during the attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' For example, if video processing involves sub-sampling the video into clips of 32 frames and cropping the frames into 224×224 crops, the attacker could pass to the network an attacked video of a temporal length of 32 frames and a spatial size 224×224, Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Effect of the Number of Poisoned Frames (UCF-101).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Different colors refer to different number of frames poisoned dur- ing the training of the attacked model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Training the model with a single poisoned frame performs best for various choices of the number of frames poisoned during evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Frame Lag Motion Blur SIG BadNet FTrojan Elimination Rate(%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='00 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='21 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='77 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='12 Sacrifice Rate(%) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='08 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='82 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='17 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='25 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='00 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Activation Clustering Defense (UCF-101).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Whereas Activation Clustering provides partial success in defending against image backdoor attacks, it fails completely against natural attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' hence bypassing sub-sampling and cropping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' However, a system could force the user to input a video of a partic- ular length, possibly greater than the length of the sub- sampled clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' This raises an important question regarding how many frames the attacker should poison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Clearly, the smaller the number of frames the attacker poisons, the less detectable the attack is, but does the attack remain effective?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' In Figure 4, we show the attack success rate of backdoor- attacked models trained on clips of 1, 8, 16, and 32 frames, and a randomly sampled number of poisoned frames (out of 32 total frames) when evaluated on clips of 1, 8, 16, and 32 poisoned frames (out of 32 total frames).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Random refers to training on a varying number of poisoned frames per clip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Note that training the model against the worst-case scenario (single frame), which mimics the case where only one of the poisoned frames is sub-sampled, provides the best guar- antees for achieving a high attack success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Defenses Against Video Backdoor Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' We explore the effect of extending some of the existing 2D backdoor defenses against video backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Optimization- based defenses are extremely costly when extended to the video domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' For example, Neural Cleanse (NC) [68], I- BAU [83], and TABOR [23] involve a trigger reconstruc- tion phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The trigger space is now bigger in the presence of the temporal dimension, and therefore, instead of opti- mizing for a 224×224×3 trigger, the defender has to search for a 32×224×224×3 trigger (assuming 32 frame clips are used), which is both costly and hard to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The attacker has the spatial and temporal dimensions to design and em- BadNet SIG 100 75 # Poisoned Frames ASR(%) (Training) 1 50 8 16 25 32 Random 0 8 16 32 1 1 8 16 32 # Poisoned Frames (EvaluationFigure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' STRIP Defense (UCF-101).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Whereas the entropy of image backdoor attacks is very low compared to that of clean sam- ples, the proposed natural backdoor attacks have a natural distri- bution of entropies similar to that of clean samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Pruning Defense (Kinetics-Sounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Pruning is com- pletely ineffective against image backdoor attacks extended to the video domain and natural video backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Even though the clean accuracy has dropped to random, the attack success rate is maintained at very high levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' bed their attack in, and, therefore, reverse engineering the trigger is quite hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' We consider three well-known defenses that introduce no computational overhead when adopted to the video domain, namely Activation Cluster (AC) [8], STRIP [20], and prun- ing [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' AC computes the activations of a neural network on clean samples (from the test set) and an inspection set of interest which may be poisoned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' AC then applies PCA to reduce the dimension of the activations, after which the projected activations are clustered into two classes and com- pared to the activations of the clean set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' STRIP blends clean samples with the samples of a possibly poisoned inspec- tion set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The entropy of the predicted probabilities is then checked for any abnormalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Unlike clean samples, poi- soned samples tend to have a low entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Pruning suggests that the backdoor is usually embedded in particular neurons Baseline Sine Attack High Frequency Attack CDA(%) 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='21 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='21 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='61 ASR(%) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='36 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='96 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Audio Backdoor Attacks (Kinetics-Sounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Both sine attack and the high-frequency band attack perform similarly to baseline in terms of CDA while being able to achieve high ASR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' in the network that are only activated in the presence of the trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Therefore, those neurons are supposed to be dor- mant as far as the test set samples, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' clean samples, are concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' This allows us to detect and prune those dor- mant neurons to eliminate the backdoor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Table 5 shows the elimination and sacrifice rates of AC when applied against some of the considered attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The elimination rate refers to the ratio of poisoned samples correctly detected as poi- soned to the total number of poisoned samples, whereas the sacrifice rate refers to the ratio of clean samples incorrectly detected as poisoned to the total number of clean samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Whereas AC has partial success in defending against image backdoor attacks, it fails completely against the proposed natural backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Figure 5 shows that the entropy of the clean and poisoned samples of the proposed natural attacks is very similar and therefore could evade the STRIP defense, while BadNet and FTrojan are detectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Finally, Figure 6 shows that pruning the least active neurons causes a reduction in CDA without reducing ASR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' This is observed not only for the natural attacks, but also for the extended im- age backdoor attacks, hinting that image backdoor defenses are not effective in the video domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Audio Backdoor Attacks Attacks proposed against audio networks have been lim- ited to adding a low-volume one-hot-spectrum noise in the frequency domain, which leaves highly visible artifacts in the spectrogram [85] or adding a human non-audible com- ponent [33], f < 20Hz or f > 20kHz, which is non- realistic, since spectrograms usually filter out those frequen- cies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' We consider two attacks against the Kinetics-Sounds dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' the first is to add a low-amplitude sine wave com- ponent with f = 800Hz to the audio signal, and the second is to add band-limited noise 5kHz < f < 6kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The spec- trograms and the absolute difference between the attacked spectrograms and the clean spectrogram are shown in Fig- ure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Since no clear artifacts are observed in the spectro- grams, human inspection fails to label the spectrograms as attacked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The CDA and ASR rates of the backdoor-attacked models for both attacks are shown in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The attacks achieve a relatively high ASR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Audiovisual Backdoor Attacks Now, we combine video and audio attacks to build a multi-modal audiovisual backdoor attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The way we do BadNet FTrojan 1500 1500 Poisoned Clean 1000 1000 500 500 0 0 2 4 0 2 4 Frame Lag Motion Blur 600 600 400 400 200 200 0 0 0 2 4 0 2 4 EntropyBadNet Frame Lag 100 ASR ASR CDA CDA Accuracy(%) 75 50 25 0 0 25 50 75 100 0 25 50 75 100 Percentage Pruned (%Late Fusion Early Fusion Clean Audio Sine Attack High Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Attack Clean Audio Sine Attack High Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Attack Clean Video 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='25 / - 81.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='00 FTrojan 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='66 / 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='16 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='76 / 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='48 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='99 / 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='29 86.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='59 / 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='94 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='65 / 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='00 Video Corruption 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='11 / 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='57 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='24 / 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='68 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='66 / 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='94 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='59 / 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='29 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='59 / 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='00 85.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='94 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='40 / 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='58 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='65 / 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='00 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='62 / 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='00 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Audiovisual Backdoor Attacks (Kinetics-Sounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The entries in the table report the CDA(%)/ASR(%) of attacking late and early fused audiovisual networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' When a single modality is attacked, late fusion has a low ASR compared to early fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' When both modalities are attacked, the ASR of both late and early fusion are high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Clean and Attacked Audio Spectrograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The uti- lized audio backdoor attacks are not only audibly imperceptible but also leave no perceptible artifacts in the Mel spectrogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The spectrogram of each attack is followed by the absolute difference of the attacked spectrogram with the clean one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' it is by taking our attacked models from Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='3 and applying early or late fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' For early fusion, we extract video and audio features using our trained audio and video backbones, and we then train a classifier on the con- catenation of the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' In late fusion, the video and au- dio networks predict independently on the input, and then the individual logits are aggregated to produce the final pre- diction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' To answer the three questions posed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content='3, we run experiments in which both modalities are attacked and others in which only a single modality is attacked for both early and late fusion setups (Table 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' We summarize the results as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' (1) Attacking two modalities con- sistently improves ASR and even CDA in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' (2) Attacking a single modality is good enough to achieve a high ASR in the case of early fusion but not late fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' (3) Early fusion enables the best of both worlds for the at- tacker, namely, a high CDA and an almost perfect ASR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' On the other hand, late fusion experiences some serious drops in ASR in the unimodal attack setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' An interesting find- ing in these experiments is the following: if the outsourcer has the option to outsource the most expensive modality, training wise, while training other modalities in-house, ap- plying late fusion could be used as a defense mechanism, especially in the presence of more clean modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Conclusion Backdoor attacks present a serious and exploitable vul- nerability against both unimodal and multi-modal video action recognition models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' We showed how existing im- age backdoor attacks could be extended either statically or dynamically to develop powerful backdoor attacks that achieve both a high clean data accuracy and a high attack success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Besides existing image backdoor attacks, there exists a set of natural video backdoor attacks, such as mo- tion blur and frame lag, that are resilient to existing image backdoor defenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Given that videos are usually accom- panied by audio, we showed two ways in which one could attack audio classifiers in a human inaudible manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' The Clean Spectrogram 80 40 0 Sine Attack 80 40 0 Mel Frequency 80 40 0 High Frequency Attack 80 40 0 80 40 0 0 50 100 150 200 250 300 Frameattacked video and audio models are then used to train an audiovisual action recognition model by applying both early and late fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Different combinations of poisoned modal- ities are tested, concluding that: (1) poisoning two modal- ities could achieve extremely high attack success rates in both late and early fusion settings, and (2) if a single modal- ity is poisoned, unlike early fusion, late fusion could reduce the effectiveness of the backdoor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' We hope that our work reignites the attention of the community towards exploring backdoor attacks and defenses in the video domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' References [1] Humam Alwassel, Dhruv Kumar Mahajan, Lorenzo Tor- resani, Bernard Ghanem, and Du Tran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Self-supervised learning by cross-modal audio-video clustering.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' Impercep- tible backdoor attack: From input space to feature represen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' In IJCAI, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} +page_content=' 2' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'} diff --git a/3NFQT4oBgHgl3EQfGjVY/content/tmp_files/2301.13245v1.pdf.txt b/3NFQT4oBgHgl3EQfGjVY/content/tmp_files/2301.13245v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..25a01f70feb45da009140f73393c1058d46793d2 --- /dev/null +++ b/3NFQT4oBgHgl3EQfGjVY/content/tmp_files/2301.13245v1.pdf.txt @@ -0,0 +1,1622 @@ +A Safety Framework for Flow Decomposition Problems via +Integer Linear Programming +Fernando H. C. Dias1,⋆[0000−0002−6398−919X], Manuel C´aceres1,⋆[0000−0003−0235−6951], Lucia +Williams2,⋆[0000−0003−3785−0247], Brendan Mumey2,⋆⋆[0000−0001−7151−2124], and +Alexandru I. Tomescu1,⋆⋆[0000−0002−5747−8350] +1 Department of Computer Science, University of Helsinki, Finland +{fernando.cunhadias,manuel.caceres,alexandru.tomescu}@helsinki.fi +2 School of Computing, Montana State University, Bozeman, MT, USA +{lucia.williams,brendan.mumey}@montana.edu +Abstract. Many important problems in Bioinformatics (e.g., assembly or multi-assembly) admit mul- +tiple solutions, while the final objective is to report only one. A common approach to deal with this +uncertainty is finding safe partial solutions (e.g., contigs) which are common to all solutions. Previous +research on safety has focused on polynomially-time solvable problems, whereas many successful and +natural models are NP-hard to solve, leaving a lack of “safety tools” for such problems. We propose +the first method for computing all safe solutions for an NP-hard problem, minimum flow decomposi- +tion. We obtain our results by developing a “safety test” for paths based on a general Integer Linear +Programming (ILP) formulation. Moreover, we provide implementations with practical optimizations +aimed to reduce the total ILP time, the most efficient of these being based on a recursive group-testing +procedure. +Results: Experimental results on the transcriptome datasets of Shao and Kingsford (TCBB, 2017) +show that all safe paths for minimum flow decompositions correctly recover up to 90% of the full RNA +transcripts, which is at least 25% more than previously known safe paths, such as (C´aceres et al. TCBB, +2021), (Zheng et al., RECOMB 2021), (Khan et al., RECOMB 2022, ESA 2022). Moreover, despite the +NP-hardness of the problem, we can report all safe paths for 99.8% of the over 27,000 non-trivial graphs +of this dataset in only 1.5 hours. Our results suggest that, on perfect data, there is less ambiguity than +thought in the notoriously hard RNA assembly problem. +Availability: https://github.com/algbio/mfd-safety +Contact: alexandru.tomescu@helsinki.fi +Keywords: RNA assembly · Network flow · Flow decomposition · Integer linear programming · Safety +⋆ Shared first-author contribution +⋆⋆ Shared last-author contribution +arXiv:2301.13245v1 [cs.DS] 30 Jan 2023 + +1 +Introduction +In real-world scenarios where an unknown object needs to be discovered from the input data, we would like +to formulate a computational problem loosely enough so that the unknown object is indeed a solution to +the problem, but also tightly enough so that the problem does not admit many other solutions. However, +this goal is difficult in practice, and indeed, various commonly used problem formulations in Bioinformatics +still admit many solutions. While a naive approach is to just exhaustively enumerate all these solutions, a +more practical approach is to report only those sub-solutions (or partial solutions) that are common to all +solutions to the problem. +In the graph theory community such sub-solutions have been called persistent [14,21], and in the Bioin- +formatics community reliable [54], or more recently, safe [51]. The study of safe sub-solutions started in +Bioinformatics in the 1990’s [54,11,37] with those amino-acid pairs that are common to all optimal and +suboptimal alignments of two protein sequences. +In the genome assembly community, the notion of contig, namely a string that is guaranteed to appear in +any possible assembly of the reads, is at the core of most genome assemblers. This approach originated in 1995 +with the notion of unitigs [25] (non-branching paths in an assembly graph), which were progressively [42,6] +generalized to paths made up of a prefix of nodes with in-degree one followed by nodes with out-degree +one [35,24,29] (also called extended unitigs, or Y-to-V contigs). +Later, [51] formalized all such types of contigs as those safe strings that appear in all solutions to a +genome assembly problem formulation, expressed as a certain type of walk in a graph. [10,9] proposed more +efficient and unifying safety algorithms for several types of graph walks. [45] recently studied the safety of +contigs produced by state-of-the-art genome assemblers on real data. +Analogous studies were recently made also for multi-assembly problems, where several related genomic +sequences need to be assembled from a sample of mixed reads. [8] studied safe paths that appear in all +constrained path covers of a directed acyclic graph (DAG). Zheng, Ma and Kingsford studied the more +practical setting of a network flow in a DAG by finding those paths that appear in any flow decomposition +of the given network flow, under a probabilistic framework [34], or a combinatorial framework [58].3 [27] +presented a simple characterization of safe paths appearing in any flow decomposition of a given acyclic +network flow, leading to a more efficient algorithm than the one of [58], and further optimized by [28]. +Motivation. Despite the significant progress in obtaining safe algorithms for a range of different appli- +cations, current safe algorithms are limited to problems where computing a solution itself is achievable in +polynomial time. However, many natural problems are NP-hard, and safe algorithms for such problems are +fully missing. Apart from the theoretical interest, usually such NP-hard problems correspond to restrictions +of easier (polynomially-computable) problems, and thus by definition, also have longer safe sub-solutions. +As such, current safety algorithms miss data that could be reported as correct, just because they do not +constrain the solution space enough. A major reason for this lack of progress is that if a problem is NP-hard, +then its safety version is likely to be hard too. This phenomenon can be found both in classically studied NP- +hard problems — for example, computing nodes present in all maximum independent sets of an undirected +graph is NP-hard [21] — as well as in NP-hard problems studied for their application to Bioinformatics, as +we discuss further in the appendix. +We introduce our results by focusing on the flow decomposition problem. This is a classical model at the +core of multi-assembly software for RNA transcripts [33,31,5,50] and viral quasi-species genomes [3,2,44,12], +and also a standard problem with applications in other fields, such as networking [36,22,13,23] or transporta- +tion [39,38]. In its most basic optimization form, minimum flow decomposition (MFD), we are given a flow +in a graph, and we need to decompose it into a minimum number of paths with associated weights, such +that the superposition of these weighted paths gives the original flow. This is an NP-hard problem, even +when restricted to DAGs [53,22]. Various approaches have been proposed to tackle the problem, including +fixed-parameter tractable algorithms [30], approximation algorithms [36,7] and Integer Linear Programming +formulations [15,46]. +3 The problem AND-Quant from [58] actually handles a more general version of this problem. +1 + +In Bioinformatics applications, reads or contigs originating from a mixed sample of genomic sequences +with different abundances are aligned to a reference. A graph model, such as a splice graph or a variation +graph, is built from these alignments. Read abundances assigned to the nodes and edges of this graph +then correspond to a flow in case of perfect data. If this is not the case, the abundance values can either +be minimally corrected to become a flow, or one can consider variations of the problem where e.g., the +superposition of the weighted paths is closest (or within a certain range) to the edge abundances [50,5]. +Current safety algorithms for flow decompositions such as [58,27,26,28] compute paths appearing in all +possible flow decompositions (of any size), even though decompositions of minimum size are assumed to +better model the RNA assembly problem [30,48,55]. Even dropping the minimality constraint, but adding +other simple constraints easily renders the problem NP-hard (see e.g., [56]), motivating further study of +practical safe algorithms for NP-hard problems. +Contributions. Integer Linear Programming (ILP) is a general and flexible method that has been suc- +cessfully applied to solve NP-hard problems, including in Bioinformatics. In this paper, we consider graph +problems whose solution consists of a set of paths (i.e., not repeating nodes) that can be formulated in +ILP. We introduce a technique that, given an ILP formulation of such a graph problem, can enhance it +with additional variables and constraints in order to test the safety of a given set of paths. An obvious first +application of this safety test is to use it with a single path in a straightforward avoid-and-test approach, +using a standard two-pointer technique that has been used previously to find safe paths for flow decomposi- +tion. However, we find that a top-down recursive approach that uses the group-testing capability halves the +number of computationally-intensive ILP calls, resulting in a 3x speedup over the straightforward approach. +Additionally, we prove that computing all the safe paths for MFDs is an intractable problem, confirming +the above intuitive claim that if a problem is hard, then also its safety version is hard. We give this proof +in the appendix by showing that the NP-hardness reduction for MFD by [22] can be modified into a Turing +reduction from the UNIQUE 3SAT problem. +On the dataset [47] containing splice graphs from human, zebrafish and mouse transcriptomes, safe +paths for MFDs (SafeMFD) correctly recover up to 90% of the full RNA transcripts while maintaining a +99% precision, outperforming, by a wide margin (25% increase), state-of-the-art safety approaches, such as +extended unitigs [35,24,29], safe paths for constrained path covers of the edges [8], and safe paths for all +flow decompositions [28,27,26,58]. On the harder dataset by [26], SafeMFD also dominates in a significant +proportion of splice graphs (built from t ≤ 15 RNA transcripts), recovering more than 95% of the full +transcripts while maintaining a 98% precision. For larger t, precision drastically drops (91% precision in the +entire dataset), suggesting that in more complex splice graphs smaller solutions are introduced as an artifact +of the combinatorial nature of the splice graph, and the minimality condition [30,48,55] is thus incorrect in +this domain. +2 +Methods +2.1 +Preliminaries +ILP models. In this paper we use ILP models as blackboxes, with as few assumptions as possible to further +underline the generality of our approach. Let M(V, C) be an ILP model consisting of a set V of variables +and a set C of constraints on these variables, built from an input graph G = (V, E). We make only two +assumptions on M. First, that a solution to this model consists of a given number k ≥ 1 of paths P1, . . . , Pk +in G (in this paper, paths do not repeat vertices). Second, we assume that the k paths are modeled via +binary edge variables xuvi, for all (u, v) ∈ E and for all i ∈ {1, . . . , k}. More specifically, for all i ∈ {1, . . . , k}, +we require that the edges (u, v) ∈ E for which the corresponding variable xuvi equals 1 induce a path in G. +For example, if G is a DAG, it is a standard fact (see e.g., [49]) that a path from a given s ∈ V to a given +2 + +t ∈ V (an s-t path) can be expressed with the following constraints: +� +(u,v)∈E +xuvi − +� +(v,u)∈E +xvui = +� +� +� +� +� +0, +if v ∈ V \ {s, t}, +1, +if v = t, +−1, +if v = s. +(1) +If G is not a DAG, there are other types of constraints that can be added to the xuvi variables to ensure +that they induce a path; see, for example, the many formulations in [49]. We will assume that such constraints +are part of the set C of constraints of M(V, C), but their exact formulation is immaterial for our approach. In +fact, one could even add additional constraints to C to further restrict the solution space. For example, some +ILP models from [15,46] handle the case when the input also contains a set of paths (subpath constraints) +that must appear in at least one of the k solution paths. +Flow decomposition. In the flow decomposition problem we are given a flow network (V, E, f), where +G = (V, E) is a (directed) graph with unique source s ∈ V and unique sink t ∈ V , and f assigns a positive +integer flow value fuv to every edge (u, v) ∈ E. Flow conservation must hold for every node different from s +and t, namely, the sum of the flow values entering the node must equal the sum of the flow values exiting the +node. See Figure 1(a) for an example. We say that k s-t paths P1, . . . , Pk, with associated positive integer +weights w1, . . . , wk, are a flow decomposition (FD) if their superposition equals the flow f. Formally, for +every (u, v) ∈ E it must hold that +� +i∈{1,...,k} s.t. +(u,v)∈Pi +wi = fuv. +(2) +See Figures 1(b) and 1(c) for two examples. The number k of paths is also called the size of the flow +decomposition. In the minimum flow decomposition (MFD) problem, we need to find a flow decomposition +of minimum size.4 On DAGs, a flow decomposition into paths always exists [1], but in general graphs, cycles +may be necessary to decompose the flow (see e.g. [16] for different possible formulations of the problem). +For concreteness, we now describe the ILP models from [15] for finding a flow decomposition into k +weighted paths in a DAG. They consist of (i) modeling the k paths via the xuvi variables (with constraints +(1)), (ii) adding path-weight variables w1, . . . , wk, and (iii) requiring that these weighted paths form a flow +decomposition, via the following (non-linear) constraint: +� +i∈{1,...,k} +xuviwi = fuv, +∀(u, v) ∈ E. +(3) +This constraint can then be easily linearized by introducing additional variables and constraints; see e.g. [15] +for these technical details. However, as mentioned above, the precise formulation of the ILP model M for a +problem is immaterial for our method. Only the two assumptions on M made above matter for obtaining +our results. +Safety. Given a problem on a graph G whose solutions consist of k paths in G, we say that a path P is safe +if for any solution P1, . . . , Pk to the problem, there exists some i ∈ {1, . . . , k} such that P is a subpath of Pi. +If the problem is given as an ILP model M, we also say that P is safe for M. We say that P is a maximal +safe path, if P is a safe path and there is no larger safe path containing P as subpath. [27] characterized safe +paths for all FDs (not necessarily of minimum size) using the excess flow fP of a path P, defined as the flow +on the first edge of P minus the flow on the edges out-going from the internal nodes of P, and different from +the edges of P (see Figure 1(d) for an example). It holds that P is safe for all FDs if and only if fP > 0 [27]. +4 In this paper we work only with integer flow values and weights for simplicity and since this is the most studied +version of the problem, see e.g., [30]. However, the problem can also be defined with fractional weights [41], and +in this case the two problems can have different minima on the same input [53]. This fractional case can also be +modeled by ILP [15], and all the results from our paper also immediately carry over to this variant. +3 + + +8 +9 +9 +3 +5 +7 +s +t +b +a +d +e +7 + c +10 +10 +f + +s +t +b +a +d +e + c +f + +s +t +b +a +d +e + c +f +5 +3 +2 +7 +3 +4 +1 +9 +3 +3 +Figure 1 +Figure 2 +(a) A flow network with source s and sink t. + +8 +9 +9 +3 +5 +7 +s +t +b +a +d +e +7 + c +10 +10 +f + +s +t +b +a +d +e + c +f + +s +t +b +a +d +e + c +f +5 +3 +2 +7 +3 +4 +1 +9 +3 +3 +Figure 1 +Figure 2 + +8 +9 +9 +3 +5 +7 +s +t +b +a +d +e +7 + c +10 +10 +f +3 +3 + +s +t +b +a +d +e + c +f +xsai = 1 +xabi = 1 +xbci = 1 +xcdi = 1 +xdfi = 1 +Figure 3 +xadi = 0 +xeti = 0 +xdei = 0 +xfti = 1 +xsbi = 0 +xbdi = 0 +(b) An MFD into 4 paths of weights 5,3,7,2, respec- +tively. The green dashed path is a subpath of the +orange path. + +8 +9 +9 +3 +5 +7 +s +t +b +a +d +e +7 + c +10 +10 +f + +s +t +b +a +d +e + c +f + +s +t +b +a +d +e + c +f +5 +3 +2 +7 +3 +4 +1 +9 +3 +3 +Figure 1 +Figure 2 + +8 +9 +9 +3 +5 +7 +s +t +b +a +d +e +7 + c +10 +10 +f +3 +3 + +s +t +b +a +d +e + c +f +xsai = 1 +xabi = 1 +xbci = 1 +xcdi = 1 +xdfi = 1 +Figure 3 +xadi = 0 +xeti = 0 +xdei = 0 +xfti = 1 +xsbi = 0 +xbdi = 0 +(c) An MFD into 4 paths of weights 3,4,1,9, respec- +tively. The green dashed path is a subpath of the pink +path. + +8 +9 +9 +3 +5 +7 +s +t +b +a +d +e +7 + c +10 +10 +f + +s +t +b +a +d +e + c +f + +s +t +b +a +d +e + c +f +5 +3 +2 +7 +3 +4 +1 +9 +3 +3 +Figure 1 +Figure 2 + +8 +9 +9 +3 +5 +7 +s +t +b +a +d +e +7 + c +10 +10 +f +3 +3 + +s +t +b +a +d +e +7 + c +10 +f +3 +xsai = 1 +xabi = 1 +xbci = 1 +xcdi = 1 +xdfi = 1 +(d) The two subpaths (red and blue) of the green +dashed path that are maximal safe paths for all FDs. +Fig. 1: Flow decompositions and safe paths. The flow network in (a) admits different MFDs, in (b) and in (c). +The path (s, a, b, c, d) (dashed green) is a maximal safe path for MFDs, i.e., it is a subpath of some path +of all MFDs and it cannot be extended without losing this property. However, the path (s, a, b, c, d) is not +safe for all FDs. Indeed, its two subpaths (s, a, b) (dashed red in (d)) and (b, c, d) (dashed blue in (d)) are +maximal safe paths for all FDs. To see this, note that the excess flow of (s, a, b) is 3, while the excess flow of +(s, a, b, c) (and of (s, a, b, c, d)) is −6. +4 + +The excess flow can be computed in time linear in the length of P (assuming we have pre-computed the flow +outgoing from every node), giving thus a linear-time verification of whether P is safe. +A basic property of safe solutions is that any sub-solution of them is also safe. Computing safe paths +for MFDs can thus potentially lead to joining several safe paths for FDs, obtaining longer paths from the +unknown sequences we are trying to assemble. See Figure 1 for an example of a maximal safe path for MFDs +and two maximal subpaths of it that are safe for FDs. +2.2 +Finding Maximal Safe Paths for MFD via ILP +We now present a method for finding all maximal safe paths for MFD via ILP. The basic idea is to define an +inner “safety test” which can be repeatedly called as part of an outer algorithm over the entire instance to +find all maximal safe paths. Because calls to the ILP solver are expensive, the guiding choice for our overall +approach is to minimize the number of ILP calls. This inspires us to test the safety of a group of paths as the +inner safety test, which we achieve by augmenting our ILP model so that it can give us information about +the safety of the paths in the set. We use this to define a recursive algorithm to fully determine the safety +status of each path in a group of paths. We can then structure the safety test in either a top-down manner +(starting with long unsafe paths and shrinking them until they are safe) or a bottom-up manner (starting +with short safe paths and lengthening them until they become unsafe). +Safety test (inner algorithm) Let M(V, C) be an ILP model as discussed in Section 2.1; namely, its k +solution paths are modeled by binary variables xuvi for each (u, v) ∈ E and each i ∈ {1, . . . , k}. We assume +that M(V, C) is feasible (i.e., the problem admits at least one solution). We first show how to modify the +ILP model so that, for a given set of paths, it can tell us one of the following: (1) a set of paths that are not +safe (the remaining being of unknown status), or (2) that all paths are safe. The idea is to maximize the +number of paths that can be simultaneously avoided from the given set of paths. +Let P be a set of paths. For each path P ∈ P, we create an auxiliary binary variable γP that indicates: +γP ≡ +� +1 +if P was avoided in the solution, +0 +otherwise. +(4) +Since the model solutions are paths (i.e., not repeating nodes), we can encode whether P appears in the +solution by whether all of the ℓ − 1 edges of P appear simultaneously. Using this fact, we add a new set of +constraints R(P) that include the γP indicator variables for each path P ∈ P: +R(P) := {xv1v2i + xv2v3i + · · · + xvℓ−1vℓi +≤ ℓ − 1 − γP : ∀i ∈ {1, . . . , k}, ∀P ∈ P}. +(5) +Next, as the objective function of the ILP model, we require that it should maximize the number of +avoided paths from P, i,e., the sum of the γP variables: +max +� +P ∈P +γP . +(6) +All paths P such that γP = 1 are unsafe, since they were avoided in some minimum flow decomposition. +Conversely, if the objective value of Eq. (6) was 0, then γP = 0 for all paths in P, and it must be that all +paths in P are safe (if not, at least one path could be avoided and increase the objective). We encapsulate +this group testing ILP in a function GroupTest(M, P) that returns a set N ⊆ P with the properties that: +(1) if N = ∅, then all paths in P are safe, and (2) if N ̸= ∅, then all paths in N are unsafe (and |N| is +maximized). +We employ GroupTest(M, P) to construct a recursive procedure GetSafe(M, P) that determines all safe +paths in P, as shown in Algorithm 1. +5 + + +8 +9 +9 +3 +5 +7 +s +t +b +a +d +e +7 + c +10 +10 +f + +s +t +b +a +d +e + c +f + +s +t +b +a +d +e + c +f +5 +3 +2 +7 +3 +4 +1 +9 +3 +3 +Figure 1 +Figure 2 + +8 +9 +9 +3 +5 +7 +s +t +b +a +d +e +7 + c +10 +10 +f +3 +3 + +s +t +b +a +d +e + c +f +xsai = 1 +xabi = 1 +xbci = 1 +xcdi = 1 +xdfi = 0 +Figure 3 +xadi = 0 +xeti = 1 +xdei = 1 +xfti = 0 +xsbi = 0 +xbdi = 0 +Fig. 2: Illustration of modeling a solution path and a tested path via binary edge variables and safety +verification constraints. The ith solution path Pi is shown in orange, and a tested path P is shown in dashed +green. Constraint (5) includes xsai +xabi +xbci +xcdi +xdei ≤ 5−γP . This simplifies to γP ≤ 0, thus forcing +γP = 0, which indicates P was not avoided in the solution. +Algorithm 1: Testing a set of paths P for safety. +Input: A feasible ILP model M(V, C), and a set of paths P +Output: Those paths P ∈ P that are safe for M(V, C) +1 Procedure GetSafe(M, P) +2 +N = GroupTest(M, P) if N = ∅ then +3 +return P +4 +else +5 +return GetSafe(M, P \ N) +We note that in the special case that |P| = 1, GetSafe(M, P) makes only a single call to the ILP via +GroupTest(M, P) to determine whether not the given path is safe. With this safety test for a single path, we +can easily adapt a standard two-pointer approach as the outer algorithm to find all maximal safe paths for +MFD by starting with some MFD solution P1, . . . , Pk of M(V, C). This same procedure was used in [26] to +find all maximal safe paths for FD, using an excess flow check as the inner safety algorithm. +Find all maximal safe paths (outer algorithm) We give two algorithms for finding all maximal safe +paths. Both algorithms use a similar approach, however the first uses a top-down approach starting from the +original full solution paths and reports all safe paths (these again must be maximal safe), and then trims all +the unsafe paths to find new maximal safe paths. The second is bottom-up in that it tries to extend known +safe subpaths until they cannot be further extended (and at this point must be maximal safe). We present +the first algorithm in detail and defer discussion of the second to the appendix. +We say a set of subpaths T = {Pi[li, ri]} is a trimming core provided that for any unreported maximal +safe path P = Pi[l, r], there is a Pi[li, ri] ∈ T , where li ≤ l ≤ r ≤ ri. +We will use the original k solution paths {Pi} as our initial trimming core; the complete algorithm is +given in Algorithm 2. See Fig. 3 in the appendix for an illustration of the algorithm’s initial steps. The +algorithm first checks if any of the paths in T are safe; if so, these are reported as maximal safe. For those +paths that were unsafe, it then considers trimming one vertex from the left and one vertex from the right +to create new subpaths. Of these subpaths, some may be contained in a safe path in T ; these subpaths +can be ignored as they are not maximal safe. The algorithm recurses on those subpaths whose safety status +cannot be determined (lines 6–10). In this way, the algorithm maintains the invariant that no paths in T are +properly contained in a safe path; thus paths reported in line 4 must be maximal safe. +6 + +Algorithm 2: An algorithm to compute all maximal safe paths that can be trimmed from a +trimming core set T . +Input: An ILP model M and a trimming core set T +Output: All maximal safe paths for M that are trimmed subpaths of T +1 Procedure AllMaxSafe-TopDown(M, T ) +2 +S = GetSafe(M, T ) for Pi[li, ri] ∈ S do +3 +output Pi[li, ri] +4 +U = T \ S L = {Pi[li + 1, ri] : Pi[li, ri] ∈ U, (ri = |Pi| or Pi[li + 1, ri + 1] ∈ U)} +R = {Pi[li, ri − 1] : Pi[li, ri] ∈ U, (li = 1 or Pi[li − 1, ri − 1] ∈ U)} P = L ∪ R if P ̸= ∅ then +5 +AllMaxSafe-TopDown(M, P) +3 +Experiments +To test the performance of our methods, we computed safe paths using different safety approaches and re- +ported the quality and running time performances as described below. Additional details on the experimental +setup are given in the appendix. +Implementation details – SafeMFD. We implemented the previously described algorithms to compute +all maximal safe paths for minimum flow decompositions in Python. The implementation, SafeMFD, uses +the package NetworkX [20] for graph processing and the package gurobipy [19] to model and solve the ILPs +and it is openly available5. Our fastest variant (see Table 2 in the appendix for a comparison of running +times) implements Algorithm 2 using the group testing in Algorithm 1. We used this variant to compare +against other safety approaches. All tested variants of SafeMFD implement the following two optimizations: +1. +Before processing an input flow graph we contract it using Y-to-V contraction [51], which is known [30] +to maintain (M)FD solution paths. Moreover, since edges in the contracted graph correspond to extended +unitigs [35,24,29], source-to-sink edges are further removed from the contracted graph and reported as +safe. As such, our algorithms compute all maximal safe paths for funnels [17,26] without using the ILP. +2. +Before testing the safety of a path we check if its excess-flow [26] is positive. If this is the case, the +path is removed from the corresponding test. Having positive excess flow implies safety for all flow +decomposition and thus also safety for minimum flow decompositions. +Safety approaches tested. We compare the following state-of-the-art safety approaches: +EUnitigs: +Maximal paths made up of a prefix of nodes with in-degree one followed by nodes with out- +degree one; also called extended unitigs [51,35,24,29]. We use the C++ implementation provided by Khan +et al. [26] (which computes only the extended unitigs contained in FD paths). +SafeFlow: +Maximal safe paths for all flow decompositions [26]. We use the C++ implementation provided +by Khan et al. [26]. +SafeMFD: +Maximal safe paths for all minimum flow decompositions, as proposed in this work. Every +flow graph processed is given a time budget of 2 minutes. If a flow graph consumes its time budget, the +solution of SafeFlow is output instead. +SafeEPC: +Maximal safe paths for all constrained path covers of edges. Previous authors [8,26] have con- +sidered safe path covers of the nodes, but for a more fair comparison, we instead use path covers of edges. +To this end, we transform the input graphs by splitting every edge by adding a node in the middle and +run the C++ implementation provided by the authors of [8]. Since flow decompositions are path covers +of edges, safe paths for all edge path covers are subpaths of safe paths for MFD. However, we restrict +the path covers to those of minimum size and minimum size plus one, as recommended by the authors +of [8] to obtain good coverage results while maintaining high precision. +5 https://github.com/algbio/mfd-safety +7 + +All safety approaches require a post processing step for removing duplicates, prefixes and suffixes. We +use the C++ implementation provided by [26] for this purpose. +Datasets. We use two datasets of flow graphs inspired by RNA transcript assembly. The datasets were +created by simulating abundances on a set of transcripts and then perfectly superposing them into a splice +graphs that are guaranteed to respect flow conservation. As such, the ground truth corresponds to a flow +decomposition (not necessarily minimum). To avoid a skewed picture of our results we filtered out trivial +instances with a unique flow decomposition (or funnels, see [17,26]) from the two datasets.6 +Catfish: +Created by [48], it includes 100 simulated human, mouse and zebrafish transcriptomes using Flux- +Simulator [18] as well as 1,000 experiments from the Sequence Read Archive simulating abundances using +Salmon [40]. We took one experiment per dataset, which corresponds to 27,696 non-trivial flow graphs. +RefSim: +Created by [8] from the Ensembl [57] annotated transcripts of GRCh.104 homo sapiens reference +genome, and later augmented by Khan et al. [26] with simulated abundances using the RNASeqRead- +Simulator [32]. This dataset has 10,323 non-trivial graphs. +Quality metrics. We use the same quality metrics employed by previous multi-assembly safety approaches [8,26]. +We provide a high-level description of them for completeness. +Weighted precision of reported paths: +As opposed to normal precision, the weighted version considers +the length of the reported subpaths. It is computed as the total length of the correctly reported subpaths +divided by the total length of all reported subpaths. A reported subpath is considered correct if and only +if it is a subpath of some path in the ground truth (exact alignment of exons/nodes). +Maximum coverage of a ground truth path P: +The longest segment of P covered by some reported +subpath (exact alignment of exons/nodes), divided by |P|. +We compute the weighted precision of a graph as the average weighted precision over all reported +paths in the graph, and the maximum coverage of a graph as the average maximum coverage over all +ground truth paths in the graph. +F-Score of a graph: +Harmonic mean between weighted precision and maximum coverage of a graph, +which assigns a global score to the corresponding approach on the graph. +These metrics are computed per flow graph and reported as an average. In the case of the Catfish dataset +the metrics are computed in terms of exons (nodes), since genomic coordinates of exons are missing, whereas +in the case of the RefSim dataset the metrics are computed in terms of genomic positions, as this information +is present in the input. +4 +Results and Discussion +In the Catfish dataset, EUnitigs and SafeFlow ran in less than a second, while SafeEPC took approximately +30 seconds to compute. On the other hand, solving a harder problem, SafeMFD took approximately 1.5 +hours to compute in the rest of the dataset, timing out in only 54 graphs (we use a cutoff of 2 minutes), i.e., +only 0.2% of the entire dataset. This equates to only 0.2 seconds on average per solved graph, underlying +the scalability of our approach. +Table 1 shows that SafeMFD, on average, covers close to 90% of the ground truth paths, while maintaining +a high precision (99%). This corresponds to an increase of approximately 25% in coverage against its closest +competitor SafeFlow. SafeMFD also dominates in the combined metric of F-Score, being the only safe +approach with F-Score over 90%. Figure 4 in the appendix shows the metrics on graphs grouped by number +t of ground truth paths, indicating the dominance in coverage and F-Score of SafeMFD across all values of +t, and indicating that the decrease in precision appears for large values of t (t ≥ 12). +6 The exact datasets used in our experiments can be found at https://zenodo.org/record/7182096. +8 + +Table 1: Summary of quality metrics for both datasets. For Catfish, the metrics are computed in terms of +nodes/exons and for RefSim in terms of genomic positions; t is the number of ground truth paths. +Dataset +Graphs +Algorithm Max. Coverage Wt. Precision F-Score +Catfish +All +(100%) +EUnitigs +0.60 +1.00 +0.74 +SafeEPC +0.60 +0.99 +0.74 +SafeFlow +0.71 +1.00 +0.82 +SafeMFD +0.88 +0.99 +0.93 +RefSim +t ≤ 10 +(68%) +EUnitigs +0.72 +1.00 +0.83 +SafeEPC +0.73 +1.00 +0.84 +SafeFlow +0.84 +1.00 +0.91 +SafeMFD +0.97 +0.99 +0.98 +t ≤ 15 +(84%) +EUnitigs +0.70 +1.00 +0.82 +SafeEPC +0.71 +1.00 +0.83 +SafeFlow +0.83 +1.00 +0.90 +SafeMFD +0.96 +0.98 +0.97 +All +(100%) +EUnitigs +0.68 +1.00 +0.80 +SafeEPC +0.69 +0.99 +0.81 +SafeFlow +0.81 +1.00 +0.89 +SafeMFD +0.93 +0.91 +0.90 +In the harder RefSim dataset, EUnitigs and SafeFlow also ran in less than a second, while SafeEPC +took approximately 2 minutes. In this case, SafeMFD ran out of time in 1,562 graphs (15% of the entire +dataset); however, recall that in these experiments we allow a time budget of only 2 minutes. In the rest of +the dataset, it took approximately 7.5 hours in total, corresponding to only 3 seconds on average per graph, +again underlying that our method, even though it solves many NP-hard problems for each input graph, +overall scales sufficiently well. +Table 1 shows that again SafeMFD dominates in coverage, being the only approach obtaining coverage +over 90%, with is a 15% improvement over SafeFlow. This time its precision drops to close to 90%, and +obtaining an F-Score of 90%, very similar to its closest competitor, SafeFlow. However, recall that coverage +is computed only from correctly aligned paths, thus the drop in precision comes only from safe paths not +counting in the coverage metric. If we restrict the metrics to graphs with at most 15 ground truth paths, +which is still a significant proportion (84%) of the entire dataset, then SafeMFD has a very high precision +(98%) while improving coverage by 15% with respect to SafeFlow. Thus, the drop in precision occurs in +graphs with a large number of ground truth paths, which can also be corroborated by Figure 5 in the +appendix. +These drops in precision (both in RefSim and Catfish) for large t can be explained by the fact that a +larger number of ground truth paths produces more complex splice graphs and introduces more artificial +solutions of potentially smaller size. As such, the larger t, the less likely that the ground truth is a minimum +flow decomposition of the graph, and thus the more likely that SafeMFD reports incorrect solutions. This +motivates future work on safety not only on minimum flow decompositions but also in flow decompositions +of at most a certain size, analogously to how it is done for SafeEPC. This is still easily achievable with +our framework by just changing the ILP blackbox, and keeping everything else unchanged (e.g., the inner +and outer algorithms). Namely, instead of formulating the ILP model M(V, C) to admit solutions of exactly +optimal k paths, it can be changed to allow solutions of at most some k′ paths, with k′ greater than the +optimal k. If k′ is also greater than the number of ground truth paths in these complex graphs, then safe +paths are fully correct, meaning that we overall increase precision. +9 + +5 +Conclusion +RNA assembly is a difficult problem in practice, with even the top tools reporting low precision values. While +there are still many issues that can introduce uncertainty in practice, we can now provide a major source +of additional information during the process: which RNA fragments must be included in any parsimonious +explanation of the data? Though others have considered RNA assembly in the safety framework [58,26], we +are the first to show that safety can be practically used even when we look for optimal (i.e., minimum) size +solutions. Our experimental results show that safe paths for MFD clearly outperform other safe approaches +for the Catfish dataset, commonly used in this field. On a significant proportion of the second dataset, safe +paths for MFD still significantly outperforms other safe methods. +More generally, this is the first work to show that the safety framework can be practically applied to +NP-hard problems, where the inner algorithm is an efficient test of safety of a group of paths, and the outer +algorithm guides the applications of this test. Because our method was very successful on our test data set, +there is strong motivation to try the approach to on other NP-hard graph problems whose solutions are +sets of paths. For example, we could study other variations on MFD, such as finding flow decompositions +minimizing the longest path (NP-hard when flow values are integer [4,43]). 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Presented at RECOMB 2021. +12 + +A +Additional Figures +P1 +P2 +P3 +P4 +P1 +P2 +P3 +P4 +a) Group Testing - Initial Iteration +c) Group Testing - Second Iteration +b) Identifying Safe Paths +P2 +P2 +P4 +P4 +(a) First group test +P1 +P2 +P3 +P4 +P1 +P2 +P3 +P4 +a) Group Testing - Initial Iteration +c) Group Testing - Second Iteration +b) Identifying Safe Paths +P2 +P2 +P4 +P4 +(b) Result: {P1, P3} are safe, {P2, P4} +are unsafe +P1 +P2 +P3 +P4 +P1 +P2 +P3 +P4 +a) Group Testing - Initial Iteration +c) Group Testing - Second Iteration +b) Identifying Safe Paths +P2 +P2 +P4 +P4 +(c) Second group test +Fig. 3: Illustration of the initial group tests performed by Algorithm 2. Fig. 3(a) shows the first group test +(using Algorithm 1) on MFD solution paths {P1, P2, P3, P4}; suppose {P1, P3} were safe (Fig. 3(b)); these +are then reported as maximal safe. In this case we trim {P2, P4} on both the left and right and make the +next group test shown in Fig. 3(c). +13 + +(a) Weighted Precision +(b) Maximum Coverage +(c) F-Score +Fig. 4: Quality metrics on graphs distributed by number of paths in the ground truth for the Catfish dataset. +The metrics are computed in terms of exons/nodes. +(a) Weighted Precision +(b) Maximum Coverage +(c) F-Score +Fig. 5: Quality metrics on graphs distributed by number of paths in the ground truth for the RefSim dataset. +The metrics are computed in terms of genomic positions. +B +Additional Algorithms and Experimental Results +B.1 +The bottom-up algorithm +Algorithm 3, detailed below, uses a bottom-up group-testing strategy to find all maximal safe paths. +Definition 1. We say a set of subpaths E = {Pi[li, ri]} is an extending core provided all paths in E are safe +and for any unreported maximal safe path P = Pi[l, r], there is a Pi[li, ri] ∈ E, where l ≤ li ≤ ri ≤ r. +Note that maximal FD-safe subpaths provide an extending core (as well just the set of all single-edge +subpaths in each path). Algorithm 3 provides an algorithm to find all maximal safe paths based on group +testing, starting from an extending core. The idea is to try both left-extending (by one) and right-extending +(by one) each subpath in the core; if neither of these extensions are safe, then we know that that core subpath +must be maximal safe. Testing all extensions can done quickly using Algorithm 1. We then recurse on a new +core set consisting of those extensions that were found to be safe. +14 + +1.0 +0.8 +0.6 +0.4 +0.2 +EUnitigs +SafeEPC +SafeFlow +SafeMFD +0.0 +3 +4 +5 +6 +7 +8 +9 +10 11 12 +13 +14 +15 +# Ground truth paths1.0 +0.8 +0.6 +0.4 +0.2 +EUnitigs +SafeEPC +SafeFlow +0.0 +SafeMFD +3 +4 +5 +6 +7 +8 +9 +1011.12 +13 +14 +15 +# Ground truth paths1.0 +0.8 +0.6 +0.4 +0.2 +EUnitigs +SafeEPC +SafeFlow +0.0 +SafeMFD +3 +4 +5 +6 +7 +8 +9 +1011.12 +13 +14 +15 +# Ground truth paths1.0 +0.8 +0.6 +0.4 +0.2 +EUnitigs +SafeEPC +SafeFlow +SafeMFD +0.0 +3 +4 +5 +6 +7 +8 +9 +10 11 12 +13 +14 +15 +# Ground truth paths1.0 +0.8 +0.6 +0.4 +0.2 +EUnitigs +SafeEPC +SafeFlow +0.0 +SafeMFD +3 +4 +5 +6 +7 +8 +9 +10.11.12 +13 +14 +15 +# Ground truth paths1.0 +0.8 +0.6 +0.4 +0.2 +EUnitigs +SafeEPC +SafeFlow +0.0 +SafeMFD +3 +4 +5 +6 +7 +8 +9 +1011.12 +13 +14 +15 +# Ground truth pathsAlgorithm 3: An algorithm to output all maximal safe subpaths that can be extended from an +extending core set E. +Input: An ILP model M and an extending core set E +Output: All maximal safe paths for M that extend some path from E +1 Procedure AllMaxSafe-BottomUp(M, E) +2 +L = {Pi[li − 1, ri] : Pi[li, ri] ∈ E, li > 1} R = {Pi[li, ri + 1] : Pi[li, ri] ∈ E, ri < |Pi|} P = L ∪ R S += GetSafe(M, P) for Pi[li, ri] ∈ E do +3 +if Pi[li − 1, ri] /∈ S and Pi[li, ri + 1] /∈ S then +4 +output Pi[li, ri] +5 +if S ̸= ∅ then +6 +AllMaxSafe-BottomUp(M, S) +B.2 +The two-pointer algorithm +As we observed in Section 2.2, we can test whether a single path P is safe using one ILP call. We will +assume that this test is encapsulated as a procedure IsSafe(M, P). Once we can test whether a single path +is safe for M(V, C), we can adopt a standard approach to compute all maximal safe paths. Namely, we start +by computing one solution of M(V, C), P1, . . . , Pk and then compute maximal safe paths by a two-pointer +technique that for each path Pi, finds all maximal safe paths by just a linear number of calls to the procedure +IsSafe [26]. +This works as follows. We use two pointers, a left pointer L, and a right pointer R. Initially, L points to +the first node of path Pi and R to the second node. As long as the subpath of Pi between L and R is safe, +we move the right pointer to the next node on Pi. When this subpath is not safe, we output the subpath +between L and the previous location of R as a maximal safe path, and we start moving the left pointer to +the next node on Pi, until the subpath between L and R is safe. We stop the procedure once we reach the +end of Pi. We summarize this procedure as Algorithm 4; see also Figure 6 for an example. + +1 +8 +3 +2 +5 +6 + 4 +7 + +s +t +b +a +d +e + c +f + +s +t +b +a +d +e + c +f + +s +t +b +a +d +e + c +f +xsai + xabi + xbci + xcdi + xdfi ≤ 4 +xsai + xabi + xbci + xcdi ≤ 3 +xabi + xbci + xcdi + xdfi ≤ 3 +R +L +R +∀i ∈ {1,…, k} +∀i ∈ {1,…, k} +∀i ∈ {1,…, k} +L +L +R +(a) Current iteration + +1 +8 +3 +2 +5 +6 + 4 +7 + +s +t +b +a +d +e + c +f + +s +t +b +a +d +e + c +f + +s +t +b +a +d +e + c +f +xsai + xabi + xbci + xcdi + xdfi ≤ 4 +xsai + xabi + xbci + xcdi ≤ 3 +xabi + xbci + xcdi + xdfi ≤ 3 +R +L +R +∀i ∈ {1,…, k} +∀i ∈ {1,…, k} +∀i ∈ {1,…, k} +L +L +R +(b) Right pointer movement + +1 +8 +3 +2 +5 +6 + 4 +7 + +s +t +b +a +d +e + c +f + +s +t +b +a +d +e + c +f + +s +t +b +a +d +e + c +f +xsai + xabi + xbci + xcdi + xdfi ≤ 4 +xsai + xabi + xbci + xcdi ≤ 3 +xabi + xbci + xcdi + xdfi ≤ 3 +R +L +R +∀i ∈ {1,…, k} +∀i ∈ {1,…, k} +∀i ∈ {1,…, k} +L +L +R +(c) Left pointer movement +Fig. 6: Illustration of the two-pointer algorithm applied on a flow decomposition path Pi (in orange). In each +sub-figure, the subpath P (dashed green) between the nodes pointed by the left pointer L and the right +pointer R is tested for safety, by adding constraints S(P). In (a), IsSafe(M, P) returns True, and the right +pointer advances on Pi. In (b), IsSafe(M, P) returns False, and the previous subpath from (a) is output as +a maximal safe path. In (c), the left pointer has advanced, and the new path P is tested for safety. +15 + +Algorithm 4: The two-pointer algorithm applied to compute all maximal subpaths of a given +solution path Pi +Input: An ILP model M and one of its k solution paths, Pi = (v1, . . . , vt), t ≥ 2 +Output: All maximal safe subpaths of Pi for M +1 Procedure AllMaxSafe-TwoPointer(M, Pi) +2 +L ← 1, R ← 2 while True do +3 +while IsSafe(M, Pi[L, R]) and R ≤ t do +4 +R ← R + 1 +5 +output Pi[L, R − 1] if R > t then return; +6 +while not IsSafe(M, Pi[L, R]) do +7 +L ← L + 1 +Dataset (# Graphs) +Variant +Time (hh:mm:ss) # ILP calls +Catfish +(27,613) +TopDown +01:13:27 +124,676 +BottomUp +03:22:13 +212,774 +TwoPointer +04:21:44 +226,365 +TwoPointerBin +03:31:57 +216,540 +RefSim +(5,808) +TopDown +04:38:41 +55,450 +BottomUp +11:55:20 +76,837 +TwoPointer +13:48:00 +127,352 +TwoPointerBin +11:34:02 +119,218 +Table 2: Running times and number of ILP calls in four different variants of SafeMFD. +B.3 +Running time experiments among different variants proposed +We conducted the experiments on an isolated Linux server with AMD Ryzen Threadripper PRO 3975WX +CPU with 32 cores (64 virtual) and 504GB of RAM. Time and peak memory usage of each program were +measured with the GNU time command. SafeMFD was allowed to run Gurobi with 12 threads. All C++ +implementations were compiled with optimization level 3 (-O3 flag). Running time and peak memory is +computed and reported per dataset. +SafeMFD includes the following four variants computing maximal safe paths: +TopDown : Implements Algorithm 2 using the group testing in Algorithm 1. +BottomUp : Implements Algorithm 3 (Appendix B.1) using the group testing in Algorithm 1. +TwoPointer : Implements Algorithm 4 (Appendix B.2), the traditional two-pointer algorithm [26]. +TwoPointerBin : Same as previous variant, but it additionally replaces the linear scan employed to extend +and reduce the currently processed safe path by a binary search7. +To compare between our four different variants we first run them all on every dataset, and then filter +out those graphs that ran out of time in some variant. This way we ensure that no variant consumes its +time budget and thus our running time measurements are not skewed by the unsuccessful inputs’ timeouts. +Applying this filter we removed 83 graphs from the Catfish dataset (0.3%) and 4,515 graphs from the RefSim +dataset (43.74%). +Table 2 shows the running times and number of ILP calls of the different variants on both datasets. +TopDown clearly outperforms the rest, being at least twice as fast, and performing (roughly) half many ILP +calls. While BottomUp is analogous to TopDown, the superiority of the latter can be explained by the length +maximal safe paths. Since maximal safe paths are long it is faster to obtain them by reducing unsafe paths +7 The binary search is only applied if the search space is larger than a constant threshold set experimentally. +16 + +(TopDown) than by extending safe paths (BottomUp and both TwoPointer variants). On the other hand, +TwoPointer is the slowest variant and BottomUp and TwoPointerBin obtain similar improvements (over +TwoPointer) by following different strategies. While BottomUp reduces the number of ILP calls more than +TwoPointerBin (better appreciated in the RefSim dataset), the ILP calls of BottomUp take longer (since +BottomUp tests several paths at the same time and TwoPointerBin only one), and thus the total running +times of both is similar. This motivates future work on combining both approaches, while processing the +paths starting from unsafe (as in TopDown) for better performance. +C +Hardness of Testing MFD Safety +In this section we give a Turing-reduction from the UNIQUE 3SAT problem (U3SAT) to the problem of +determining if a given path P in a flow network G is safe for minimum flow decomposition (call this problem +MFD-SAFETY ). A 3SAT instance belongs to U3SAT if and only if it has exactly one satisfying assignment. +U3SAT has been shown to be NP-hard under randomized reductions [52], but it is open as to whether it is +NP-hard in general. +The reduction leverages the construction in [22] that reduces 3SAT to minimum flow decomposition. We +first briefly review this construction: A variable gadget (see Fig. 4 in [22]) is created for each 3SAT variable x +and a clause gadget (see Fig. 5 in [22]) is created for each 3SAT clause. Positive literals in each clause receive +flow from the left side of the corresponding variable gadget, whereas negative literals receive flow from the +right side. Theorem VI.1 in [22] establishes that a 3SAT instance is satisfiable if and only if the constructed +flow network has a minimum flow decomposition of a certain size. Any flow decomposition achieving this +size must have a specific structure; in particular, there must be a flow path of weight 4 that either travels +up the left side of the gadget (setting x to TRUE), or the right side (setting x to FALSE). +⋯ +4 +4 +4 +4 +4 +4 +⋯ +4 +4 +4 +4 +t(x) +s(x) +Fig. 7: The variable gadget from [22], showing only the weight 4 edges (other edges have weights from +{1, 2}). A key property established in [22] is that if the 3SAT instance is satisfiable then in a minimum flow +decomposition, a weight 4 flow path must travel up either the left side of the gadget (as shown), or the +right side. A left flow path indicates the variable should be set to TRUE, while right indicates FALSE. We +leverage this construction to reduce U3SAT to MFD-SAFETY. +Theorem 1. There is a polynomial time Turing-reduction from U3SAT to MFD-SAFETY. +17 + +Proof. To obtain the desired Turing-reduction algorithm, instead of checking the size of the MFD, we will +instead sequentially check the MFD-SAFETY of the aforementioned side paths traveling up the left and +right sides of each variable gadget. Provided each variable gadget has exactly one safe side path we can then +check the corresponding truth assignment to see if each clause is satisfied. If yes, we accept the instance as +belonging to U3SAT, otherwise we reject. +Suppose the instance does belong to U3SAT. In this case there is a satisfying assignment so the MFD +must have the structure as described above. Furthermore, since there is exactly one satisfying assignment, +exactly one side path of each variable gadget must be safe and so our algorithm finds it and then verifies that +the truth assignment satisfies each clause, thus accepting the instance. On the other hand, if the instance does +not belong to U3SAT it could either be unsatisfiable or have multiple satisfying assignments. If unsatisfiable, +no matter whether the safety checks pass, the corresponding assignment will not satisfy all clauses, so the +instance will be rejected. If there are multiple solutions, then any variable that can be both TRUE and +FALSE will not have a safe side path in the MFD. This means the safety check will fail and the instance +will again be rejected. +⊓⊔ +18 + diff --git a/3NFQT4oBgHgl3EQfGjVY/content/tmp_files/load_file.txt b/3NFQT4oBgHgl3EQfGjVY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a77af63b5d85c4edfef0851cf7ed0072e94e788e --- /dev/null +++ b/3NFQT4oBgHgl3EQfGjVY/content/tmp_files/load_file.txt @@ -0,0 +1,886 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf,len=885 +page_content='A Safety Framework for Flow Decomposition Problems via Integer Linear Programming Fernando H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Dias1,⋆[0000−0002−6398−919X], Manuel C´aceres1,⋆[0000−0003−0235−6951], Lucia Williams2,⋆[0000−0003−3785−0247], Brendan Mumey2,⋆⋆[0000−0001−7151−2124], and Alexandru I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Tomescu1,⋆⋆[0000−0002−5747−8350] 1 Department of Computer Science, University of Helsinki, Finland {fernando.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='cunhadias,manuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='caceres,alexandru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='tomescu}@helsinki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='fi 2 School of Computing, Montana State University, Bozeman, MT, USA {lucia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='williams,brendan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='mumey}@montana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='edu Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Many important problems in Bioinformatics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=', assembly or multi-assembly) admit mul- tiple solutions, while the final objective is to report only one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' A common approach to deal with this uncertainty is finding safe partial solutions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=', contigs) which are common to all solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Previous research on safety has focused on polynomially-time solvable problems, whereas many successful and natural models are NP-hard to solve, leaving a lack of “safety tools” for such problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We propose the first method for computing all safe solutions for an NP-hard problem, minimum flow decomposi- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We obtain our results by developing a “safety test” for paths based on a general Integer Linear Programming (ILP) formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Moreover, we provide implementations with practical optimizations aimed to reduce the total ILP time, the most efficient of these being based on a recursive group-testing procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Results: Experimental results on the transcriptome datasets of Shao and Kingsford (TCBB, 2017) show that all safe paths for minimum flow decompositions correctly recover up to 90% of the full RNA transcripts, which is at least 25% more than previously known safe paths, such as (C´aceres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' TCBB, 2021), (Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=', RECOMB 2021), (Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=', RECOMB 2022, ESA 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Moreover, despite the NP-hardness of the problem, we can report all safe paths for 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='8% of the over 27,000 non-trivial graphs of this dataset in only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='5 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Our results suggest that, on perfect data, there is less ambiguity than thought in the notoriously hard RNA assembly problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Availability: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='com/algbio/mfd-safety Contact: alexandru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='tomescu@helsinki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='fi Keywords: RNA assembly · Network flow · Flow decomposition · Integer linear programming · Safety ⋆ Shared first-author contribution ⋆⋆ Shared last-author contribution arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='13245v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='DS] 30 Jan 2023 1 Introduction In real-world scenarios where an unknown object needs to be discovered from the input data, we would like to formulate a computational problem loosely enough so that the unknown object is indeed a solution to the problem, but also tightly enough so that the problem does not admit many other solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' However, this goal is difficult in practice, and indeed, various commonly used problem formulations in Bioinformatics still admit many solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' While a naive approach is to just exhaustively enumerate all these solutions, a more practical approach is to report only those sub-solutions (or partial solutions) that are common to all solutions to the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' In the graph theory community such sub-solutions have been called persistent [14,21], and in the Bioin- formatics community reliable [54], or more recently, safe [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' The study of safe sub-solutions started in Bioinformatics in the 1990’s [54,11,37] with those amino-acid pairs that are common to all optimal and suboptimal alignments of two protein sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' In the genome assembly community, the notion of contig, namely a string that is guaranteed to appear in any possible assembly of the reads, is at the core of most genome assemblers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' This approach originated in 1995 with the notion of unitigs [25] (non-branching paths in an assembly graph), which were progressively [42,6] generalized to paths made up of a prefix of nodes with in-degree one followed by nodes with out-degree one [35,24,29] (also called extended unitigs, or Y-to-V contigs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Later, [51] formalized all such types of contigs as those safe strings that appear in all solutions to a genome assembly problem formulation, expressed as a certain type of walk in a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' [10,9] proposed more efficient and unifying safety algorithms for several types of graph walks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' [45] recently studied the safety of contigs produced by state-of-the-art genome assemblers on real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Analogous studies were recently made also for multi-assembly problems, where several related genomic sequences need to be assembled from a sample of mixed reads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' [8] studied safe paths that appear in all constrained path covers of a directed acyclic graph (DAG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Zheng, Ma and Kingsford studied the more practical setting of a network flow in a DAG by finding those paths that appear in any flow decomposition of the given network flow, under a probabilistic framework [34], or a combinatorial framework [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='3 [27] presented a simple characterization of safe paths appearing in any flow decomposition of a given acyclic network flow, leading to a more efficient algorithm than the one of [58], and further optimized by [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Despite the significant progress in obtaining safe algorithms for a range of different appli- cations, current safe algorithms are limited to problems where computing a solution itself is achievable in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' However, many natural problems are NP-hard, and safe algorithms for such problems are fully missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Apart from the theoretical interest, usually such NP-hard problems correspond to restrictions of easier (polynomially-computable) problems, and thus by definition, also have longer safe sub-solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' As such, current safety algorithms miss data that could be reported as correct, just because they do not constrain the solution space enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' A major reason for this lack of progress is that if a problem is NP-hard, then its safety version is likely to be hard too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' This phenomenon can be found both in classically studied NP- hard problems — for example, computing nodes present in all maximum independent sets of an undirected graph is NP-hard [21] — as well as in NP-hard problems studied for their application to Bioinformatics, as we discuss further in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We introduce our results by focusing on the flow decomposition problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' This is a classical model at the core of multi-assembly software for RNA transcripts [33,31,5,50] and viral quasi-species genomes [3,2,44,12], and also a standard problem with applications in other fields, such as networking [36,22,13,23] or transporta- tion [39,38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' In its most basic optimization form, minimum flow decomposition (MFD), we are given a flow in a graph, and we need to decompose it into a minimum number of paths with associated weights, such that the superposition of these weighted paths gives the original flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' This is an NP-hard problem, even when restricted to DAGs [53,22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Various approaches have been proposed to tackle the problem, including fixed-parameter tractable algorithms [30], approximation algorithms [36,7] and Integer Linear Programming formulations [15,46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 3 The problem AND-Quant from [58] actually handles a more general version of this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 1 In Bioinformatics applications, reads or contigs originating from a mixed sample of genomic sequences with different abundances are aligned to a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' A graph model, such as a splice graph or a variation graph, is built from these alignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Read abundances assigned to the nodes and edges of this graph then correspond to a flow in case of perfect data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' If this is not the case, the abundance values can either be minimally corrected to become a flow, or one can consider variations of the problem where e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=', the superposition of the weighted paths is closest (or within a certain range) to the edge abundances [50,5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Current safety algorithms for flow decompositions such as [58,27,26,28] compute paths appearing in all possible flow decompositions (of any size), even though decompositions of minimum size are assumed to better model the RNA assembly problem [30,48,55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Even dropping the minimality constraint, but adding other simple constraints easily renders the problem NP-hard (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=', [56]), motivating further study of practical safe algorithms for NP-hard problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Integer Linear Programming (ILP) is a general and flexible method that has been suc- cessfully applied to solve NP-hard problems, including in Bioinformatics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' In this paper, we consider graph problems whose solution consists of a set of paths (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=', not repeating nodes) that can be formulated in ILP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We introduce a technique that, given an ILP formulation of such a graph problem, can enhance it with additional variables and constraints in order to test the safety of a given set of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' An obvious first application of this safety test is to use it with a single path in a straightforward avoid-and-test approach, using a standard two-pointer technique that has been used previously to find safe paths for flow decomposi- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' However, we find that a top-down recursive approach that uses the group-testing capability halves the number of computationally-intensive ILP calls, resulting in a 3x speedup over the straightforward approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Additionally, we prove that computing all the safe paths for MFDs is an intractable problem, confirming the above intuitive claim that if a problem is hard, then also its safety version is hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We give this proof in the appendix by showing that the NP-hardness reduction for MFD by [22] can be modified into a Turing reduction from the UNIQUE 3SAT problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' On the dataset [47] containing splice graphs from human, zebrafish and mouse transcriptomes, safe paths for MFDs (SafeMFD) correctly recover up to 90% of the full RNA transcripts while maintaining a 99% precision, outperforming, by a wide margin (25% increase), state-of-the-art safety approaches, such as extended unitigs [35,24,29], safe paths for constrained path covers of the edges [8], and safe paths for all flow decompositions [28,27,26,58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' On the harder dataset by [26], SafeMFD also dominates in a significant proportion of splice graphs (built from t ≤ 15 RNA transcripts), recovering more than 95% of the full transcripts while maintaining a 98% precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' For larger t, precision drastically drops (91% precision in the entire dataset), suggesting that in more complex splice graphs smaller solutions are introduced as an artifact of the combinatorial nature of the splice graph, and the minimality condition [30,48,55] is thus incorrect in this domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 2 Methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='1 Preliminaries ILP models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' In this paper we use ILP models as blackboxes, with as few assumptions as possible to further underline the generality of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Let M(V, C) be an ILP model consisting of a set V of variables and a set C of constraints on these variables, built from an input graph G = (V, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We make only two assumptions on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' First, that a solution to this model consists of a given number k ≥ 1 of paths P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' , Pk in G (in this paper, paths do not repeat vertices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Second, we assume that the k paths are modeled via binary edge variables xuvi, for all (u, v) ∈ E and for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' More specifically, for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' , k}, we require that the edges (u, v) ∈ E for which the corresponding variable xuvi equals 1 induce a path in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' For example, if G is a DAG, it is a standard fact (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=', [49]) that a path from a given s ∈ V to a given 2 t ∈ V (an s-t path) can be expressed with the following constraints: � (u,v)∈E xuvi − � (v,u)∈E xvui = � � � � � 0, if v ∈ V \\ {s, t}, 1, if v = t, −1, if v = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' (1) If G is not a DAG, there are other types of constraints that can be added to the xuvi variables to ensure that they induce a path;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' see, for example, the many formulations in [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We will assume that such constraints are part of the set C of constraints of M(V, C), but their exact formulation is immaterial for our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' In fact, one could even add additional constraints to C to further restrict the solution space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' For example, some ILP models from [15,46] handle the case when the input also contains a set of paths (subpath constraints) that must appear in at least one of the k solution paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Flow decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' In the flow decomposition problem we are given a flow network (V, E, f), where G = (V, E) is a (directed) graph with unique source s ∈ V and unique sink t ∈ V , and f assigns a positive integer flow value fuv to every edge (u, v) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Flow conservation must hold for every node different from s and t, namely, the sum of the flow values entering the node must equal the sum of the flow values exiting the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' See Figure 1(a) for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We say that k s-t paths P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' , Pk, with associated positive integer weights w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' , wk, are a flow decomposition (FD) if their superposition equals the flow f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Formally, for every (u, v) ∈ E it must hold that � i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=',k} s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' (u,v)∈Pi wi = fuv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' (2) See Figures 1(b) and 1(c) for two examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' The number k of paths is also called the size of the flow decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' In the minimum flow decomposition (MFD) problem, we need to find a flow decomposition of minimum size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='4 On DAGs, a flow decomposition into paths always exists [1], but in general graphs, cycles may be necessary to decompose the flow (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' [16] for different possible formulations of the problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' For concreteness, we now describe the ILP models from [15] for finding a flow decomposition into k weighted paths in a DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' They consist of (i) modeling the k paths via the xuvi variables (with constraints (1)), (ii) adding path-weight variables w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' , wk, and (iii) requiring that these weighted paths form a flow decomposition, via the following (non-linear) constraint: � i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=',k} xuviwi = fuv, ∀(u, v) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' (3) This constraint can then be easily linearized by introducing additional variables and constraints;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' [15] for these technical details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' However, as mentioned above, the precise formulation of the ILP model M for a problem is immaterial for our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Only the two assumptions on M made above matter for obtaining our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Given a problem on a graph G whose solutions consist of k paths in G, we say that a path P is safe if for any solution P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' , Pk to the problem, there exists some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' , k} such that P is a subpath of Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' If the problem is given as an ILP model M, we also say that P is safe for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We say that P is a maximal safe path, if P is a safe path and there is no larger safe path containing P as subpath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' [27] characterized safe paths for all FDs (not necessarily of minimum size) using the excess flow fP of a path P, defined as the flow on the first edge of P minus the flow on the edges out-going from the internal nodes of P, and different from the edges of P (see Figure 1(d) for an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' It holds that P is safe for all FDs if and only if fP > 0 [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 4 In this paper we work only with integer flow values and weights for simplicity and since this is the most studied version of the problem, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=', [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' However, the problem can also be defined with fractional weights [41], and in this case the two problems can have different minima on the same input [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' This fractional case can also be modeled by ILP [15], and all the results from our paper also immediately carry over to this variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 3 8 9 9 3 5 7 s t b a d e 7 c 10 10 f s t b a d e c f s t b a d e c f 5 3 2 7 3 4 1 9 3 3 Figure 1 Figure 2 (a) A flow network with source s and sink t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 8 9 9 3 5 7 s t b a d e 7 c 10 10 f s t b a d e c f s t b a d e c f 5 3 2 7 3 4 1 9 3 3 Figure 1 Figure 2 8 9 9 3 5 7 s t b a d e 7 c 10 10 f 3 3 s t b a d e c f xsai = 1 xabi = 1 xbci = 1 xcdi = 1 xdfi = 1 Figure 3 xadi = 0 xeti = 0 xdei = 0 xfti = 1 xsbi = 0 xbdi = 0 (b) An MFD into 4 paths of weights 5,3,7,2, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' The green dashed path is a subpath of the orange path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 8 9 9 3 5 7 s t b a d e 7 c 10 10 f s t b a d e c f s t b a d e c f 5 3 2 7 3 4 1 9 3 3 Figure 1 Figure 2 8 9 9 3 5 7 s t b a d e 7 c 10 10 f 3 3 s t b a d e c f xsai = 1 xabi = 1 xbci = 1 xcdi = 1 xdfi = 1 Figure 3 xadi = 0 xeti = 0 xdei = 0 xfti = 1 xsbi = 0 xbdi = 0 (c) An MFD into 4 paths of weights 3,4,1,9, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' The green dashed path is a subpath of the pink path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 8 9 9 3 5 7 s t b a d e 7 c 10 10 f s t b a d e c f s t b a d e c f 5 3 2 7 3 4 1 9 3 3 Figure 1 Figure 2 8 9 9 3 5 7 s t b a d e 7 c 10 10 f 3 3 s t b a d e 7 c 10 f 3 xsai = 1 xabi = 1 xbci = 1 xcdi = 1 xdfi = 1 (d) The two subpaths (red and blue) of the green dashed path that are maximal safe paths for all FDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 1: Flow decompositions and safe paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' The flow network in (a) admits different MFDs, in (b) and in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' The path (s, a, b, c, d) (dashed green) is a maximal safe path for MFDs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=', it is a subpath of some path of all MFDs and it cannot be extended without losing this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' However, the path (s, a, b, c, d) is not safe for all FDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Indeed, its two subpaths (s, a, b) (dashed red in (d)) and (b, c, d) (dashed blue in (d)) are maximal safe paths for all FDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' To see this, note that the excess flow of (s, a, b) is 3, while the excess flow of (s, a, b, c) (and of (s, a, b, c, d)) is −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 4 The excess flow can be computed in time linear in the length of P (assuming we have pre-computed the flow outgoing from every node), giving thus a linear-time verification of whether P is safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' A basic property of safe solutions is that any sub-solution of them is also safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Computing safe paths for MFDs can thus potentially lead to joining several safe paths for FDs, obtaining longer paths from the unknown sequences we are trying to assemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' See Figure 1 for an example of a maximal safe path for MFDs and two maximal subpaths of it that are safe for FDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='2 Finding Maximal Safe Paths for MFD via ILP We now present a method for finding all maximal safe paths for MFD via ILP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' The basic idea is to define an inner “safety test” which can be repeatedly called as part of an outer algorithm over the entire instance to find all maximal safe paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Because calls to the ILP solver are expensive, the guiding choice for our overall approach is to minimize the number of ILP calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' This inspires us to test the safety of a group of paths as the inner safety test, which we achieve by augmenting our ILP model so that it can give us information about the safety of the paths in the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We use this to define a recursive algorithm to fully determine the safety status of each path in a group of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We can then structure the safety test in either a top-down manner (starting with long unsafe paths and shrinking them until they are safe) or a bottom-up manner (starting with short safe paths and lengthening them until they become unsafe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Safety test (inner algorithm) Let M(V, C) be an ILP model as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' namely, its k solution paths are modeled by binary variables xuvi for each (u, v) ∈ E and each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We assume that M(V, C) is feasible (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=', the problem admits at least one solution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We first show how to modify the ILP model so that, for a given set of paths, it can tell us one of the following: (1) a set of paths that are not safe (the remaining being of unknown status), or (2) that all paths are safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' The idea is to maximize the number of paths that can be simultaneously avoided from the given set of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Let P be a set of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' For each path P ∈ P, we create an auxiliary binary variable γP that indicates: γP ≡ � 1 if P was avoided in the solution, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' (4) Since the model solutions are paths (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=', not repeating nodes), we can encode whether P appears in the solution by whether all of the ℓ − 1 edges of P appear simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Using this fact, we add a new set of constraints R(P) that include the γP indicator variables for each path P ∈ P: R(P) := {xv1v2i + xv2v3i + · · · + xvℓ−1vℓi ≤ ℓ − 1 − γP : ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' , k}, ∀P ∈ P}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' (5) Next, as the objective function of the ILP model, we require that it should maximize the number of avoided paths from P, i,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=', the sum of the γP variables: max � P ∈P γP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' (6) All paths P such that γP = 1 are unsafe, since they were avoided in some minimum flow decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Conversely, if the objective value of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' (6) was 0, then γP = 0 for all paths in P, and it must be that all paths in P are safe (if not, at least one path could be avoided and increase the objective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We encapsulate this group testing ILP in a function GroupTest(M, P) that returns a set N ⊆ P with the properties that: (1) if N = ∅, then all paths in P are safe, and (2) if N ̸= ∅, then all paths in N are unsafe (and |N| is maximized).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We employ GroupTest(M, P) to construct a recursive procedure GetSafe(M, P) that determines all safe paths in P, as shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 5 8 9 9 3 5 7 s t b a d e 7 c 10 10 f s t b a d e c f s t b a d e c f 5 3 2 7 3 4 1 9 3 3 Figure 1 Figure 2 8 9 9 3 5 7 s t b a d e 7 c 10 10 f 3 3 s t b a d e c f xsai = 1 xabi = 1 xbci = 1 xcdi = 1 xdfi = 0 Figure 3 xadi = 0 xeti = 1 xdei = 1 xfti = 0 xsbi = 0 xbdi = 0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 2: Illustration of modeling a solution path and a tested path via binary edge variables and safety verification constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' The ith solution path Pi is shown in orange, and a tested path P is shown in dashed green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Constraint (5) includes xsai +xabi +xbci +xcdi +xdei ≤ 5−γP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' This simplifies to γP ≤ 0, thus forcing γP = 0, which indicates P was not avoided in the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Algorithm 1: Testing a set of paths P for safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Input: A feasible ILP model M(V, C), and a set of paths P Output: Those paths P ∈ P that are safe for M(V, C) 1 Procedure GetSafe(M, P) 2 N = GroupTest(M, P) if N = ∅ then 3 return P 4 else 5 return GetSafe(M, P \\ N) We note that in the special case that |P| = 1, GetSafe(M, P) makes only a single call to the ILP via GroupTest(M, P) to determine whether not the given path is safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' With this safety test for a single path, we can easily adapt a standard two-pointer approach as the outer algorithm to find all maximal safe paths for MFD by starting with some MFD solution P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' , Pk of M(V, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' This same procedure was used in [26] to find all maximal safe paths for FD, using an excess flow check as the inner safety algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Find all maximal safe paths (outer algorithm) We give two algorithms for finding all maximal safe paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Both algorithms use a similar approach, however the first uses a top-down approach starting from the original full solution paths and reports all safe paths (these again must be maximal safe), and then trims all the unsafe paths to find new maximal safe paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' The second is bottom-up in that it tries to extend known safe subpaths until they cannot be further extended (and at this point must be maximal safe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We present the first algorithm in detail and defer discussion of the second to the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We say a set of subpaths T = {Pi[li, ri]} is a trimming core provided that for any unreported maximal safe path P = Pi[l, r], there is a Pi[li, ri] ∈ T , where li ≤ l ≤ r ≤ ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We will use the original k solution paths {Pi} as our initial trimming core;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' the complete algorithm is given in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 3 in the appendix for an illustration of the algorithm’s initial steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' The algorithm first checks if any of the paths in T are safe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' if so, these are reported as maximal safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' For those paths that were unsafe, it then considers trimming one vertex from the left and one vertex from the right to create new subpaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Of these subpaths, some may be contained in a safe path in T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' these subpaths can be ignored as they are not maximal safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' The algorithm recurses on those subpaths whose safety status cannot be determined (lines 6–10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' In this way, the algorithm maintains the invariant that no paths in T are properly contained in a safe path;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' thus paths reported in line 4 must be maximal safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 6 Algorithm 2: An algorithm to compute all maximal safe paths that can be trimmed from a trimming core set T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Input: An ILP model M and a trimming core set T Output: All maximal safe paths for M that are trimmed subpaths of T 1 Procedure AllMaxSafe-TopDown(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' T ) 2 S = GetSafe(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' T ) for Pi[li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' ri] ∈ S do 3 output Pi[li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' ri] 4 U = T \\ S L = {Pi[li + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' ri] : Pi[li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' ri] ∈ U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' (ri = |Pi| or Pi[li + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' ri + 1] ∈ U)} R = {Pi[li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' ri − 1] : Pi[li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' ri] ∈ U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' (li = 1 or Pi[li − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' ri − 1] ∈ U)} P = L ∪ R if P ̸= ∅ then 5 AllMaxSafe-TopDown(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' P) 3 Experiments To test the performance of our methods,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' we computed safe paths using different safety approaches and re- ported the quality and running time performances as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Additional details on the experimental setup are given in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Implementation details – SafeMFD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We implemented the previously described algorithms to compute all maximal safe paths for minimum flow decompositions in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' The implementation, SafeMFD, uses the package NetworkX [20] for graph processing and the package gurobipy [19] to model and solve the ILPs and it is openly available5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Our fastest variant (see Table 2 in the appendix for a comparison of running times) implements Algorithm 2 using the group testing in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We used this variant to compare against other safety approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' All tested variants of SafeMFD implement the following two optimizations: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Before processing an input flow graph we contract it using Y-to-V contraction [51], which is known [30] to maintain (M)FD solution paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Moreover, since edges in the contracted graph correspond to extended unitigs [35,24,29], source-to-sink edges are further removed from the contracted graph and reported as safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' As such, our algorithms compute all maximal safe paths for funnels [17,26] without using the ILP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Before testing the safety of a path we check if its excess-flow [26] is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' If this is the case, the path is removed from the corresponding test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Having positive excess flow implies safety for all flow decomposition and thus also safety for minimum flow decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Safety approaches tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We compare the following state-of-the-art safety approaches: EUnitigs: Maximal paths made up of a prefix of nodes with in-degree one followed by nodes with out- degree one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' also called extended unitigs [51,35,24,29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We use the C++ implementation provided by Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' [26] (which computes only the extended unitigs contained in FD paths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' SafeFlow: Maximal safe paths for all flow decompositions [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We use the C++ implementation provided by Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' SafeMFD: Maximal safe paths for all minimum flow decompositions, as proposed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Every flow graph processed is given a time budget of 2 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' If a flow graph consumes its time budget, the solution of SafeFlow is output instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' SafeEPC: Maximal safe paths for all constrained path covers of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Previous authors [8,26] have con- sidered safe path covers of the nodes, but for a more fair comparison, we instead use path covers of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' To this end, we transform the input graphs by splitting every edge by adding a node in the middle and run the C++ implementation provided by the authors of [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Since flow decompositions are path covers of edges, safe paths for all edge path covers are subpaths of safe paths for MFD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' However, we restrict the path covers to those of minimum size and minimum size plus one, as recommended by the authors of [8] to obtain good coverage results while maintaining high precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 5 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='com/algbio/mfd-safety 7 All safety approaches require a post processing step for removing duplicates, prefixes and suffixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We use the C++ implementation provided by [26] for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We use two datasets of flow graphs inspired by RNA transcript assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' The datasets were created by simulating abundances on a set of transcripts and then perfectly superposing them into a splice graphs that are guaranteed to respect flow conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' As such, the ground truth corresponds to a flow decomposition (not necessarily minimum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' To avoid a skewed picture of our results we filtered out trivial instances with a unique flow decomposition (or funnels, see [17,26]) from the two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='6 Catfish: Created by [48], it includes 100 simulated human, mouse and zebrafish transcriptomes using Flux- Simulator [18] as well as 1,000 experiments from the Sequence Read Archive simulating abundances using Salmon [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We took one experiment per dataset, which corresponds to 27,696 non-trivial flow graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' RefSim: Created by [8] from the Ensembl [57] annotated transcripts of GRCh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='104 homo sapiens reference genome, and later augmented by Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' [26] with simulated abundances using the RNASeqRead- Simulator [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' This dataset has 10,323 non-trivial graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Quality metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We use the same quality metrics employed by previous multi-assembly safety approaches [8,26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We provide a high-level description of them for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Weighted precision of reported paths: As opposed to normal precision, the weighted version considers the length of the reported subpaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' It is computed as the total length of the correctly reported subpaths divided by the total length of all reported subpaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' A reported subpath is considered correct if and only if it is a subpath of some path in the ground truth (exact alignment of exons/nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Maximum coverage of a ground truth path P: The longest segment of P covered by some reported subpath (exact alignment of exons/nodes), divided by |P|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We compute the weighted precision of a graph as the average weighted precision over all reported paths in the graph, and the maximum coverage of a graph as the average maximum coverage over all ground truth paths in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' F-Score of a graph: Harmonic mean between weighted precision and maximum coverage of a graph, which assigns a global score to the corresponding approach on the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' These metrics are computed per flow graph and reported as an average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' In the case of the Catfish dataset the metrics are computed in terms of exons (nodes), since genomic coordinates of exons are missing, whereas in the case of the RefSim dataset the metrics are computed in terms of genomic positions, as this information is present in the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 4 Results and Discussion In the Catfish dataset, EUnitigs and SafeFlow ran in less than a second, while SafeEPC took approximately 30 seconds to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' On the other hand, solving a harder problem, SafeMFD took approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='5 hours to compute in the rest of the dataset, timing out in only 54 graphs (we use a cutoff of 2 minutes), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=', only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='2% of the entire dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' This equates to only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='2 seconds on average per solved graph, underlying the scalability of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Table 1 shows that SafeMFD, on average, covers close to 90% of the ground truth paths, while maintaining a high precision (99%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' This corresponds to an increase of approximately 25% in coverage against its closest competitor SafeFlow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' SafeMFD also dominates in the combined metric of F-Score, being the only safe approach with F-Score over 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Figure 4 in the appendix shows the metrics on graphs grouped by number t of ground truth paths, indicating the dominance in coverage and F-Score of SafeMFD across all values of t, and indicating that the decrease in precision appears for large values of t (t ≥ 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 6 The exact datasets used in our experiments can be found at https://zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='org/record/7182096.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 8 Table 1: Summary of quality metrics for both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' For Catfish, the metrics are computed in terms of nodes/exons and for RefSim in terms of genomic positions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' t is the number of ground truth paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Dataset Graphs Algorithm Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Coverage Wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Precision F-Score Catfish All (100%) EUnitigs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='74 SafeEPC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='74 SafeFlow 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='71 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='82 SafeMFD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='93 RefSim t ≤ 10 (68%) EUnitigs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='72 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='83 SafeEPC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='73 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='84 SafeFlow 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='84 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='91 SafeMFD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='98 t ≤ 15 (84%) EUnitigs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='82 SafeEPC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='71 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='83 SafeFlow 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='83 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='90 SafeMFD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='97 All (100%) EUnitigs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='68 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='80 SafeEPC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='81 SafeFlow 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='81 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='89 SafeMFD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='90 In the harder RefSim dataset, EUnitigs and SafeFlow also ran in less than a second, while SafeEPC took approximately 2 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' In this case, SafeMFD ran out of time in 1,562 graphs (15% of the entire dataset);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' however, recall that in these experiments we allow a time budget of only 2 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' In the rest of the dataset, it took approximately 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='5 hours in total, corresponding to only 3 seconds on average per graph, again underlying that our method, even though it solves many NP-hard problems for each input graph, overall scales sufficiently well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Table 1 shows that again SafeMFD dominates in coverage, being the only approach obtaining coverage over 90%, with is a 15% improvement over SafeFlow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' This time its precision drops to close to 90%, and obtaining an F-Score of 90%, very similar to its closest competitor, SafeFlow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' However, recall that coverage is computed only from correctly aligned paths, thus the drop in precision comes only from safe paths not counting in the coverage metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' If we restrict the metrics to graphs with at most 15 ground truth paths, which is still a significant proportion (84%) of the entire dataset, then SafeMFD has a very high precision (98%) while improving coverage by 15% with respect to SafeFlow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Thus, the drop in precision occurs in graphs with a large number of ground truth paths, which can also be corroborated by Figure 5 in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' These drops in precision (both in RefSim and Catfish) for large t can be explained by the fact that a larger number of ground truth paths produces more complex splice graphs and introduces more artificial solutions of potentially smaller size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' As such, the larger t, the less likely that the ground truth is a minimum flow decomposition of the graph, and thus the more likely that SafeMFD reports incorrect solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' This motivates future work on safety not only on minimum flow decompositions but also in flow decompositions of at most a certain size, analogously to how it is done for SafeEPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' This is still easily achievable with our framework by just changing the ILP blackbox, and keeping everything else unchanged (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=', the inner and outer algorithms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Namely, instead of formulating the ILP model M(V, C) to admit solutions of exactly optimal k paths, it can be changed to allow solutions of at most some k′ paths, with k′ greater than the optimal k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' If k′ is also greater than the number of ground truth paths in these complex graphs, then safe paths are fully correct, meaning that we overall increase precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 9 5 Conclusion RNA assembly is a difficult problem in practice, with even the top tools reporting low precision values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' While there are still many issues that can introduce uncertainty in practice, we can now provide a major source of additional information during the process: which RNA fragments must be included in any parsimonious explanation of the data?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Though others have considered RNA assembly in the safety framework [58,26], we are the first to show that safety can be practically used even when we look for optimal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=', minimum) size solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Our experimental results show that safe paths for MFD clearly outperform other safe approaches for the Catfish dataset, commonly used in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' On a significant proportion of the second dataset, safe paths for MFD still significantly outperforms other safe methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' More generally, this is the first work to show that the safety framework can be practically applied to NP-hard problems, where the inner algorithm is an efficient test of safety of a group of paths, and the outer algorithm guides the applications of this test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Because our method was very successful on our test data set, there is strong motivation to try the approach to on other NP-hard graph problems whose solutions are sets of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' For example, we could study other variations on MFD, such as finding flow decompositions minimizing the longest path (NP-hard when flow values are integer [4,43]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' The approach given in this paper can also be directly extended to find decompositions into both cycles and paths [16], though not trails and walks, because they repeat edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We could also formulate a safety test for classic NP-hard graph problems like Hamiltonian path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Acknowledgements This work was partially funded by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 851093, SAFEBIO), partially by the Academy of Finland (grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 322595, 352821, 346968), and partially by the US National Science Foundation (NSF) (grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 1759522, 1920954).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Ravindra K Ahuja, Thomas L Magnanti, and James B Orlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Network flows.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Andrew D Yates, Premanand Achuthan, Wasiu Akanni, James Allen, Jamie Allen, Jorge Alvarez-Jarreta, M Rid- wan Amode, Irina M Armean, Andrey G Azov, Ruth Bennett, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Ensembl 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Nucleic acids research, 48(D1):D682–D688, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Hongyu Zheng, Cong Ma, and Carl Kingsford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Deriving ranges of optimal estimated transcript expression due to nonidentifiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Journal of Computational Biology, 29(2):121–139, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Presented at RECOMB 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 12 A Additional Figures P1 P2 P3 P4 P1 P2 P3 P4 a) Group Testing - Initial Iteration c) Group Testing - Second Iteration b) Identifying Safe Paths P2 P2 P4 P4 (a) First group test P1 P2 P3 P4 P1 P2 P3 P4 a) Group Testing - Initial Iteration c) Group Testing - Second Iteration b) Identifying Safe Paths P2 P2 P4 P4 (b) Result: {P1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' P3} are safe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' {P2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' P4} are unsafe P1 P2 P3 P4 P1 P2 P3 P4 a) Group Testing - Initial Iteration c) Group Testing - Second Iteration b) Identifying Safe Paths P2 P2 P4 P4 (c) Second group test Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 3: Illustration of the initial group tests performed by Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 3(a) shows the first group test (using Algorithm 1) on MFD solution paths {P1, P2, P3, P4};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' suppose {P1, P3} were safe (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 3(b));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' these are then reported as maximal safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' In this case we trim {P2, P4} on both the left and right and make the next group test shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 13 (a) Weighted Precision (b) Maximum Coverage (c) F-Score Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 4: Quality metrics on graphs distributed by number of paths in the ground truth for the Catfish dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' The metrics are computed in terms of exons/nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' (a) Weighted Precision (b) Maximum Coverage (c) F-Score Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 5: Quality metrics on graphs distributed by number of paths in the ground truth for the RefSim dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' The metrics are computed in terms of genomic positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' B Additional Algorithms and Experimental Results B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='1 The bottom-up algorithm Algorithm 3, detailed below, uses a bottom-up group-testing strategy to find all maximal safe paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We say a set of subpaths E = {Pi[li, ri]} is an extending core provided all paths in E are safe and for any unreported maximal safe path P = Pi[l, r], there is a Pi[li, ri] ∈ E, where l ≤ li ≤ ri ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Note that maximal FD-safe subpaths provide an extending core (as well just the set of all single-edge subpaths in each path).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Algorithm 3 provides an algorithm to find all maximal safe paths based on group testing, starting from an extending core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' The idea is to try both left-extending (by one) and right-extending (by one) each subpath in the core;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' if neither of these extensions are safe, then we know that that core subpath must be maximal safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Testing all extensions can done quickly using Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We then recurse on a new core set consisting of those extensions that were found to be safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='2 EUnitigs SafeEPC SafeFlow SafeMFD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='0 3 4 5 6 7 8 9 10 11 12 13 14 15 # Ground truth paths1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='2 EUnitigs SafeEPC SafeFlow 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='0 SafeMFD 3 4 5 6 7 8 9 1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='12 13 14 15 # Ground truth paths1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='2 EUnitigs SafeEPC SafeFlow 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='0 SafeMFD 3 4 5 6 7 8 9 1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='12 13 14 15 # Ground truth paths1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='2 EUnitigs SafeEPC SafeFlow SafeMFD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='0 3 4 5 6 7 8 9 10 11 12 13 14 15 # Ground truth paths1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='2 EUnitigs SafeEPC SafeFlow 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='0 SafeMFD 3 4 5 6 7 8 9 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='12 13 14 15 # Ground truth paths1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='2 EUnitigs SafeEPC SafeFlow 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='0 SafeMFD 3 4 5 6 7 8 9 1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='12 13 14 15 # Ground truth pathsAlgorithm 3: An algorithm to output all maximal safe subpaths that can be extended from an extending core set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Input: An ILP model M and an extending core set E Output: All maximal safe paths for M that extend some path from E 1 Procedure AllMaxSafe-BottomUp(M, E) 2 L = {Pi[li − 1, ri] : Pi[li, ri] ∈ E, li > 1} R = {Pi[li, ri + 1] : Pi[li, ri] ∈ E, ri < |Pi|} P = L ∪ R S = GetSafe(M, P) for Pi[li, ri] ∈ E do 3 if Pi[li − 1, ri] /∈ S and Pi[li, ri + 1] /∈ S then 4 output Pi[li, ri] 5 if S ̸= ∅ then 6 AllMaxSafe-BottomUp(M, S) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='2 The two-pointer algorithm As we observed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='2, we can test whether a single path P is safe using one ILP call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We will assume that this test is encapsulated as a procedure IsSafe(M, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Once we can test whether a single path is safe for M(V, C), we can adopt a standard approach to compute all maximal safe paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Namely, we start by computing one solution of M(V, C), P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' , Pk and then compute maximal safe paths by a two-pointer technique that for each path Pi, finds all maximal safe paths by just a linear number of calls to the procedure IsSafe [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' This works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We use two pointers, a left pointer L, and a right pointer R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Initially, L points to the first node of path Pi and R to the second node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' As long as the subpath of Pi between L and R is safe, we move the right pointer to the next node on Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' When this subpath is not safe, we output the subpath between L and the previous location of R as a maximal safe path, and we start moving the left pointer to the next node on Pi, until the subpath between L and R is safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We stop the procedure once we reach the end of Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We summarize this procedure as Algorithm 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' see also Figure 6 for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 1 8 3 2 5 6 4 7 s t b a d e c f s t b a d e c f s t b a d e c f xsai + xabi + xbci + xcdi + xdfi ≤ 4 xsai + xabi + xbci + xcdi ≤ 3 xabi + xbci + xcdi + xdfi ≤ 3 R L R ∀i ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='…,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' k} ∀i ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='…,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' k} ∀i ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='…,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' k} L L R (a) Current iteration 1 8 3 2 5 6 4 7 s t b a d e c f s t b a d e c f s t b a d e c f xsai + xabi + xbci + xcdi + xdfi ≤ 4 xsai + xabi + xbci + xcdi ≤ 3 xabi + xbci + xcdi + xdfi ≤ 3 R L R ∀i ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='…,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' k} ∀i ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='…,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' k} ∀i ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='…,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' k} L L R (b) Right pointer movement 1 8 3 2 5 6 4 7 s t b a d e c f s t b a d e c f s t b a d e c f xsai + xabi + xbci + xcdi + xdfi ≤ 4 xsai + xabi + xbci + xcdi ≤ 3 xabi + xbci + xcdi + xdfi ≤ 3 R L R ∀i ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='…,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' k} ∀i ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='…,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' k} ∀i ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='…,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' k} L L R (c) Left pointer movement Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 6: Illustration of the two-pointer algorithm applied on a flow decomposition path Pi (in orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' In each sub-figure, the subpath P (dashed green) between the nodes pointed by the left pointer L and the right pointer R is tested for safety, by adding constraints S(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' In (a), IsSafe(M, P) returns True, and the right pointer advances on Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' In (b), IsSafe(M, P) returns False, and the previous subpath from (a) is output as a maximal safe path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' In (c), the left pointer has advanced, and the new path P is tested for safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 15 Algorithm 4: The two-pointer algorithm applied to compute all maximal subpaths of a given solution path Pi Input: An ILP model M and one of its k solution paths, Pi = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' , vt), t ≥ 2 Output: All maximal safe subpaths of Pi for M 1 Procedure AllMaxSafe-TwoPointer(M, Pi) 2 L ← 1, R ← 2 while True do 3 while IsSafe(M, Pi[L, R]) and R ≤ t do 4 R ← R + 1 5 output Pi[L, R − 1] if R > t then return;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 6 while not IsSafe(M, Pi[L, R]) do 7 L ← L + 1 Dataset (# Graphs) Variant Time (hh:mm:ss) # ILP calls Catfish (27,613) TopDown 01:13:27 124,676 BottomUp 03:22:13 212,774 TwoPointer 04:21:44 226,365 TwoPointerBin 03:31:57 216,540 RefSim (5,808) TopDown 04:38:41 55,450 BottomUp 11:55:20 76,837 TwoPointer 13:48:00 127,352 TwoPointerBin 11:34:02 119,218 Table 2: Running times and number of ILP calls in four different variants of SafeMFD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='3 Running time experiments among different variants proposed We conducted the experiments on an isolated Linux server with AMD Ryzen Threadripper PRO 3975WX CPU with 32 cores (64 virtual) and 504GB of RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Time and peak memory usage of each program were measured with the GNU time command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' SafeMFD was allowed to run Gurobi with 12 threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' All C++ implementations were compiled with optimization level 3 (-O3 flag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Running time and peak memory is computed and reported per dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' SafeMFD includes the following four variants computing maximal safe paths: TopDown : Implements Algorithm 2 using the group testing in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' BottomUp : Implements Algorithm 3 (Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='1) using the group testing in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' TwoPointer : Implements Algorithm 4 (Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='2), the traditional two-pointer algorithm [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' TwoPointerBin : Same as previous variant, but it additionally replaces the linear scan employed to extend and reduce the currently processed safe path by a binary search7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' To compare between our four different variants we first run them all on every dataset, and then filter out those graphs that ran out of time in some variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' This way we ensure that no variant consumes its time budget and thus our running time measurements are not skewed by the unsuccessful inputs’ timeouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Applying this filter we removed 83 graphs from the Catfish dataset (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='3%) and 4,515 graphs from the RefSim dataset (43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='74%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Table 2 shows the running times and number of ILP calls of the different variants on both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' TopDown clearly outperforms the rest, being at least twice as fast, and performing (roughly) half many ILP calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' While BottomUp is analogous to TopDown, the superiority of the latter can be explained by the length maximal safe paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Since maximal safe paths are long it is faster to obtain them by reducing unsafe paths 7 The binary search is only applied if the search space is larger than a constant threshold set experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 16 (TopDown) than by extending safe paths (BottomUp and both TwoPointer variants).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' On the other hand, TwoPointer is the slowest variant and BottomUp and TwoPointerBin obtain similar improvements (over TwoPointer) by following different strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' While BottomUp reduces the number of ILP calls more than TwoPointerBin (better appreciated in the RefSim dataset), the ILP calls of BottomUp take longer (since BottomUp tests several paths at the same time and TwoPointerBin only one), and thus the total running times of both is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' This motivates future work on combining both approaches, while processing the paths starting from unsafe (as in TopDown) for better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' C Hardness of Testing MFD Safety In this section we give a Turing-reduction from the UNIQUE 3SAT problem (U3SAT) to the problem of determining if a given path P in a flow network G is safe for minimum flow decomposition (call this problem MFD-SAFETY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' A 3SAT instance belongs to U3SAT if and only if it has exactly one satisfying assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' U3SAT has been shown to be NP-hard under randomized reductions [52], but it is open as to whether it is NP-hard in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' The reduction leverages the construction in [22] that reduces 3SAT to minimum flow decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We first briefly review this construction: A variable gadget (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 4 in [22]) is created for each 3SAT variable x and a clause gadget (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 5 in [22]) is created for each 3SAT clause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Positive literals in each clause receive flow from the left side of the corresponding variable gadget, whereas negative literals receive flow from the right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Theorem VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content='1 in [22] establishes that a 3SAT instance is satisfiable if and only if the constructed flow network has a minimum flow decomposition of a certain size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Any flow decomposition achieving this size must have a specific structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' in particular, there must be a flow path of weight 4 that either travels up the left side of the gadget (setting x to TRUE), or the right side (setting x to FALSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' ⋯ 4 4 4 4 4 4 ⋯ 4 4 4 4 t(x) s(x) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 7: The variable gadget from [22], showing only the weight 4 edges (other edges have weights from {1, 2}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' A key property established in [22] is that if the 3SAT instance is satisfiable then in a minimum flow decomposition, a weight 4 flow path must travel up either the left side of the gadget (as shown), or the right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' A left flow path indicates the variable should be set to TRUE, while right indicates FALSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' We leverage this construction to reduce U3SAT to MFD-SAFETY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' There is a polynomial time Turing-reduction from U3SAT to MFD-SAFETY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' To obtain the desired Turing-reduction algorithm, instead of checking the size of the MFD, we will instead sequentially check the MFD-SAFETY of the aforementioned side paths traveling up the left and right sides of each variable gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Provided each variable gadget has exactly one safe side path we can then check the corresponding truth assignment to see if each clause is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' If yes, we accept the instance as belonging to U3SAT, otherwise we reject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Suppose the instance does belong to U3SAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' In this case there is a satisfying assignment so the MFD must have the structure as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' Furthermore, since there is exactly one satisfying assignment, exactly one side path of each variable gadget must be safe and so our algorithm finds it and then verifies that the truth assignment satisfies each clause, thus accepting the instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' On the other hand, if the instance does not belong to U3SAT it could either be unsatisfiable or have multiple satisfying assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' If unsatisfiable, no matter whether the safety checks pass, the corresponding assignment will not satisfy all clauses, so the instance will be rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' If there are multiple solutions, then any variable that can be both TRUE and FALSE will not have a safe side path in the MFD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' This means the safety check will fail and the instance will again be rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} +page_content=' ⊓⊔ 18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'} diff --git a/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf b/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..5780fad0844863d5ef864a9703a8759eda9463c5 --- /dev/null +++ b/49E1T4oBgHgl3EQf6QXo/content/2301.03522v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:80fb761f474b7bc057b81daac9459b1d167947c304674b262e0682e0faf75eec +size 133493 diff --git a/49E1T4oBgHgl3EQf6QXo/vector_store/index.faiss b/49E1T4oBgHgl3EQf6QXo/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..6852ca7619b547f8c5d384387dd87cd33d48321a --- /dev/null +++ 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Dec 2022 +Isometric path antichain covers: beyond hyperbolic graphs∗ +Dibyayan Chakraborty† +Florent Foucaud‡ +January 3, 2023 +Abstract +The isometric path antichain cover number of a graph G, denoted by ipacc (G), is a graph pa- +rameter that was recently introduced to provide a constant factor approximation algorithm for Iso- +metric Path Cover, whose objective is to cover all vertices of a graph with a minimum number +of isometric paths (i.e. shortest paths between their end-vertices). This parameter was previously +shown to be bounded for chordal graphs and, more generally, for graphs of bounded chordality and +bounded treelength. In this paper, we show that the isometric path antichain cover number remains +bounded for graphs in three seemingly unrelated graph classes, namely, hyperbolic graphs, (theta, +prism, pyramid)-free graphs, and outerstring graphs. Hyperbolic graphs are extensively studied in +Metric Graph Theory. The class of (theta, prism, pyramid)-free graphs are extensively studied in +Structural Graph Theory, e.g. in the context of the Strong Perfect Graph Theorem. The class of +outerstring graphs is studied in Geometric Graph Theory and Computational Geometry. Our results +imply a constant factor approximation algorithm for Isometric Path Cover on all the above graph +classes. Our results also show that the distance functions of these (structurally) different graph classes +are more similar than previously thought. +1 +Introduction +A path is isometric if it is a shortest path between its endpoints. An isometric path cover of a graph G +is a set of isometric paths such that each vertex of G belongs to at least one of the paths. The isometric +path number of G is the smallest size of an isometric path cover of G. Given a graph G and an integer k, +the objective of the algorithmic problem Isometric Path Cover is to decide if there exists an isometric +path cover of cardinality at most k. Isometric Path Cover has been introduced and studied in the +context of pursuit-evasion games [1, 2] and used in the context of Product Structure Theorems [15]. +The goal of this paper is to continue the study of approximation algorithms for Isometric Path +Cover on several graph classes. +We do so by continuing the study of a recently introduced graph +parameter which seems interesting in its own right, as it encapsulates several previously unrelated graph +classes. +Isometric Path Cover has also been studied from a structural point of view: the cardinalities +of the optimal solution have been determined for square grids [17], hypercubes [18], complete r-partite +graphs [24] and Cartesian products of complete graphs [24], and it was recently proved that the pathwidth +of a graph is always upper-bounded by the size of its smallest isometric path cover [16]. However, until +recently the algorithmic aspects of Isometric Path Cover remained unexplored. The problem is easy +to solve on trees and more generally, on block graphs [23] but remains hard on chordal graph, i.e. graphs +without any induced cycle of length at least 4 [7]. It can be approximated in polynomial time within a +factor of log(d) for graphs of diameter d by a greedy algorithm [27] and solved in polynomial time for every +∗This +research +was +partially +financed +by +the +IFCAM +project +“Applications +of +graph +homomorphisms” +(MA/IFCAM/18/39), the ANR project GRALMECO (ANR-21-CE48-0004) and the French government IDEX-ISITE ini- +tiative 16-IDEX-0001 (CAP 20-25). +†Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France +‡Université Clermont-Auvergne, CNRS, Mines de Saint-Étienne, Clermont-Auvergne-INP, LIMOS, 63000 Clermont- +Ferrand, France +1 + +fixed value of k by an XP algorithm [16]. In a quest to find constant factor approximation algorithms +for Isometric Path Cover, Chakraborty et al. [7] introduced a parameter called the isometric path +antichain cover number of graphs, denoted by ipacc (G) (see Section 2 for a definition) and proved a +result directly implying the following (see [7, Proposition 10]). +Proposition 1 ([7]). For a graph G, if ipacc (G) ≤ c, then Isometric Path Cover admits a polynomial- +time c-approximation algorithm on G. +Proposition 1 is proved by a simple approximation algorithm described as follows. For each vertex r +of the graph, perform a Breadth-First Search at this vertex. Remove edges joining any vertices at the +same distance from r, and orient all edges towards r. The resulting directed acyclic graph can be seen as +the Hasse diagram of a poset. Compute a chain covering of that poset using classic methods related to +Dilworth’s theorem. The chains are the isometric paths of the solution. Keep the smallest of all solutions +over all choices of r. +Using Proposition 1, the above algorithm was shown to be a constant factor approximation algo- +rithm for many graph classes, including interval graphs, chordal graphs, and more generally, graphs with +bounded treelength. Indeed, on all these graph classes, the isometric path antichain cover number is +shown to be bounded by a constant (note that one does not need to compute this parameter for the +algorithm to function: it serves only in the analysis of the approximation ratio of the algorithm). As +noted in [7], this parameter may be unbounded on general graphs, for example for the class of hypercubes +or square grids. +In this paper, we continue to study the boundedness of the isometric path antichain cover number +of various graph classes. Specifically, we consider three structurally unrelated graph classes, namely, +hyperbolic graphs, (theta, prism, pyramid)-free graphs, and outerstring graphs, which extends the above +work to strictly larger graph classes. +Hyperbolic graphs: A graph G is said to be δ-hyperbolic [19] if for any four vertices u, v, x, y, the two +larger of the three distance sums d (u, v)+d (x, y), d (u, x)+d (v, y) and d (u, y)+d (v, x) differ by at most +2δ. A graph class G is hyperbolic if there exists a constant δ such that every graph G ∈ G is δ-hyperbolic. +This parameter was first introduced by Gromov in the context of automatic groups [19] in relation with +their Cayley graphs. The hyperbolicity of a tree is 0, and in general, hyperbolicity seems to measure +how much the distance function of a graph deviates from a tree metric. Many structurally defined graph +classes like chordal graphs, cocomparability graphs, asteroidal-triple free graphs, graphs with bounded +chordality or treelength are hyperbolic graphs [8, 21]. Moreover, hyperbolicity has been found to capture +important properties of several large practical graphs such as the Internet [26] or database relations [31]. +Due to its importance in discrete mathematics, algorithms, metric graph theory, researchers have studied +various algorithmic aspects of hyperbolic graphs [8, 12, 9, 13]. Note that graphs with diameter 2 are +hyperbolic, which may contain any graph as an induced subgraph. +(theta, prism, pyramid)-free graphs: A theta is a graph made of three vertex-disjoint induced paths +P1 = a . . . b, P2 = a . . . b, P3 = a . . . b of lengths at least 2, and such that no edges exist between the paths +except the three edges incident to a and the three edges incident to b. See Figure 2 for an illustration. A +pyramid is a graph made of three induced paths P1 = a . . . b1, P2 = a . . . b2, P3 = a . . . b3, two of which +have lengths at least 2, vertex-disjoint except at a, and such that b1b2b3 is a triangle and no edges exist +between the paths except those of the triangle and the three edges incident to a. A prism is a graph +made of three vertex-disjoint induced paths P1 = a1 . . . b1, P2 = a2 . . . b2, P3 = a3 . . . b3 of lengths at +least 1, such that a1a2a3 and b1b2b3 are triangles and no edges exist between the paths except those of +the two triangles. A graph G is (theta, pyramid, prism)-free if G does not contain any induced subgraph +isomorphic to a theta, pyramid or prism. A graph is a 3-path configuration if it is a theta, pyramid or +prism. The study of 3-path configurations dates back to the works of Watkins and Meisner [32] in 1967 +and plays “special roles” in the proof of the celebrated Strong Perfect Graph Theorem [10, 14, 28, 30]. +Important graph classes like chordal graphs, circular arc graphs, universally-signable graphs [11] exclude +all 3-path configurations. Popular graph classes like perfect graphs, even hole-free graphs exclude some +2 + +Bounded isometric path +antichain cover number +bounded hyperbolicity * +(t-theta, t-prism, t-pyramid)- +free * +Outerstring * +circle * +(theta,prism,pyramid)- +free * +Universally signable * +bounded tree-length +bounded chordality +bounded diameter +chordal +AT-free +Interval +circular arc * +Permutation +Figure 1: Inclusion diagram for graph classes discussed here (and related ones). If a class A has an +upward path to class B, then A is included in B. For graphs in the gray classes, the complexity of +Isometric Path Cover is open; for all other graph classes, it is NP-complete. For all shown graph +classes, Isometric Path Cover is constant-factor approximable in polynomial time. Constant factor +approximation algorithms for Isometric Path Cover on graph classes marked with * are contributions +of this paper. +b +a +b1 +b2 +b3 +a +b1 +b2 +b3 +a1 +a2 +a3 +(a) +(b) +(c) +(d) +Figure 2: (a) Theta, (b) Pyramid, (c) Prism, (d) Outerstrings. The figure shows that the graph K2,3, +which is also a theta, is an outerstring graph. +of the 3-path configurations. Note that, (theta, prism, pyramid)-free graphs are not hyperbolic. To see +this, consider a cycle C of order n. Clearly, C excludes all 3-path configurations and has hyperbolicity +Ω(n). +Outerstring graphs: A set S of simple curves on the plane is grounded if there exists a horizontal line +containing one endpoint of each of the curves in S. A graph G is an outerstring graph if there is a collection +C of grounded simple curves and a bijection between V (G) and C such that two curves in S if and only +if the corresponding vertices are adjacent in G. See Figure 2(d) for an illustration. The term “outerstring +graph” was first used in the early 90’s [22] in the context of studying intersection graphs of simple curves +on the plane. Many well-known graph classes like chordal graphs, circular arc graphs, circle graphs +(intersection graphs of chords of a circle), or cocomparability graphs are also outerstring graphs and +thus, motivated researchers from the geometric graph theory and computational geometry communities +to study algorithmic and structural aspects of outerstring graphs and its subclasses [4, 5, 6, 20, 25]. Note +that, in general, outerstring graphs may contain a prism, pyramid or theta as an induced subgraph. +Moreover, cycles of arbitrary order are outerstring graphs, implying that outerstring graphs are not +hyperbolic. +It is clear from the above discussion that the classes of hyperbolic graphs, (theta, prism, pyramid)-free +3 + +graphs, and outerstring graphs are pairwise incomparable (with respect to the containment relationship). +1.1 +Our contributions +The main contribution of this paper is to show that the isometric path antichain cover number (see +Section 2 for a definition) remains bounded on hyperbolic graphs, (theta, pyramid, prism)-free graphs, +and outerstring graphs. Specifically, we prove the following theorems. +Theorem 2. Let G be a graph with hyperbolicity δ. Then, ipacc (G) ≤ 12δ + 6. +Theorem 3. Let G be a (theta, pyramid, prism)-free graph. Then, ipacc (G) ≤ 71. +Theorem 4. Let G be an outerstring graph. Then, ipacc (G) ≤ 95. +To the best of our knowledge, the isometric path antichain cover number being bounded (by con- +stant(s)) is the only known non-trivial property shared by any two or all three of these graph classes. +To provide a unified proof of Theorem 3 and 4, we study a more general graph class called (t-theta, +t-pyramid, t-prism)-free graphs [29] (see Section 4 for definition). When t = 1, (t-theta, t-pyramid, t- +prism)-free graphs are exactly (theta, prism, pyramid)-free graphs. Moreover, we show that all outerstring +graphs are (4-theta, 4-pyramid, 4-prism)-free graphs (Lemma 16). We prove the following. +Theorem 5. For t ≥ 1, let G be a (t-theta, t-pyramid, t-prism)-free graph. Then ipacc (G) ≤ 8t + 63. +Due to Proposition 1 and the above theorems, we also have the following corollary. +Corollary 6. There is an approximation algorithm for Isometric Path Cover with approximation +ratio +(a) 12δ + 6 on δ-hyperbolic graphs, +(b) 73 on (theta, prism, pyramid)-free graphs, and +(c) 95 on outerstring graphs. +(d) 8t + 63 on (t-theta, t-pyramid, t-prism)-free graphs. +Organisation: In Section 2, we introduce the recall some definitions and some results. In Section 3 we +prove Theorem 2. In Section 4, we prove Theorems 3 and 5. In Section 5, we prove Theorem 4. We +conclude in Section 6. +2 +Definitions and preliminary observations +In this section, we formally recall the definition of isometric path antichain cover number of graphs from +[7] and some related observations. A sequence of distinct vertices forms a path P if any two consecutive +vertices are adjacent. Whenever we fix a path P of G, we shall refer to the subgraph formed by the +edges between the consecutive vertices of P. The length of a path P, denoted by |P|, is the number of +its vertices minus one. A path is induced if there are no graph edges joining non-consecutive vertices. +In a directed graph, a directed path is a path in which all arcs are oriented in the same direction. For a +path P of a graph G between two vertices u and v, the vertices V (P) \ {u, v} are internal vertices of P. +A path between two vertices u and v is called a (u, v)-path. Similarly, we have the notions of isometric +(u, v)-path and induced (u, v)-path. For a vertex r of G and a set S of vertices of G, the distance of S from +r, denoted as d (r, S), is the minimum of the distance between any vertex of S and r. For a subgraph +H of G, the distance of H w.r.t. r is d (r, V (H)). Formally, we have d (r, S) = min{d (r, v) : v ∈ S} and +d (r, H) = d (r, V (H)). +4 + +For a graph G and a vertex r ∈ V (G), consider the following operations on G. First, remove all +edges xy from G such that d (r, x) = d (r, y). +Let G′ +r be the resulting graph. +Then, for each edge +e = xy ∈ E(G′ +r) with d (r, x) = d (r, y) − 1, orient e from y to x. Let −→ +Gr be the directed acyclic graph +formed after applying the above operation on G′. Note that this digraph can easily be computed in linear +time using a Breadth-First Search (BFS) traversal with starting vertex r. +The following definition is inspired by the terminology of posets (as the graph −→ +Gr can be seen as the +Hasse diagram of a poset). +Definition 7. For a graph G and a vertex r ∈ V (G), two vertices x, y ∈ V (G) are antichain vertices if +there are no directed paths from x to y or from y to x in −→ +Gr. A set X of vertices of G is an antichain +set if any two vertices in X are antichain vertices. The cardinality of the largest antichain set in −→ +Gr will +be denoted by β +�−→ +Gr +� +. The cardinality of the largest antichain set of G, is defined as +β (G) = min +� +β +�−→ +Gr +� +: r ∈ V (G) +� +Definition 8 ([7]). Let r be a vertex of a graph G. For a subgraph H, Ar (H) shall denote the maximum +antichain set of H in −→ +Gr. The isometric path antichain cover number of −→ +Gr, denoted by ipacc +�−→ +Gr +� +, is +defined as follows: +ipacc +�−→ +Gr +� += max {|Ar (P) |: P is an isometric path} +The isometric path antichain cover number of graph G, denoted as ipacc (G), is defined as the minimum +over all possible antichain covers of its associated directed acyclic graphs: +ipacc (G) = min +� +ipacc +�−→ +Gr +� +: r ∈ V (G) +� +We recall the proof of the following proposition from [7] which will be used heavily in this paper. +Proposition 9 ([7]). Let G be a graph and r, an arbitrary vertex of G. Consider the directed acyclic +graph −→ +Gr, and let P be an isometric path between two vertices x and y in G. Then |P| ≥ |d (r, x) − +d (r, y) | + |Ar (P) | − 1. +Proof. Orient the edges of P from y to x in G. First, observe that P must contain a set E1 of oriented +edges such that |E1| = |d (r, y) − d (r, x) | and for any −→ +ab ∈ E1, d (r, a) = d (r, b) + 1. Let the vertices of +the largest antichain set of P in −→ +Gr, i.e., Ar (P), be ordered as a1, a2, . . . , at according to their occurrence +while traversing P from y to x. For i ∈ [2, t], let Pi be the subpath of P between ai−1 and ai. Observe that +for any i ∈ [2, t], since ai and ai−1 are antichain vertices, there must exist an oriented edge −→ +bici ∈ E(Pi) +such that either d (r, bi) = d (r, ci) or d (r, bi) = d (r, ci) − 1. +Let E2 = {bici}i∈[2,t]. +Observe that +E1 ∩ E2 = ∅ and therefore |P| ≥ |E1| + |E2| = |d (r, y) − d (r, x) | + |Ar (P) | − 1. +3 +Proof of Theorem 2 +In this section, we shall show that isometric path antichain cover number of graphs with hyperbolicity +at most δ is at most 12δ + 6. To achieve our goal we need to recall a few definitions from the literature. +For three vertices x, y, z of a graph G, a geodesic triangle [3], denoted as ∆(x, y, z) is the union P(x, y) ∪ +P(y, z)∪P(x, z) of three isometric paths connecting these vertices. A geodesic triangle ∆(x, y, z) is called +ρ-slim if for any vertex u ∈ P(x, y) the distance d (u, P(y, z) ∪ P(x, z)) is at most ρ. The smallest value +of ρ for which every geodesic triangle of G is ρ-slim is called the slimness of G and is denoted by sl (G). +In the following lemma, we shall show that if the isometric path antichain cover number of a graph is +large then so is the slimness of the graph. +Lemma 10. For any graph G, ipacc (G) ≤ 4sl (G) + 2. +5 + +u +v +c +c′ +Figure 3: An example of a 4-fat turtle. Let C be the cycle induced by the black vertices, P be the path +induced by the white vertices. Then the tuple (4, C, P, c, c′) defines a 4-fat turtle. +Proof. Let ρ = sl (G). +Aiming for a contradiction, let r be a vertex of G such that there exists an +isometric path P such that |Ar (P) | ≥ 4ρ + 3. Let the vertices of Ar (P) be named and ordered as +a1, a2, . . . , a2ρ+2, . . . , a4ρ+3 as they are encountered while traversing P from one end-vertex to the other. +Let x = a1, y = a4ρ+3. Let −→ +Px be an oriented path from x to r in −→ +Gr. Observe that Px, the path of +G obtained by removing the orientation of −→ +Px, is an (x, r)-isometric path. Let −→ +Py be an oriented path +from y to r in −→ +Gr. Similarly, Py, the path of G obtained by removing the orientation of −→ +Py, is an (y, r)- +isometric path. Observe that P, Px, Py form a geodesic triangle with x, r, y as end-vertices. Consider the +vertex z = a2ρ+2 on the path P. Since ρ = sl (G), there exists a vertex w ∈ V (Px) ∪ V (Py) such that +d (w, z) ≤ ρ. Without loss of generality, assume w ∈ V (Px). Then, d (x, z) ≤ d (x, w) + d (w, z). By +using that d (r, z) ≤ d (r, w) + d (w, z) ≤ d (r, w) + ρ, we get d (x, z) ≤ |d (r, x) − d (r, z) | + 2ρ. But this +contradicts Proposition 9, due to which we have d (x, z) ≥ |d (r, x) − d (r, z) | + 2ρ + 1. +Now we shall use the following result. +Proposition 11 ([3]). For any graph G, sl (G) ≤ 3hb (G). +Proposition 11 and Lemma 10, imply the theorem. +4 +Proofs of Theorem 3 and 5 +In this section, we shall prove Theorems 3 and 5. First we shall define the notions of t-theta, t-prism, +and t-pyramid [29]. +For an integer t ≥ 1, a t-prism is a graph made of three vertex-disjoint induced paths P1 = a1 . . . b1, +P2 = a2 . . . b2, P3 = a3 . . . b3 of lengths at least t, such that a1a2a3 and b1b2b3 are triangles and no edges +exist between the paths except those of the two triangles. For an integer t ≥ 1, a t-pyramid is a graph +made of three induced paths P1 = a . . . b1, P2 = a . . . b2, P3 = a . . . b3 of lengths at least t, two of which +have lengths at least t + 1, they are pairwise vertex-disjoint except at a, such that b1b2b3 is a triangle +and no edges exist between the paths except those of the triangle and the three edges incident to a. For +an integer t ≥ 1, a t-theta is a graph made of three internally vertex-disjoint induced paths P1 = a . . . b, +P2 = a . . . b, P3 = a . . . b of lengths at least t+1, and such that no edges exist between the paths except the +three edges incident to a and the three edges incident to b. A graph G is (t-theta, t-pyramid, t-prism)-free +if G does not contain any induced subgraph isomorphic to a t-theta, t-pyramid or t-prism. When t = 1, +(t-theta, t-pyramid, t-prism)-free graphs are exactly (theta, prism, pyramid)-free graphs. +Now we shall show that the isometric path antichain cover number of (t-theta, t-pyramid, t-prism)- +free graphs are bounded above by a linear function on t. We shall show that, when the isometric path +antichain cover number of a graph is large, the existence of a structure called “t-fat turtle” (defined +below) as an induced subgraph is forced, which, cannot be present in a ((t − 1)-theta, (t − 1)-pyramid, +(t − 1)-prism)-free graph. +Definition 12. For an integer t ≥ 1, a “t-fat turtle” consists of a cycle C and an induced (u, v)-path P +of length at least t such that all of the following hold. +6 + +(a) V (P) ∩ V (C) = ∅, +(b) For any vertex w ∈ (V (P) \ {u, v}), N(w) ∩ V (C) = ∅ and both u and v have at least one neighbour +in C. +(c) For any vertex w ∈ N(u) ∩ V (C) and w′ ∈ N(v) ∩ V (C), the distance between w and w′ in C is at +least t, +(d) There exist two vertices {c, c′} ⊂ V (C) and two distinct components Cu, Cv of C − {c, c′} such that +N(u) ∩ V (C) ⊆ V (Cu) and N(v) ∩ V (C) ⊆ V (Cv). +The tuple (t, C, P, c, c′) defines the t-fat turtle. See Figure 3 for an example. +In the following observation, we show that any (t-theta, t-pyramid,t-prism)-free graph cannot contain +a (t + 1)-fat turtle as an induced subgraph. +Lemma 13. For some integer t ≥ 1, let G be a graph containing a (t + 1)-fat turtle as an induced +subgraph. Then G is not (t-theta, t-pyramid, t-prism)-free. +Proof. Let (t+1, C, P, c, c′) be a (t+1)-fat turtle in G. Let the vertices of C be named c = a0, a1, . . . , ak = +c′, ak+1, . . . , a|V (C)| as they are encountered while traversing C starting from c in a counter-clockwise +manner. Denote by u, v the end-vertices of P. By definition, there exist two distinct components Cu, Cv of +C−{c, c′} such that N(u)∩V (C) ⊆ V (Cu) and N(v)∩V (C) ⊆ V (Cv). Without loss of generality, assume +V (Cu) = {a1, a2, . . . , ak−1} and V (Cv) = {ak+1, ak+2, . . . , a|V (C)|}. Let i− and i+ be the minimum and +maximum indices such that ai− and ai+ are adjacent to u. Let j− and j+ be the minimum and maximum +indices such that aj− and aj+ are adjacent to v. By definition, i− ≤ i+ < j− ≤ j+. Let P1 be the +(ai−, aj+)-subpath of C containing c. Let P2 be the (ai+, aj−)-subpath of C that contains c′. Observe +that P1 and P2 have length at least t (by definition). Now we show that P, P1, P2 together form one of +theta, pyramid or prism. If ai− = ai+ and aj− = aj+, then P, P1, P2 form a t-theta. If i− ≤ i+ − 2 and +j− ≤ j+ − 2, then also P, P1, P2 form a t-theta. If j− = j+ − 1 and i− = i+ − 1, then P, P1, P2 form a +t-prism. In any other case, P, P1, P2 form a t-pyramid. +In the remainder of this section, we shall prove that there exists a linear function f(t) such that if +the isometric path antichain cover number of a graph is more than f(t), then G is forced to contain a +(t + 1)-fat turtle as an induced subgraph, and therefore is not (t-theta, t-pyramid,t-prism)-free. We shall +use the following observation. +Observation 14. Let G be a graph, r be an arbitrary vertex, P be an isometric (u, v)-path in G and Q +be a subpath of an isometric (v, r)-path in G such that one endpoint of Q is v. Let P ′ be the maximum +(u, w)-subpath of P such that no internal vertex of P ′ is a neighbour of some vertex of Q. We have that +|Ar (P ′) | ≥ |Ar (P) | − 3. +Proof. Suppose |Ar (P ′) | ≤ |Ar (P) | − 4 and consider the (w, v)-subpath, say P ′′, of P. Observe that +|Ar (P ′′) | ≥ 4. Now let w′ be a vertex of Q which is a neighbour of w. Observe that |d (r, w)−d (r, w′) | ≤ 1 +and therefore d (w, v) = |E(P ′′)| ≤ |d (r, w)−d (r, v) |+2. But this contradicts Proposition 9, which implies +that the length of P ′′ is at least |d (r, w) − d (r, v) | + 3. +Lemma 15. For an integer t ≥ 1, let G be a graph with ipacc (G) ≥ 8t + 64. Then G has a (t + 1)-fat +turtle as an induced subgraph. +Proof. Let r be a vertex of G such that ipacc +�−→ +Gr +� +is at least 8t+64. Then there exists an isometric path +P such that |Ar (P) | ≥ 8t+ 64. Let the two endpoints of P be a and b. (See Figure 4.) Let u be a vertex +of P such that d (r, u) = d (r, P). Let Pau be the (a, u)-subpath of P and Pbu be the (b, u)-subpath of P. +Both Pau and Pbu are isometric paths and observe that either |Ar (Pau) | ≥ 4t+32 or |Ar (Pbu) | ≥ 4t+32. +Without loss of generality, assume that |Ar (Pbu) | ≥ 4t + 32. Let Qr +b be an isometric (b, r)-path in G. +7 + +r +z2 +w2 +u +z +z1 +w +b +w1 +c +(= a2t+13) +x +c1 +a +c2 +T (c1, c2) +≥ t +≥ t +≥ t +Qr +b +Qr +u +Figure 4: Illustration of the notations used in the proof of Lemma 15. +Let Ruw be the maximum (u, w)-subpath, of Pbu such that no internal vertex of Ruw is a neighbour +of Qr +b. Note that Ruw is an isometric path and w has a neighbour in Qr +b. Applying Observation 14, we +have the following: +Claim 15.1. |Ar (Ruw) | ≥ 4t + 29. +Let Qr +u be any isometric (u, r)-path of G and let Rzw be the maximum (z, w)-subpath of Ruw such +that no internal vertex of Rzw has a neighbour in Qr +u. Observe that Rzw is an isometric path, and z has +a neighbour in Qr +u. Again applying Observation 14, we have the following: +Claim 15.2. |Ar (Rzw) | ≥ 4t + 26. +Let a1, a2, . . . , ak be the vertices of Ar (Rzw) ordered according to their appearance while traversing +Rzw from z to w. Due to Claim 15.2, we have that k ≥ 4t + 26. Let c = a2t+13 and Qr +c denote an +isometric (c, r)-path. Let T (r, c1) be the maximum subpath of Qr +c such that no internal vertex of T (r, c1) +is adjacent to any vertex of Rzw. +Claim 15.3. Let x be a neighbor of c1 in Rzw, X be the (x, b)-subpath of Pub and Y be the (x, u)-subpath +of Pub. Then |Ar (X) | ≥ 2t + 11 and |Ar (Y ) | ≥ 2t + 11. +Proof. Let Rcw denote the (c, w)-subpath of Rzw. Observe that |Ar (Rcw) | ≥ 2t + 14. First, consider +the case when x lies in the (z, c)-subpath of Rzw. In this case, Rcw is a subpath of X and therefore +|Ar (X) | ≥ 2t + 14. Now consider the case when x lies in Rcw. In this case, applying Observation 14, +we have that |Ar (X) | ≥ |Ar (Rcw) | − 3 ≥ 2t + 11. Using a similar argument, we have that |Ar (Y ) | ≥ +2t + 11. +Let T (c1, c2) be the maximum (c1, c2)-subpath of T (c1, r) such that no internal vertex of T (c1, c2) is +adjacent to a vertex of Qr +b or Qr +u. We have the following claim. +Claim 15.4. The length of T (c1, c2) is at least t + 3. +Proof. Assume that the length of T (c1, c2) is at most t + 2 and x be a neighbour of c1 in Rzw. Observe +that all vertices of Rzw are at distance at least d (r, u) i.e. d (r, Rzw) ≥ d (r, u), since d (r, u) = d (r, P). +Hence, +(+) d (r, x) ≥ d (r, u) and d (r, c1) ≥ d (r, u) − 1. +8 + +Now, suppose c2 has a neighbor c3 in Qr +u. Hence d (c3, x) ≤ d (c3, c2) + d (c2, c1) + d (c1, x) ≤ t + 4. +Now, using (+) and the fact that c3 lies on an isometric (r, u)-path (Qr +u), we have that d (c3, u) ≤ t + 4. +Therefore, d (u, x) ≤ d (c3, u) + d (c3, x) ≤ 2t + 8. But this contradicts Proposition 9 and Claim 15.3, as +they together imply that d (u, x) is at least d (r, x) − d (r, u) + 2t + 10≥ 2t + 10. +Hence, c2 must have a neighbour c3 in Qr +b. First, assume that d (r, x) ≥ d (r, b). Then, as d (c3, x) ≤ +d (c3, c2) + d (c2, c1) + d (c1, x) ≤ t + 4 and c3 lies on an isometric (r, b)-path (Qr +b), we have that d (x, b) ≤ +2t + 8. But again this contradicts Proposition 9 and Claim 15.3, as they together imply that the length +of d (x, b) is at least d (r, x) − d (r, u) + 2t + 10. Now, assume that d (r, x) < d (r, b). Let b′ be a vertex of +Qr +b such that d (r, b′) = d (r, x). Using a similar argumentation as before, we have that d (x, b′) ≤ 2t + 8. +Hence, d (x, b) ≤ d (x, b′) + d (b′, b) ≤ d (r, b) − d (r, x) + 2t + 8. But this contradicts Proposition 9 which, +due to Claim 15.3, implies that d (x, b) ≥ d (r, b) − d (r, x) + 2t + 10. +The path T (c1, c2) forms the first ingredient to extract a (t + 1)-fat turtle. Let z1 be the neighbor of +z in Qr +u and w1 be the neighbour of w in Qr +b. We have the following claim. +Claim 15.5. The vertices w1 and z1 are non adjacent. +Proof. Recall that z1 lies in Qr +u and d (r, z) ≥ d (r, u). Hence z1 must be a neighbor of u. If w1 and z1 are +adjacent, then observe that d (u, b) ≤ d (r, b) − d (r, w1) + 2 ≤. This implies d (u, b) ≤ d (r, b) − d (r, u)+ 3. +But this shall again contradict Proposition 9. +Now we shall construct a (w1, z1)-path as follows: Consider the maximum (w1, w2)-subpath, say +T (w1, w2), of Qr +b such that no internal vertex of T (w1, w2) has a neighbour in Qr +u. Similarly, consider the +maximum (z1, z2)-subpath, say T (z1, z2), of Qr +b such that no internal vertex of T (z1, z2) is a neighbor of +w2. Let T be the path obtained by taking the union of T (w1, w2) and T (z1, z2). Observe that z2 must +be a neighbour of w2 and T is an induced (w1, z1)-path. The definitions of T and Rzw imply that their +union induces a cycle Z. Here we have the second and final ingredient to extract the (t + 1)-fat turtle. +Suppose that c2 has a neighbour in T . Let T ′ be the maximum subpath of T (c1, c2) which is vertex- +disjoint from Z. Due to Claim 15.4, the length of T ′ is at least t + 1. Let e1 and e2 be the end-vertices +of T ′. Observe the following. +• Each of e1 and e2 has at least one neighbor in Z. +• Z −{z, w} contains two distinct components C1, C2 such that for i ∈ {1, 2}, N(ei)∩V (Z) ⊆ V (Ci). +• For a vertex e′ +1 ∈ N(e1) ∩ V (Z) and e′ +2 ∈ N(e2) ∩ V (Z), the distance between e′ +1 and e′ +2 is at least +t + 1. This statement follows from Claim 15.3. +Hence, we have that the tuple (t + 1, Z, T ′, z, w) defines a (t + 1)-fat turtle. Now consider the case +when c2 does not have a neighbor in T . By definition, c2 has at least one neighbor in Qr +u or Qr +b. Without +loss of generality, assume that c2 has a neighbor c3 in Qr +u such that the (z2, c3)-subpath, say, T ′′ of Qr +u +has no neighbor of c2 other than c3. Observe that the path T ∗ = (T ′ ∪ (T ′′ − {z2})) is vertex-disjoint +from Z and has length at least t + 1. Let e1, e2 be the two end-vertices of T ∗. Observe the following. +• Each of e1 and e2 has at least one neighbor in Z. +• Z −{z, w} contains two distinct components C1, C2 such that for i ∈ {1, 2}, N(ei)∩V (Z) ⊆ V (Ci). +• For a vertex e′ +1 ∈ N(e1) ∩ V (Z) and e′ +2 ∈ N(e2) ∩ V (Z), the distance between e′ +1 and e′ +2 is at least +t + 1. This statement follows from Claim 15.3. +Hence, (t + 1, Z, T ∗, z, w) is a (t + 1)-fat turtle +Proof of Theorem 5 and 3: Lemma 13 and 15 together imply the theorems. +9 + +5 +Proof of Theorem 4 +Next we shall show that outerstring graphs are (4-theta, 4-prism, 4-pyramid)-free. +Lemma 16. Let G be an outerstring graph. Then, G is (4-theta, 4-prism, 4-pyramid)-free. +Proof. To prove the lemma, we shall need to recall a few definitions and results from the literature. A +graph G is a string graph if there is a collection S of simple curves on the plane and a bijection between +V (G) and S such that two curves in S intersect if and only if the corresponding vertices are adjacent in +G. Let G be a graph with an edge e. The graph G \ e is obtained by contracting the edge e into a single +vertex. Observe that string graphs are closed under edge contraction [22]. We shall use the following +result. +Proposition 17 ([22]). Let G be an outerstring graph with an edge e. Then G\e is an outerstring graph. +A full subdivision of a graph means replacing each edge of G with a new path of length at least two. +We shall use the following result implied from Theorem 1 of [22]. +Proposition 18 ([22]). Let G be a string graph. Then G does not contain a full subdivision of K3,3 as +an induced subgraph. +For a graph G, the graph G+ is constructed by introducing a new apex vertex a and connecting a +with all vertices of G by new copies of paths of length at least 2. We shall use the following result of +Biedl et al. [4]. +Proposition 19 (Lemma 1, [4]). A graph G is an outerstring graph if and only if G+ is a string graph. +Now we are ready to prove the lemma. +Let G be an outerstring graph. Assume for the sake of +contradiction that G contains an induced subgraph H which is a 4-theta, 4-pyramid, or a 4-prism. Since +every induced subgraph of an outerstring graph is also an outerstring graph, we have that H is an +outerstring graph. Let E be the set of edges of H whose both endpoints are part of some triangle. Now +consider the graph H1 = H \ E which is obtained by contracting all edges in E. By Proposition 17, H1 +is an outerstring graph and it is easy to check that H1 is a 3-theta. Let u and v be the vertices of H1 +with degree 3 and w1, w2, w3 be the set of mutually non-adjacent vertices such that for each i ∈ {1, 2, 3} +d (u, wi) = 2 and d (v, wi) ≥ 2. Since H1 is a 3-theta, w1, w2, w3 exist. Now consider the graph H+ +1 and +a be the new apex vertex. Due to Proposition 19, we have that H+ +1 is a string graph. But notice that, +for each pair of vertices in {x, y} ⊂ {w1, w2, w3, u, v, a}, there exists a unique path of length at least 2 +connecting x, y. This implies that H+ +1 (which is a string graph) contains a full subdivision of K3,3, which +contradicts Proposition 18. +Proof of Theorem 4: Lemma 16 and Theorem 5 together imply the theorem. +6 +Conclusion +In this paper, we derived upper bounds on the isometric path antichain cover number of three seemingly +(structurally) different classes of graphs, namely hyperbolic graphs, (theta,pyramid,prism)-free graphs +and outerstring graphs. We have not made any efforts in reducing the constants in our bounds. In +particular, we believe that a careful analysis of the structure of outerstring graphs would help in reducing +its isometric path antichain cover number. (Note that outerstring graphs may contain a theta, 2-pyramid +or a 2-prism). We note that the isometric path antichain cover number of a (n × n)-grid is Ω(n), which +implies that the isometric path antichain cover number of planar graphs (which are also string graphs) +is not bounded. Similarly, we note that the isometric path antichain cover number of G1, G2 and G3 +are unbounded where G1 denotes the class of (theta, prism)-free graphs, G2 denotes the class of (prism, +pyramid)-free graphs and G3 denotes the class of (theta, pyramid)-free graphs. An interesting direction +10 + +of research is to generalise the properties of hyperbolic graphs to graphs with bounded isometric path +antichain cover number. +We also note that recognizing graphs with a given value of isometric path antichain cover number +might be computationally hard. This problem does not seem to be in NP: to certify that a graph has +isometric path antichain cover number at most k, (intuitively) one would need to check, for all possible +isometric paths, that it does not contain any antichain of size k + 1 (with respect to all possible roots r). +On the contrary, it is in coNP: to certify that its isometric path antichain cover number is not at most k, +one may exhibit, for every possible root r, one isometric path and one antichain of size k + 1 contained +in the path. Checking the validity of this certificate can be done in polynomial time. We do not know if +the problem is coNP-hard. Nevertheless, this parameter seems interesting from a structural graph theory +point of view, since it encapsulates several seemingly unrelated graph classes with, as a consequence, +common algorithmic behaviours of these classes (recall that the value of the parameter does not need to +be computed for the approximation algorithm to work). Using our framework, perhaps other common +properties of these classes could be exhibited? +Our results imply a constant factor approximation algorithm for Isometric Path Cover on hyper- +bolic graphs, (theta, pyramid, prism)-free graphs and outerstring graphs. However, the existence of a +constant factor approximation algorithm for Isometric Path Cover on general graphs is not known +(it was observed that the algorithm from [7] also used here, can have non-constant approximation ratios, +for example on hypercube graphs, whose isometric path antichain cover numbers are unbounded). +Polynomial-time solvability of Isometric Path Cover on restricted graph classes like split graphs, +interval graphs, planar graphs etc. also remains unknown, see [7]. +Acknowledgement: We thank Nicolas Trotignon for suggesting us to study the class of (t-theta, t- +pyramid, t-prism)-free graphs. +References +[1] I. Abraham, C. Gavoille, A. Gupta, O. Neiman, and K. Talwar. Cops, robbers, and threatening +skeletons: Padded decomposition for minor-free graphs. SIAM Journal on Computing, 48(3):1120– +1145, 2019. +[2] M. Aigner and M. Fromme. A game of cops and robbers. Discrete Applied Mathematics, 8(1):1–12, +1984. +[3] J. M. Alonso, T. Brady, D. Cooper, V. 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Canadian Journal of Mathe- +matics, 19:1319–1328, 1967. +13 + diff --git a/4NAyT4oBgHgl3EQfcPdw/content/tmp_files/load_file.txt b/4NAyT4oBgHgl3EQfcPdw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1d48de9f738e3dc47612738b288da817c964fbdb --- /dev/null +++ b/4NAyT4oBgHgl3EQfcPdw/content/tmp_files/load_file.txt @@ -0,0 +1,698 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf,len=697 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content='00278v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content='CO] 31 Dec 2022 Isometric path antichain covers: beyond hyperbolic graphs∗ Dibyayan Chakraborty† Florent Foucaud‡ January 3, 2023 Abstract The isometric path antichain cover number of a graph G, denoted by ipacc (G), is a graph pa- rameter that was recently introduced to provide a constant factor approximation algorithm for Iso- metric Path Cover, whose objective is to cover all vertices of a graph with a minimum number of isometric paths (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' shortest paths between their end-vertices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' This parameter was previously shown to be bounded for chordal graphs and, more generally, for graphs of bounded chordality and bounded treelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' In this paper, we show that the isometric path antichain cover number remains bounded for graphs in three seemingly unrelated graph classes, namely, hyperbolic graphs, (theta, prism, pyramid)-free graphs, and outerstring graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Hyperbolic graphs are extensively studied in Metric Graph Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' The class of (theta, prism, pyramid)-free graphs are extensively studied in Structural Graph Theory, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' in the context of the Strong Perfect Graph Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' The class of outerstring graphs is studied in Geometric Graph Theory and Computational Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Our results imply a constant factor approximation algorithm for Isometric Path Cover on all the above graph classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Our results also show that the distance functions of these (structurally) different graph classes are more similar than previously thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' 1 Introduction A path is isometric if it is a shortest path between its endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' An isometric path cover of a graph G is a set of isometric paths such that each vertex of G belongs to at least one of the paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' The isometric path number of G is the smallest size of an isometric path cover of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Given a graph G and an integer k, the objective of the algorithmic problem Isometric Path Cover is to decide if there exists an isometric path cover of cardinality at most k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Isometric Path Cover has been introduced and studied in the context of pursuit-evasion games [1, 2] and used in the context of Product Structure Theorems [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' The goal of this paper is to continue the study of approximation algorithms for Isometric Path Cover on several graph classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' We do so by continuing the study of a recently introduced graph parameter which seems interesting in its own right, as it encapsulates several previously unrelated graph classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Isometric Path Cover has also been studied from a structural point of view: the cardinalities of the optimal solution have been determined for square grids [17], hypercubes [18], complete r-partite graphs [24] and Cartesian products of complete graphs [24], and it was recently proved that the pathwidth of a graph is always upper-bounded by the size of its smallest isometric path cover [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' However, until recently the algorithmic aspects of Isometric Path Cover remained unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' The problem is easy to solve on trees and more generally, on block graphs [23] but remains hard on chordal graph, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' graphs without any induced cycle of length at least 4 [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' It can be approximated in polynomial time within a factor of log(d) for graphs of diameter d by a greedy algorithm [27] and solved in polynomial time for every ∗This research was partially financed by the IFCAM project “Applications of graph homomorphisms” (MA/IFCAM/18/39), the ANR project GRALMECO (ANR-21-CE48-0004) and the French government IDEX-ISITE ini- tiative 16-IDEX-0001 (CAP 20-25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' †Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France ‡Université Clermont-Auvergne, CNRS, Mines de Saint-Étienne, Clermont-Auvergne-INP, LIMOS, 63000 Clermont- Ferrand, France 1 fixed value of k by an XP algorithm [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' In a quest to find constant factor approximation algorithms for Isometric Path Cover, Chakraborty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' [7] introduced a parameter called the isometric path antichain cover number of graphs, denoted by ipacc (G) (see Section 2 for a definition) and proved a result directly implying the following (see [7, Proposition 10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Proposition 1 ([7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' For a graph G, if ipacc (G) ≤ c, then Isometric Path Cover admits a polynomial- time c-approximation algorithm on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Proposition 1 is proved by a simple approximation algorithm described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' For each vertex r of the graph, perform a Breadth-First Search at this vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Remove edges joining any vertices at the same distance from r, and orient all edges towards r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' The resulting directed acyclic graph can be seen as the Hasse diagram of a poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Compute a chain covering of that poset using classic methods related to Dilworth’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' The chains are the isometric paths of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Keep the smallest of all solutions over all choices of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Using Proposition 1, the above algorithm was shown to be a constant factor approximation algo- rithm for many graph classes, including interval graphs, chordal graphs, and more generally, graphs with bounded treelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Indeed, on all these graph classes, the isometric path antichain cover number is shown to be bounded by a constant (note that one does not need to compute this parameter for the algorithm to function: it serves only in the analysis of the approximation ratio of the algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' As noted in [7], this parameter may be unbounded on general graphs, for example for the class of hypercubes or square grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' In this paper, we continue to study the boundedness of the isometric path antichain cover number of various graph classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Specifically, we consider three structurally unrelated graph classes, namely, hyperbolic graphs, (theta, prism, pyramid)-free graphs, and outerstring graphs, which extends the above work to strictly larger graph classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Hyperbolic graphs: A graph G is said to be δ-hyperbolic [19] if for any four vertices u, v, x, y, the two larger of the three distance sums d (u, v)+d (x, y), d (u, x)+d (v, y) and d (u, y)+d (v, x) differ by at most 2δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' A graph class G is hyperbolic if there exists a constant δ such that every graph G ∈ G is δ-hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' This parameter was first introduced by Gromov in the context of automatic groups [19] in relation with their Cayley graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' The hyperbolicity of a tree is 0, and in general, hyperbolicity seems to measure how much the distance function of a graph deviates from a tree metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Many structurally defined graph classes like chordal graphs, cocomparability graphs, asteroidal-triple free graphs, graphs with bounded chordality or treelength are hyperbolic graphs [8, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Moreover, hyperbolicity has been found to capture important properties of several large practical graphs such as the Internet [26] or database relations [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Due to its importance in discrete mathematics, algorithms, metric graph theory, researchers have studied various algorithmic aspects of hyperbolic graphs [8, 12, 9, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Note that graphs with diameter 2 are hyperbolic, which may contain any graph as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' (theta, prism, pyramid)-free graphs: A theta is a graph made of three vertex-disjoint induced paths P1 = a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' b, P2 = a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' b, P3 = a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' b of lengths at least 2, and such that no edges exist between the paths except the three edges incident to a and the three edges incident to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' See Figure 2 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' A pyramid is a graph made of three induced paths P1 = a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' b1, P2 = a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' b2, P3 = a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' b3, two of which have lengths at least 2, vertex-disjoint except at a, and such that b1b2b3 is a triangle and no edges exist between the paths except those of the triangle and the three edges incident to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' A prism is a graph made of three vertex-disjoint induced paths P1 = a1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' b1, P2 = a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' b2, P3 = a3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' b3 of lengths at least 1, such that a1a2a3 and b1b2b3 are triangles and no edges exist between the paths except those of the two triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' A graph G is (theta, pyramid, prism)-free if G does not contain any induced subgraph isomorphic to a theta, pyramid or prism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' A graph is a 3-path configuration if it is a theta, pyramid or prism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' The study of 3-path configurations dates back to the works of Watkins and Meisner [32] in 1967 and plays “special roles” in the proof of the celebrated Strong Perfect Graph Theorem [10, 14, 28, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Important graph classes like chordal graphs, circular arc graphs, universally-signable graphs [11] exclude all 3-path configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Popular graph classes like perfect graphs, even hole-free graphs exclude some 2 Bounded isometric path antichain cover number bounded hyperbolicity * (t-theta, t-prism, t-pyramid)- free * Outerstring * circle * (theta,prism,pyramid)- free * Universally signable * bounded tree-length bounded chordality bounded diameter chordal AT-free Interval circular arc * Permutation Figure 1: Inclusion diagram for graph classes discussed here (and related ones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' If a class A has an upward path to class B, then A is included in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' For graphs in the gray classes, the complexity of Isometric Path Cover is open;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' for all other graph classes, it is NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' For all shown graph classes, Isometric Path Cover is constant-factor approximable in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Constant factor approximation algorithms for Isometric Path Cover on graph classes marked with * are contributions of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' b a b1 b2 b3 a b1 b2 b3 a1 a2 a3 (a) (b) (c) (d) Figure 2: (a) Theta, (b) Pyramid, (c) Prism, (d) Outerstrings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' The figure shows that the graph K2,3, which is also a theta, is an outerstring graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' of the 3-path configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Note that, (theta, prism, pyramid)-free graphs are not hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' To see this, consider a cycle C of order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Clearly, C excludes all 3-path configurations and has hyperbolicity Ω(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Outerstring graphs: A set S of simple curves on the plane is grounded if there exists a horizontal line containing one endpoint of each of the curves in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' A graph G is an outerstring graph if there is a collection C of grounded simple curves and a bijection between V (G) and C such that two curves in S if and only if the corresponding vertices are adjacent in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' See Figure 2(d) for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' The term “outerstring graph” was first used in the early 90’s [22] in the context of studying intersection graphs of simple curves on the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Many well-known graph classes like chordal graphs, circular arc graphs, circle graphs (intersection graphs of chords of a circle), or cocomparability graphs are also outerstring graphs and thus, motivated researchers from the geometric graph theory and computational geometry communities to study algorithmic and structural aspects of outerstring graphs and its subclasses [4, 5, 6, 20, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Note that, in general, outerstring graphs may contain a prism, pyramid or theta as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Moreover, cycles of arbitrary order are outerstring graphs, implying that outerstring graphs are not hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' It is clear from the above discussion that the classes of hyperbolic graphs, (theta, prism, pyramid)-free 3 graphs, and outerstring graphs are pairwise incomparable (with respect to the containment relationship).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content='1 Our contributions The main contribution of this paper is to show that the isometric path antichain cover number (see Section 2 for a definition) remains bounded on hyperbolic graphs, (theta, pyramid, prism)-free graphs, and outerstring graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Specifically, we prove the following theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let G be a graph with hyperbolicity δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Then, ipacc (G) ≤ 12δ + 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let G be a (theta, pyramid, prism)-free graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Then, ipacc (G) ≤ 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let G be an outerstring graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Then, ipacc (G) ≤ 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' To the best of our knowledge, the isometric path antichain cover number being bounded (by con- stant(s)) is the only known non-trivial property shared by any two or all three of these graph classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' To provide a unified proof of Theorem 3 and 4, we study a more general graph class called (t-theta, t-pyramid, t-prism)-free graphs [29] (see Section 4 for definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' When t = 1, (t-theta, t-pyramid, t- prism)-free graphs are exactly (theta, prism, pyramid)-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Moreover, we show that all outerstring graphs are (4-theta, 4-pyramid, 4-prism)-free graphs (Lemma 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' We prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' For t ≥ 1, let G be a (t-theta, t-pyramid, t-prism)-free graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Then ipacc (G) ≤ 8t + 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Due to Proposition 1 and the above theorems, we also have the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' There is an approximation algorithm for Isometric Path Cover with approximation ratio (a) 12δ + 6 on δ-hyperbolic graphs, (b) 73 on (theta, prism, pyramid)-free graphs, and (c) 95 on outerstring graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' (d) 8t + 63 on (t-theta, t-pyramid, t-prism)-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Organisation: In Section 2, we introduce the recall some definitions and some results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' In Section 3 we prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' In Section 4, we prove Theorems 3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' In Section 5, we prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' We conclude in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' 2 Definitions and preliminary observations In this section, we formally recall the definition of isometric path antichain cover number of graphs from [7] and some related observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' A sequence of distinct vertices forms a path P if any two consecutive vertices are adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Whenever we fix a path P of G, we shall refer to the subgraph formed by the edges between the consecutive vertices of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' The length of a path P, denoted by |P|, is the number of its vertices minus one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' A path is induced if there are no graph edges joining non-consecutive vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' In a directed graph, a directed path is a path in which all arcs are oriented in the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' For a path P of a graph G between two vertices u and v, the vertices V (P) \\ {u, v} are internal vertices of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' A path between two vertices u and v is called a (u, v)-path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Similarly, we have the notions of isometric (u, v)-path and induced (u, v)-path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' For a vertex r of G and a set S of vertices of G, the distance of S from r, denoted as d (r, S), is the minimum of the distance between any vertex of S and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' For a subgraph H of G, the distance of H w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' r is d (r, V (H)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Formally, we have d (r, S) = min{d (r, v) : v ∈ S} and d (r, H) = d (r, V (H)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' 4 For a graph G and a vertex r ∈ V (G), consider the following operations on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' First, remove all edges xy from G such that d (r, x) = d (r, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let G′ r be the resulting graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Then, for each edge e = xy ∈ E(G′ r) with d (r, x) = d (r, y) − 1, orient e from y to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let −→ Gr be the directed acyclic graph formed after applying the above operation on G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Note that this digraph can easily be computed in linear time using a Breadth-First Search (BFS) traversal with starting vertex r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' The following definition is inspired by the terminology of posets (as the graph −→ Gr can be seen as the Hasse diagram of a poset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' For a graph G and a vertex r ∈ V (G), two vertices x, y ∈ V (G) are antichain vertices if there are no directed paths from x to y or from y to x in −→ Gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' A set X of vertices of G is an antichain set if any two vertices in X are antichain vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' The cardinality of the largest antichain set in −→ Gr will be denoted by β �−→ Gr � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' The cardinality of the largest antichain set of G, is defined as β (G) = min � β �−→ Gr � : r ∈ V (G) � Definition 8 ([7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let r be a vertex of a graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' For a subgraph H, Ar (H) shall denote the maximum antichain set of H in −→ Gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' The isometric path antichain cover number of −→ Gr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' denoted by ipacc �−→ Gr � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' is defined as follows: ipacc �−→ Gr � = max {|Ar (P) |: P is an isometric path} The isometric path antichain cover number of graph G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' denoted as ipacc (G),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' is defined as the minimum over all possible antichain covers of its associated directed acyclic graphs: ipacc (G) = min � ipacc �−→ Gr � : r ∈ V (G) � We recall the proof of the following proposition from [7] which will be used heavily in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Proposition 9 ([7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let G be a graph and r, an arbitrary vertex of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Consider the directed acyclic graph −→ Gr, and let P be an isometric path between two vertices x and y in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Then |P| ≥ |d (r, x) − d (r, y) | + |Ar (P) | − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Orient the edges of P from y to x in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' First, observe that P must contain a set E1 of oriented edges such that |E1| = |d (r, y) − d (r, x) | and for any −→ ab ∈ E1, d (r, a) = d (r, b) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let the vertices of the largest antichain set of P in −→ Gr, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=', Ar (P), be ordered as a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' , at according to their occurrence while traversing P from y to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' For i ∈ [2, t], let Pi be the subpath of P between ai−1 and ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Observe that for any i ∈ [2, t], since ai and ai−1 are antichain vertices, there must exist an oriented edge −→ bici ∈ E(Pi) such that either d (r, bi) = d (r, ci) or d (r, bi) = d (r, ci) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let E2 = {bici}i∈[2,t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Observe that E1 ∩ E2 = ∅ and therefore |P| ≥ |E1| + |E2| = |d (r, y) − d (r, x) | + |Ar (P) | − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' 3 Proof of Theorem 2 In this section, we shall show that isometric path antichain cover number of graphs with hyperbolicity at most δ is at most 12δ + 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' To achieve our goal we need to recall a few definitions from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' For three vertices x, y, z of a graph G, a geodesic triangle [3], denoted as ∆(x, y, z) is the union P(x, y) ∪ P(y, z)∪P(x, z) of three isometric paths connecting these vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' A geodesic triangle ∆(x, y, z) is called ρ-slim if for any vertex u ∈ P(x, y) the distance d (u, P(y, z) ∪ P(x, z)) is at most ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' The smallest value of ρ for which every geodesic triangle of G is ρ-slim is called the slimness of G and is denoted by sl (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' In the following lemma, we shall show that if the isometric path antichain cover number of a graph is large then so is the slimness of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' For any graph G, ipacc (G) ≤ 4sl (G) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' 5 u v c c′ Figure 3: An example of a 4-fat turtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let C be the cycle induced by the black vertices, P be the path induced by the white vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Then the tuple (4, C, P, c, c′) defines a 4-fat turtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let ρ = sl (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Aiming for a contradiction, let r be a vertex of G such that there exists an isometric path P such that |Ar (P) | ≥ 4ρ + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let the vertices of Ar (P) be named and ordered as a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' , a2ρ+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' , a4ρ+3 as they are encountered while traversing P from one end-vertex to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let x = a1, y = a4ρ+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let −→ Px be an oriented path from x to r in −→ Gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Observe that Px, the path of G obtained by removing the orientation of −→ Px, is an (x, r)-isometric path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let −→ Py be an oriented path from y to r in −→ Gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Similarly, Py, the path of G obtained by removing the orientation of −→ Py, is an (y, r)- isometric path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Observe that P, Px, Py form a geodesic triangle with x, r, y as end-vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Consider the vertex z = a2ρ+2 on the path P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Since ρ = sl (G), there exists a vertex w ∈ V (Px) ∪ V (Py) such that d (w, z) ≤ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Without loss of generality, assume w ∈ V (Px).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Then, d (x, z) ≤ d (x, w) + d (w, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' By using that d (r, z) ≤ d (r, w) + d (w, z) ≤ d (r, w) + ρ, we get d (x, z) ≤ |d (r, x) − d (r, z) | + 2ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' But this contradicts Proposition 9, due to which we have d (x, z) ≥ |d (r, x) − d (r, z) | + 2ρ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Now we shall use the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Proposition 11 ([3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' For any graph G, sl (G) ≤ 3hb (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Proposition 11 and Lemma 10, imply the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' 4 Proofs of Theorem 3 and 5 In this section, we shall prove Theorems 3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' First we shall define the notions of t-theta, t-prism, and t-pyramid [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' For an integer t ≥ 1, a t-prism is a graph made of three vertex-disjoint induced paths P1 = a1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' b1, P2 = a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' b2, P3 = a3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' b3 of lengths at least t, such that a1a2a3 and b1b2b3 are triangles and no edges exist between the paths except those of the two triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' For an integer t ≥ 1, a t-pyramid is a graph made of three induced paths P1 = a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' b1, P2 = a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' b2, P3 = a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' b3 of lengths at least t, two of which have lengths at least t + 1, they are pairwise vertex-disjoint except at a, such that b1b2b3 is a triangle and no edges exist between the paths except those of the triangle and the three edges incident to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' For an integer t ≥ 1, a t-theta is a graph made of three internally vertex-disjoint induced paths P1 = a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' b, P2 = a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' b, P3 = a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' b of lengths at least t+1, and such that no edges exist between the paths except the three edges incident to a and the three edges incident to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' A graph G is (t-theta, t-pyramid, t-prism)-free if G does not contain any induced subgraph isomorphic to a t-theta, t-pyramid or t-prism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' When t = 1, (t-theta, t-pyramid, t-prism)-free graphs are exactly (theta, prism, pyramid)-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Now we shall show that the isometric path antichain cover number of (t-theta, t-pyramid, t-prism)- free graphs are bounded above by a linear function on t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' We shall show that, when the isometric path antichain cover number of a graph is large, the existence of a structure called “t-fat turtle” (defined below) as an induced subgraph is forced, which, cannot be present in a ((t − 1)-theta, (t − 1)-pyramid, (t − 1)-prism)-free graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Definition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' For an integer t ≥ 1, a “t-fat turtle” consists of a cycle C and an induced (u, v)-path P of length at least t such that all of the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' 6 (a) V (P) ∩ V (C) = ∅, (b) For any vertex w ∈ (V (P) \\ {u, v}), N(w) ∩ V (C) = ∅ and both u and v have at least one neighbour in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' (c) For any vertex w ∈ N(u) ∩ V (C) and w′ ∈ N(v) ∩ V (C), the distance between w and w′ in C is at least t, (d) There exist two vertices {c, c′} ⊂ V (C) and two distinct components Cu, Cv of C − {c, c′} such that N(u) ∩ V (C) ⊆ V (Cu) and N(v) ∩ V (C) ⊆ V (Cv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' The tuple (t, C, P, c, c′) defines the t-fat turtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' See Figure 3 for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' In the following observation, we show that any (t-theta, t-pyramid,t-prism)-free graph cannot contain a (t + 1)-fat turtle as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' For some integer t ≥ 1, let G be a graph containing a (t + 1)-fat turtle as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Then G is not (t-theta, t-pyramid, t-prism)-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let (t+1, C, P, c, c′) be a (t+1)-fat turtle in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let the vertices of C be named c = a0, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' , ak = c′, ak+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' , a|V (C)| as they are encountered while traversing C starting from c in a counter-clockwise manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Denote by u, v the end-vertices of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' By definition, there exist two distinct components Cu, Cv of C−{c, c′} such that N(u)∩V (C) ⊆ V (Cu) and N(v)∩V (C) ⊆ V (Cv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Without loss of generality, assume V (Cu) = {a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' , ak−1} and V (Cv) = {ak+1, ak+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' , a|V (C)|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let i− and i+ be the minimum and maximum indices such that ai− and ai+ are adjacent to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let j− and j+ be the minimum and maximum indices such that aj− and aj+ are adjacent to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' By definition, i− ≤ i+ < j− ≤ j+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let P1 be the (ai−, aj+)-subpath of C containing c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let P2 be the (ai+, aj−)-subpath of C that contains c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Observe that P1 and P2 have length at least t (by definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Now we show that P, P1, P2 together form one of theta, pyramid or prism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' If ai− = ai+ and aj− = aj+, then P, P1, P2 form a t-theta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' If i− ≤ i+ − 2 and j− ≤ j+ − 2, then also P, P1, P2 form a t-theta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' If j− = j+ − 1 and i− = i+ − 1, then P, P1, P2 form a t-prism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' In any other case, P, P1, P2 form a t-pyramid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' In the remainder of this section, we shall prove that there exists a linear function f(t) such that if the isometric path antichain cover number of a graph is more than f(t), then G is forced to contain a (t + 1)-fat turtle as an induced subgraph, and therefore is not (t-theta, t-pyramid,t-prism)-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' We shall use the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Observation 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let G be a graph, r be an arbitrary vertex, P be an isometric (u, v)-path in G and Q be a subpath of an isometric (v, r)-path in G such that one endpoint of Q is v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let P ′ be the maximum (u, w)-subpath of P such that no internal vertex of P ′ is a neighbour of some vertex of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' We have that |Ar (P ′) | ≥ |Ar (P) | − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Suppose |Ar (P ′) | ≤ |Ar (P) | − 4 and consider the (w, v)-subpath, say P ′′, of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Observe that |Ar (P ′′) | ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Now let w′ be a vertex of Q which is a neighbour of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Observe that |d (r, w)−d (r, w′) | ≤ 1 and therefore d (w, v) = |E(P ′′)| ≤ |d (r, w)−d (r, v) |+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' But this contradicts Proposition 9, which implies that the length of P ′′ is at least |d (r, w) − d (r, v) | + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' For an integer t ≥ 1, let G be a graph with ipacc (G) ≥ 8t + 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Then G has a (t + 1)-fat turtle as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let r be a vertex of G such that ipacc �−→ Gr � is at least 8t+64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Then there exists an isometric path P such that |Ar (P) | ≥ 8t+ 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let the two endpoints of P be a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' (See Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=') Let u be a vertex of P such that d (r, u) = d (r, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let Pau be the (a, u)-subpath of P and Pbu be the (b, u)-subpath of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Both Pau and Pbu are isometric paths and observe that either |Ar (Pau) | ≥ 4t+32 or |Ar (Pbu) | ≥ 4t+32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Without loss of generality, assume that |Ar (Pbu) | ≥ 4t + 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let Qr b be an isometric (b, r)-path in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' 7 r z2 w2 u z z1 w b w1 c (= a2t+13) x c1 a c2 T (c1, c2) ≥ t ≥ t ≥ t Qr b Qr u Figure 4: Illustration of the notations used in the proof of Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let Ruw be the maximum (u, w)-subpath, of Pbu such that no internal vertex of Ruw is a neighbour of Qr b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Note that Ruw is an isometric path and w has a neighbour in Qr b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Applying Observation 14, we have the following: Claim 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' |Ar (Ruw) | ≥ 4t + 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let Qr u be any isometric (u, r)-path of G and let Rzw be the maximum (z, w)-subpath of Ruw such that no internal vertex of Rzw has a neighbour in Qr u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Observe that Rzw is an isometric path, and z has a neighbour in Qr u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Again applying Observation 14, we have the following: Claim 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' |Ar (Rzw) | ≥ 4t + 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' , ak be the vertices of Ar (Rzw) ordered according to their appearance while traversing Rzw from z to w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Due to Claim 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content='2, we have that k ≥ 4t + 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let c = a2t+13 and Qr c denote an isometric (c, r)-path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let T (r, c1) be the maximum subpath of Qr c such that no internal vertex of T (r, c1) is adjacent to any vertex of Rzw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Claim 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let x be a neighbor of c1 in Rzw, X be the (x, b)-subpath of Pub and Y be the (x, u)-subpath of Pub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Then |Ar (X) | ≥ 2t + 11 and |Ar (Y ) | ≥ 2t + 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let Rcw denote the (c, w)-subpath of Rzw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Observe that |Ar (Rcw) | ≥ 2t + 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' First, consider the case when x lies in the (z, c)-subpath of Rzw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' In this case, Rcw is a subpath of X and therefore |Ar (X) | ≥ 2t + 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Now consider the case when x lies in Rcw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' In this case, applying Observation 14, we have that |Ar (X) | ≥ |Ar (Rcw) | − 3 ≥ 2t + 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Using a similar argument, we have that |Ar (Y ) | ≥ 2t + 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let T (c1, c2) be the maximum (c1, c2)-subpath of T (c1, r) such that no internal vertex of T (c1, c2) is adjacent to a vertex of Qr b or Qr u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' We have the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Claim 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' The length of T (c1, c2) is at least t + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Assume that the length of T (c1, c2) is at most t + 2 and x be a neighbour of c1 in Rzw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Observe that all vertices of Rzw are at distance at least d (r, u) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' d (r, Rzw) ≥ d (r, u), since d (r, u) = d (r, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Hence, (+) d (r, x) ≥ d (r, u) and d (r, c1) ≥ d (r, u) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' 8 Now, suppose c2 has a neighbor c3 in Qr u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Hence d (c3, x) ≤ d (c3, c2) + d (c2, c1) + d (c1, x) ≤ t + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Now, using (+) and the fact that c3 lies on an isometric (r, u)-path (Qr u), we have that d (c3, u) ≤ t + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Therefore, d (u, x) ≤ d (c3, u) + d (c3, x) ≤ 2t + 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' But this contradicts Proposition 9 and Claim 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content='3, as they together imply that d (u, x) is at least d (r, x) − d (r, u) + 2t + 10≥ 2t + 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Hence, c2 must have a neighbour c3 in Qr b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' First, assume that d (r, x) ≥ d (r, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Then, as d (c3, x) ≤ d (c3, c2) + d (c2, c1) + d (c1, x) ≤ t + 4 and c3 lies on an isometric (r, b)-path (Qr b), we have that d (x, b) ≤ 2t + 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' But again this contradicts Proposition 9 and Claim 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content='3, as they together imply that the length of d (x, b) is at least d (r, x) − d (r, u) + 2t + 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Now, assume that d (r, x) < d (r, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let b′ be a vertex of Qr b such that d (r, b′) = d (r, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Using a similar argumentation as before, we have that d (x, b′) ≤ 2t + 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Hence, d (x, b) ≤ d (x, b′) + d (b′, b) ≤ d (r, b) − d (r, x) + 2t + 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' But this contradicts Proposition 9 which, due to Claim 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content='3, implies that d (x, b) ≥ d (r, b) − d (r, x) + 2t + 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' The path T (c1, c2) forms the first ingredient to extract a (t + 1)-fat turtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let z1 be the neighbor of z in Qr u and w1 be the neighbour of w in Qr b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' We have the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Claim 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' The vertices w1 and z1 are non adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Recall that z1 lies in Qr u and d (r, z) ≥ d (r, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Hence z1 must be a neighbor of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' If w1 and z1 are adjacent, then observe that d (u, b) ≤ d (r, b) − d (r, w1) + 2 ≤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' This implies d (u, b) ≤ d (r, b) − d (r, u)+ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' But this shall again contradict Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Now we shall construct a (w1, z1)-path as follows: Consider the maximum (w1, w2)-subpath, say T (w1, w2), of Qr b such that no internal vertex of T (w1, w2) has a neighbour in Qr u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Similarly, consider the maximum (z1, z2)-subpath, say T (z1, z2), of Qr b such that no internal vertex of T (z1, z2) is a neighbor of w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let T be the path obtained by taking the union of T (w1, w2) and T (z1, z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Observe that z2 must be a neighbour of w2 and T is an induced (w1, z1)-path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' The definitions of T and Rzw imply that their union induces a cycle Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Here we have the second and final ingredient to extract the (t + 1)-fat turtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Suppose that c2 has a neighbour in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let T ′ be the maximum subpath of T (c1, c2) which is vertex- disjoint from Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Due to Claim 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content='4, the length of T ′ is at least t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let e1 and e2 be the end-vertices of T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Observe the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Each of e1 and e2 has at least one neighbor in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Z −{z, w} contains two distinct components C1, C2 such that for i ∈ {1, 2}, N(ei)∩V (Z) ⊆ V (Ci).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' For a vertex e′ 1 ∈ N(e1) ∩ V (Z) and e′ 2 ∈ N(e2) ∩ V (Z), the distance between e′ 1 and e′ 2 is at least t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' This statement follows from Claim 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Hence, we have that the tuple (t + 1, Z, T ′, z, w) defines a (t + 1)-fat turtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Now consider the case when c2 does not have a neighbor in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' By definition, c2 has at least one neighbor in Qr u or Qr b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Without loss of generality, assume that c2 has a neighbor c3 in Qr u such that the (z2, c3)-subpath, say, T ′′ of Qr u has no neighbor of c2 other than c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Observe that the path T ∗ = (T ′ ∪ (T ′′ − {z2})) is vertex-disjoint from Z and has length at least t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let e1, e2 be the two end-vertices of T ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Observe the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Each of e1 and e2 has at least one neighbor in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Z −{z, w} contains two distinct components C1, C2 such that for i ∈ {1, 2}, N(ei)∩V (Z) ⊆ V (Ci).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' For a vertex e′ 1 ∈ N(e1) ∩ V (Z) and e′ 2 ∈ N(e2) ∩ V (Z), the distance between e′ 1 and e′ 2 is at least t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' This statement follows from Claim 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Hence, (t + 1, Z, T ∗, z, w) is a (t + 1)-fat turtle Proof of Theorem 5 and 3: Lemma 13 and 15 together imply the theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' 9 5 Proof of Theorem 4 Next we shall show that outerstring graphs are (4-theta, 4-prism, 4-pyramid)-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let G be an outerstring graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Then, G is (4-theta, 4-prism, 4-pyramid)-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' To prove the lemma, we shall need to recall a few definitions and results from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' A graph G is a string graph if there is a collection S of simple curves on the plane and a bijection between V (G) and S such that two curves in S intersect if and only if the corresponding vertices are adjacent in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let G be a graph with an edge e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' The graph G \\ e is obtained by contracting the edge e into a single vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Observe that string graphs are closed under edge contraction [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' We shall use the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Proposition 17 ([22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let G be an outerstring graph with an edge e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Then G\\e is an outerstring graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' A full subdivision of a graph means replacing each edge of G with a new path of length at least two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' We shall use the following result implied from Theorem 1 of [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Proposition 18 ([22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let G be a string graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Then G does not contain a full subdivision of K3,3 as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' For a graph G, the graph G+ is constructed by introducing a new apex vertex a and connecting a with all vertices of G by new copies of paths of length at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' We shall use the following result of Biedl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Proposition 19 (Lemma 1, [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' A graph G is an outerstring graph if and only if G+ is a string graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Now we are ready to prove the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let G be an outerstring graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Assume for the sake of contradiction that G contains an induced subgraph H which is a 4-theta, 4-pyramid, or a 4-prism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Since every induced subgraph of an outerstring graph is also an outerstring graph, we have that H is an outerstring graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let E be the set of edges of H whose both endpoints are part of some triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Now consider the graph H1 = H \\ E which is obtained by contracting all edges in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' By Proposition 17, H1 is an outerstring graph and it is easy to check that H1 is a 3-theta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Let u and v be the vertices of H1 with degree 3 and w1, w2, w3 be the set of mutually non-adjacent vertices such that for each i ∈ {1, 2, 3} d (u, wi) = 2 and d (v, wi) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Since H1 is a 3-theta, w1, w2, w3 exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Now consider the graph H+ 1 and a be the new apex vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Due to Proposition 19, we have that H+ 1 is a string graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' But notice that, for each pair of vertices in {x, y} ⊂ {w1, w2, w3, u, v, a}, there exists a unique path of length at least 2 connecting x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' This implies that H+ 1 (which is a string graph) contains a full subdivision of K3,3, which contradicts Proposition 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Proof of Theorem 4: Lemma 16 and Theorem 5 together imply the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' 6 Conclusion In this paper, we derived upper bounds on the isometric path antichain cover number of three seemingly (structurally) different classes of graphs, namely hyperbolic graphs, (theta,pyramid,prism)-free graphs and outerstring graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' We have not made any efforts in reducing the constants in our bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' In particular, we believe that a careful analysis of the structure of outerstring graphs would help in reducing its isometric path antichain cover number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' (Note that outerstring graphs may contain a theta, 2-pyramid or a 2-prism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' We note that the isometric path antichain cover number of a (n × n)-grid is Ω(n), which implies that the isometric path antichain cover number of planar graphs (which are also string graphs) is not bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Similarly, we note that the isometric path antichain cover number of G1, G2 and G3 are unbounded where G1 denotes the class of (theta, prism)-free graphs, G2 denotes the class of (prism, pyramid)-free graphs and G3 denotes the class of (theta, pyramid)-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' An interesting direction 10 of research is to generalise the properties of hyperbolic graphs to graphs with bounded isometric path antichain cover number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' We also note that recognizing graphs with a given value of isometric path antichain cover number might be computationally hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' This problem does not seem to be in NP: to certify that a graph has isometric path antichain cover number at most k, (intuitively) one would need to check, for all possible isometric paths, that it does not contain any antichain of size k + 1 (with respect to all possible roots r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' On the contrary, it is in coNP: to certify that its isometric path antichain cover number is not at most k, one may exhibit, for every possible root r, one isometric path and one antichain of size k + 1 contained in the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Checking the validity of this certificate can be done in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' We do not know if the problem is coNP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Nevertheless, this parameter seems interesting from a structural graph theory point of view, since it encapsulates several seemingly unrelated graph classes with, as a consequence, common algorithmic behaviours of these classes (recall that the value of the parameter does not need to be computed for the approximation algorithm to work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Using our framework, perhaps other common properties of these classes could be exhibited?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Our results imply a constant factor approximation algorithm for Isometric Path Cover on hyper- bolic graphs, (theta, pyramid, prism)-free graphs and outerstring graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' However, the existence of a constant factor approximation algorithm for Isometric Path Cover on general graphs is not known (it was observed that the algorithm from [7] also used here, can have non-constant approximation ratios, for example on hypercube graphs, whose isometric path antichain cover numbers are unbounded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Polynomial-time solvability of Isometric Path Cover on restricted graph classes like split graphs, interval graphs, planar graphs etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' also remains unknown, see [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Acknowledgement: We thank Nicolas Trotignon for suggesting us to study the class of (t-theta, t- pyramid, t-prism)-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' References [1] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Abraham, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Gavoille, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Gupta, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Neiman, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Talwar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'} +page_content=' Cops, 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+Radicalisation, and Politicised Hate Speech +JAROD GOVERS, ORKA Lab, Department of Software Engineering, University of Waikato, NZ +PHILIP FELDMAN, ASRC Federal, US +AARON DANT, ASRC Federal, US +PANOS PATROS, ORKA Lab, Department of Software Engineering, University of Waikato, NZ +Social media is a modern person’s digital voice to project and engage with new ideas and mobilise communities— +a power shared with extremists. Given the societal risks of unvetted content-moderating algorithms for +Extremism, Radicalisation, and Hate speech (ERH) detection, responsible software engineering must understand +the who, what, when, where, and why such models are necessary to protect user safety and free expression. +Hence, we propose and examine the unique research field of ERH context mining to unify disjoint studies. +Specifically, we evaluate the start-to-finish design process from socio-technical definition-building and dataset +collection strategies to technical algorithm design and performance. Our 2015-2021 51-study Systematic +Literature Review (SLR) provides the first cross-examination of textual, network, and visual approaches +to detecting extremist affiliation, hateful content, and radicalisation towards groups and movements. We +identify consensus-driven ERH definitions and propose solutions to existing ideological and geographic +biases, particularly due to the lack of research in Oceania/Australasia. Our hybridised investigation on Natural +Language Processing, Community Detection, and visual-text models demonstrates the dominating performance +of textual transformer-based algorithms. We conclude with vital recommendations for ERH context mining +researchers and propose an uptake roadmap with guidelines for researchers, industries, and governments to +enable a safer cyberspace. +CCS Concepts: • Computing methodologies → Discourse, dialogue and pragmatics; Lexical semantics; +Information extraction; Machine learning; Natural language processing; Knowledge representa- +tion and reasoning; Image and video acquisition; • Applied computing → Sociology; • Social and profes- +sional topics → Hate speech; Technology and censorship; Political speech; Governmental regulations. +Additional Key Words and Phrases: extremism, radicalisation, machine learning, community detection, natural +language processing, neural networks, hate speech, sociolinguistics +ACM Reference Format: +Jarod Govers, Philip Feldman, Aaron Dant, and Panos Patros. 2021. Down the Rabbit Hole: Detecting Online +Extremism, Radicalisation, and Politicised Hate Speech. ACM Comput. Surv. 00, 0, Article 000 (December 2021), +35 pages. https://doi.org/tobereplacedwhenassignedDOI +Authors’ addresses: Jarod Govers, jg199@students.waikato.ac.nz, ORKA Lab, Department of Software Engineering, University +of Waikato, Gate 1, Knighton Road, Hamilton, Waikato, NZ, 3216; Philip Feldman, philip.feldman@asrcfederal.com, ASRC +Federal, Beltsville, Maryland, US; Aaron Dant, aaron.dant@asrcfederal.com, ASRC Federal, Beltsville, Maryland, US; Panos +Patros, panos.patros@waikato.ac.nz, ORKA Lab, Department of Software Engineering, University of Waikato, Gate 1, +Knighton Road, Hamilton, Waikato, NZ, 3216. +Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee +provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and +the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. +Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires +prior specific permission and/or a fee. Request permissions from permissions@acm.org. +© 2021 Association for Computing Machinery. +0360-0300/2021/12-ART000 $15.00 +https://doi.org/tobereplacedwhenassignedDOI +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. +arXiv:2301.11579v1 [cs.SI] 27 Jan 2023 + +000:2 +Govers et al. +Contents +Abstract +1 +Contents +2 +1 +Introduction +3 +1.1 +Motivation and Contributions +3 +1.2 +Structure +4 +2 +Social Context to Social Network Analysis +4 +2.1 +Extremism and Radicalisation Decoupled +4 +2.2 +Hate Speech Decoupled +5 +3 +Systematic Literature Review Design and Protocol +6 +3.1 +Trends and Shortfalls in Prior SLRs +6 +3.2 +Research Questions +7 +3.3 +Databases +7 +3.4 +Search Strings +7 +3.5 +Inclusion and Exclusion Criteria +8 +4 +Key Research Question (RQ) Findings +9 +4.1 +Summary of the Social ERH Definitions Used by Researchers +9 +4.2 +Summary of the Data Collection, Processing, and Annotation Processes +9 +4.3 +Summary of the State-of-the-art Computational ERH Detection Models +10 +4.4 +Summary of ERH Models’ Classification Performance +10 +4.5 +Geographic Trends—Islamophobia and Exclusion in the Academic Community +11 +5 +Socio-technical Context in Research—Consensus-driven Definitions +12 +5.1 +Researchers’ Consensus-driven Definitions for ERH Concepts +13 +5.2 +Correlation Between Definitions and Algorithmic Approach +14 +6 +Building ERH Datasets—Collection, Processing, and Annotation +15 +6.1 +Prominent Platforms, Pulling, and Populations +15 +6.2 +Feature Extraction Techniques +17 +6.3 +Data Filtering +18 +6.4 +Key Takeaways for Dataset Domain, Pre-processing, and Annotation +19 +7 +Community Detection, Text, and Image ERH Detection Algorithms +20 +7.1 +Observed Non-deep Machine Learning Algorithms (MLAs) +20 +7.2 +Deep Learning Algorithms (DLAs) +22 +8 +Model Performance Evaluation, Validation, and Challenges +24 +8.1 +Benchmark Dataset Performance (Inter-study Evaluation) +25 +8.2 +Community Detection Performance +25 +9 +Future Research Directions +25 +9.1 +Ideological Isomorphism—a Novel Framework for Radicalisation Detection +25 +9.2 +Morphological Mapping and Consensus-building—a novel computationally- +grounded framework for extremism detection +26 +9.3 +Outwards Dissemination––‘traditional’ hate speech detection updated +27 +9.4 +Uptake Roadmap for Researchers, Industry, and Government +28 +10 +Conclusion +28 +References +30 +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech +000:3 +1 +INTRODUCTION +Online social media empowers users to communicate with friends and the wider world, organise +events and movements, and engage with communities all at the palm of our hands. Social media +platforms are a frequent aid for modern political exchanges and organisation [115], with extremes +amplified by algorithmic recommendation systems, such as on Twitter [49], TikTok [87], and +YouTube [63]. Furthermore, the semantic expression of ideas differs between social media platforms, +with Twitter’s short 280 character limit resulting in more narcissistic and aggressive content +compared to Facebook [75], and anonymous platforms such as 4Chan instilling a vitriolic ‘group +think’ in political threads/‘boards’ [70]. Hateful, emotive, and ‘click-worthy’ content permeates +virtual discourse, which can radicalise users down an ideological rabbit hole towards real-world +violent action. The individuals behind the 2014 Isla Vista and 2019 Christchurch shootings appeared +as individual ‘lone-wolf attacks’ without an allegiance. However, investigations found a deep +network of perverse and violent communities across social media [40, 106]. Likewise, exploiting +social media to plan politically motivated attacks towards the civilian population to coerce political +change (i.e., terrorism) delegitimises democracies, social cohesion, and physical/mental health [28, +58, 79]. +As a response, social media platforms employ text and visual content-moderation systems to +detect and remove hate speech, extremism, and radicalising content. This paper offers a state-of-the- +art Systematic Literature Review (SLR) on the definitions, data collection, annotation, processing, +and model considerations for Extremism, Radicalisation, and Hate speech (ERH) detection. +1.1 +Motivation and Contributions +Existing ERH literature reviews exist as independent microcosms, often focusing on specific types +of models, typically text-only Natural Language Processing (NLP) models or non-textual network +analysis via community detection models. Studies seldom cross-examine models and evaluate the +performance between non-textual network analysis (a ‘who-knows-who’ approach), textual, and/or +multimedia approaches for ERH detection. While we identified ten prior literature reviews for +hateful content detection, none consider the similarities and definitional nuance between ERH +concepts and what Extremism, Hate Speech, or Radicalisation means in practice by researchers [2, +5, 20, 42, 43, 51, 82, 97, 110, 114]. Evaluating the consensus for ERH definitions, dataset collection +and extraction techniques, model choice and performance are all essential to create ethical models +without injurious censorship or blowback. +Through understanding the groups, beliefs, data, and algorithms behind existing content- +moderation models—we can reliably critique often overlooked social concepts, such as algorithmic +bias, and ensure compliance between social definitions and computational practice. Hence, this new +field of ERH context mining extracts the context to classifications—enabling researchers, industries, +and governments to assess the state of social discourse. +Given the rise of state-sponsored disinformation campaigns to undermine democratic institutions +and social media campaigns, the time is now for ERH research within politicised discussions. +The three core contributions for this paper are: +(1) The establishment of consensus-driven working definitions for Extremism, Radicalisation, +and Hate Speech within the novel field of ERH context mining—and a proposed frame- +work/roadmap for future researchers, social media platforms, and government advisors. +(2) The critical examination of existing textual (NLP), network (community detection), and +hybrid text-image datasets. +(3) The identification and cross-examination of the state-of-the-art models’ performance on +benchmark datasets and relevant challenges with the current ERH detection metrics. +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +000:4 +Govers et al. +1.2 +Structure +For a high-level summary of this SLR’s findings, refer to Section 4 on Key Research Question Findings, +and Section 9 for Recommendations for Future Work. These key summaries condense and contextual- +ise the 51 studies observed between 2015-2021, which we use to build our proposed computational +ERH definitions and technological roadmap for researchers, industry, and government in Section 9. +For a holistic understanding, we present a social context to our motivations in Section 2. Related +work and areas our SLR improves on are outlined in Section 3.1. Section 3 outlines the systematic +protocol used to collect the 51 studies between 2015-2021. Further to the summaries presented in +Section 4, we present an in-depth analysis and cross-examination of studies definitions of ERH +concepts in Section 5, approaches for collecting and processing data in Section 6, algorithmic +approaches for classification in Section 7, and their performance in Section 8. We conclude with +recommendations for future SLRs, and studies in Section 9, and conclusions in Section 10. +2 +SOCIAL CONTEXT TO SOCIAL NETWORK ANALYSIS +Analysing social media requires the socio-technical considerations of what constitutes hate speech, +extremism, and radicalisation (ERH). To detect such concepts, computational models can investigate +multimodal sources—including textual meaning and intent through Natural Language Processing +(NLP), computer vision for images, and evaluating user relationships through community detection. +Hence, this section decouples and analyses ERH’s social background and definitions. +2.1 +Extremism and Radicalisation Decoupled +Extremism’s definition appears in two main flavours: politically fringe belief systems outside the +social contract or violence supporting organisational affiliation. The Anti-Defamation League (ADL) +frames extremism as a concept "used to describe religious, social or political belief systems that exist +substantially outside of belief systems more broadly accepted in society" [6]. For instance, under the +ADL’s definition, extremism can be a peaceful positive force for mainstreaming subjugated beliefs, +such as for civil rights movements. This construct of a socially mainstream belief constitutes the +Overton window [73]—and is not the target for content moderation. +Conceptually, extremism typically involves hostility towards an apparent ‘foreign’ group based +on an opposing characteristic or ideology. Core tenants of extremism can stem from political +trauma, power vacuums and opportunity, alongside societal detachment and exclusion [58, 78]. +Hence, extremism often relies on defending and congregating people(s) around a central ideology, +whose followers and devotees are considered ‘in-group’ [116]. Extremists unify through hostility +and a perceived injustice from an ‘out-group’ of people(s) that do not conform to the extremist +narrative—typically in a ‘us vs. them’ manner [28, 78, 116]. Hence, extremism detection algorithms +can use non-textual relationships as an identifying factor via clustering users into communities +(i.e., community detection) [20, 110]. Thus, extremism can simply reduce to any form of a fringe +group whose identity represents the vocal antithesis of another group. +There is no one conceptual factor to make an extremist. Extremism can also emanate from +political securitisation—whereby state actors transform a specific referent object (such as Buzan’s +five dimensions of society: societal, military, political, economic, and environmental security [25]; or +individuals and groups [28]) towards matters of national security, requiring extraordinary political +measures [25]. As the state normalises policies into matters of existential national security, society +can adapt and ideate decisions to ones of existential ‘life or death’ nature [25]. +For example, the ‘Great Replacement’ conspiracy theory claims that non-European immigrants +and children are “colonizers" or "occupiers”, and an “internal enemy”—-with the intent to securitise +migration, race, religion, and culture into wars with wording to invoke fears of a fifth column or +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech +000:5 +racial invasion/replacement [22, 106]. The Christchurch Shooter took direct interest in securitising +migrants as an extreme military threat, as far to name his manifesto after the conspiracy [106]. +Extremism is not a strictly demographic ‘majority vs minority’ concern, as it encapsulates +movements demanding radical change and earmarked by a sense of rewarding personal and social +relationships, self-esteem, and belief of a wider purpose against a perceived adversarial force [58]. +Exploiting desires for vengeance and hostility are also key recruitment strategies [20, 58, 70]. +Outside of political, cultural, and socio-economic factors, mental health and media are intrinsically +inalienable contributing factors [28, 58, 79]. Likewise, repeated media reports of footage and body +counts can gamify and normalise extremism as a macabre sport for notoriety [20, 79]. +Within industry, Facebook, Twitter, YouTube, and the European Union frame extremism as +a form of indirect or direct support for civilian-oriented and politically motivated violence for +coercive change [34]. Facebook expands this industry-wide consensus to include Militarised Social +Movements and "violence-inducing conspiracy networks such as QAnon" [37]. +Radicalisation focuses on the process of ideological movement towards a different belief, which +the EU frames as a "phased and complex process in which an individual or a group embraces a +radical ideology or belief that accepts, uses or condones violence" [35]. Terrorism consists of +politically motivated violence towards the civilian population to coerce, intimidate, or force specific +political objectives, as an end-point for violent radicalisation to project extremism [25, 28]. Borum +delineates the passive ideological movement of radicalisation from active decisions to engage in +‘action pathways’ consisting of physical terrorism, or hate crimes [20]. +Political radicalisation towards increasingly aggrandising groups can also manifest in Roe’s two +sides of nationalism: positive socio-cultural and negative ethnic/racial nationalism [98]. These +balancing forces create a form of societal security dilemma whereby the actions of one society to +strengthen its identity can cause a reaction in another societal group, weakening security between +all groups-—a radicalising spiral which can manifest into a polarised ‘culture war’ [70, 98]. However, +integration over assimilation can inversely undermining culture, self-expression and group cohesion, +leading to alienation and oppression by the dominant political or normative force [28, 98]. +Nonetheless, ERH detection does not offer a panacea to combating global terrorism, nor does +surveillance offer a ‘catch-all’ solution. In the case of the livestreamed Christchurch shooter, the +New Zealand Security Intelligence Service concluded that “the most likely (if not only) way NZSIS +could have discovered [the individual]’s plans to conduct what is now known of [the individual]’s +plans to conduct his terrorist attacks would have been via his manifesto.” [106, p. 105]. However, +the individual did not disseminate this until immediately before the attack, and his 8Chan posts +did not pass the criteria to garner a search warrant [106, p. 105]. Hence, extremism detection is an +evolutionary arms race between effective and ethical defences vs. new tactics to evade detection. +2.2 +Hate Speech Decoupled +Obtaining viewpoint neutrality to categorise hate speech is challenging due to human biases and +the risk of hate speech undermining liberties through mainstreaming intolerance—the paradox of +tolerance where a society tolerant without limit may have their rights seized by those projected +intolerance [92]. Popper encapsulates this challenge by formulating that "if we are not prepared +to defend a tolerant society against the onslaught of the intolerant, then the tolerant will be +destroyed, and tolerance with them" [92]. Defining clear hate speech restrictions are needed to +protect expression rights and victim groups rights and safety [11, 117]. +The European Union defines hate speech as "all conduct publicly inciting to violence or hatred +directed against a group of persons or a member of such a group defined by reference to race, colour, +religion, descent or national or ethnic origin." [34]. Whereas, the U.S. Department of Justice frames +that: "A hate crime is a traditional offence like murder, arson, or vandalism with an added element +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +000:6 +Govers et al. +of bias... [consisting of a] criminal offence against a person or property motivated in whole or in +part by an offender’s bias against a race, religion, disability, sexual orientation, ethnicity, gender, or +gender identity." [38] Notably, governmental laws may differ from industry content moderation +policies via the omission of sexual, gender, religious or disability protections, and may include +threats of violence and non-violent but insulting speech. +The United Nations outlines the international consensus on hate speech as "any kind of commu- +nication in speech, writing or behaviour, that attacks or uses pejorative or discriminatory language +with reference to a person or a group on the basis of who they are, in other words, based on their +religion, ethnicity, nationality, race, colour, descent, gender or other identity factor." [117, p. 2] +What all these definitions have in common is that they all involve speech directed at a portion +of the population based on a protected class. +3 +SYSTEMATIC LITERATURE REVIEW DESIGN AND PROTOCOL +This SLR investigates the state-of-the-art approaches, datasets, ethical, socio-legal, and technical +implementations used for extremism, radicalisation, and politicised hate speech detection. We +conduct a preliminary review of prior ERH-related SLRs to establish the trends and research gaps. +For the purposes of our SLR’s design, and to embed Open-Source Intelligence (OSINT) and Social +Media Intelligence (SOCMINT) principles, we define social media data as any online medium where +users can interactively communicate, exchange or influence others. We accept external data sources, +such as manifesto or news sites if interactive—such as via comment sections. Furthermore, we +propose and use a novel quality assessment criteria to filter irrelevant or ambiguous studies. +3.1 +Trends and Shortfalls in Prior SLRs +Searching for Extremism, Radicalisation, Hate speech (ERH) and related terms, resulted in ten +literature reviews ranging from January 2011 to April 2021 [2, 5, 20, 42, 43, 51, 82, 97, 110, 114]. +Aldera et al. observed only one survey before 2011 (covering 2003-2011) and another in 2013, +indicating the limited, exclusionary, but developing nature of reviews in this ERH detection area [5]. +Prior SLRs seldom delineated or elaborated on Extremism, Hate Speech and Radicalisation. Neither +“extremism”, "radicalism" [2, 5, 20, 42, 43, 110, 114] or “hate speech” oriented SLRs [51, 82, 97] +cross-reference each other despite 26.3% of the data reviewed in the “hate speech” oriented review +by Adek et al. encompassing hate speech in a political context [114]. This lack of overlap presents +an industry-wide challenge for social media companies who may oversee developments in ‘hate +speech’ detection which could transfer to a ‘extremism/radicalisation detection’ model. +SLRs prior to 2015 found that deep learning approaches (DLAs), such as Convolutional Neural +Networks (CNN) and Long Short-Term Memory (LSTM), resulted in 5-20% lower F1-scores than +non-deep approaches (e.g., Naïve Bayes, Support Vector Machines, and Random Forest classifiers) [2, +5, 42, 51, 97, 114]. DLAs post-2015 indicated a pivotal change towards higher-performing language +transformers such as Bidirectional Encoder Representations from Transformers (BERT) models [33]. +3.1.1 +Domains and Criteria. +No review delineated or removed studies that did not use English social media data. This presents +three areas of concern for researchers when attempting to compare the performance of models: +(1) Results may not be comparable, if they use culture-specific lexical datasets, or language +models trained on other languages. +(2) Linguistic differences and conveyance in language—as what may be culturally appro- +priate for the majority class may appear offensive to minority groups and vice-versa. +(3) The choice of language(s) influences the distribution of target groups—with a bias +towards Islamic extremism given its global reach in both Western (predominantly ISIS) and +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech +000:7 +Eastern countries (e.g., with studied online movements in the Russian Caucasus Region [77]). +It is worth investigating whether Gaikwad et al.’s finding that 64% of studies solely target +‘Jihadism’ corroborates with our study, which targets only English data [42, p. 17]. +Our SLR incorporates the key approaches of dataset evaluation (including their accessibility, +labels, source and target group(s)), data collection and scraping approaches, Machine Learning and +Deep Learning algorithms, pre-processing techniques, research biases, and socio-legal contexts. +Unlike prior SLRs, our SLR conceptualises all elements for ERH context mining—consisting of a +user’s ideological radicalisation to an extremist position, and then projected via hate speech. +3.2 +Research Questions +Our Research Questions (RQ) investigate the full process of ERH Context Mining—incl. data collec- +tion, annotation, pre-processing, model generation and evaluation. These RQs consist of: +(1) What are the working definitions for classifying online Extremism, Radicalisation, and Hate +Speech? +(2) What are the methodologies for collecting, processing and annotating datasets? +(3) What are the computational classification methods for ERH detection? +(4) What are the highest performing models, and what challenges exist in cross-examining them? +Given the overlap of studies across prior SLRs targeting extremism or radicalisation or hate +speech, RQ1 addresses the similarities and differences between researchers’ definitions of ERH +concepts and their computational classification approach. We dissect the ERH component of ERH +context mining and propose consensus-driven working definitions. +RQ2 addresses the vital context for ERH models—the data used and features extracted or filtered +out from it. Furthermore, identifying frequently used benchmark datasets provides the basis for +critical appraisal of the state-of-the-art algorithmic approaches in the community detection, multi- +media, and NLP spheres in RQ3/4. Covering algorithmic approaches is not in itself novel. However, +we consider novel, niche, and overlooked features relevant for an ERH model to make accurate +classifications—namely, bot/troll detection, transfer learning, the role of bias, and a hybridised eval- +uation of NLP and non-textual community detection models. We also consider critical challenges, +choice of metrics, and performance considerations not observed in prior SLRs. +3.3 +Databases +Given the cross-disciplinary, global and socio-technical concepts for ERH detection, we queried the +following range of software engineering, computer science, crime and security studies databases. +• ProQuest (with the “peer-reviewed” filter, including queries to the below databases) +• Association for Computing Machinery (ACM) Digital Library +• SpringerLink +• ResearchGate +• Wiley +• Institute of Electrical and Electronics Engineers (IEEE) Xplore +• Association for Computational Linguistics Portal +• Public Library of Science (PLOS) ONE Database +• Google Scholar—as a last line to capture other journals missed in the above searched databases +3.4 +Search Strings +The first round of study collection included automated database search strings. A second round +included a targeted manual search strategy with dissected keyword combinations to expand +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +000:8 +Govers et al. +coverage. All results were added to the Title and Abstract Screening list. The following database +search query also included time filters (2015-2021) and peer-review-only filters: +(“artificial␣intelligence”␣OR␣“machine␣learning”␣OR␣“data␣mining”␣OR␣“natural +language␣processing”␣OR␣“multiclass␣classification”␣OR␣“model”␣OR␣“analysis”␣OR +“intelligence”␣OR␣“modelling”␣OR␣“detection”) +AND␣(“hate␣speech”␣OR␣“radicalisation”␣OR␣“radicalization”␣OR␣“extremism”) +AND␣(“social␣media”␣OR␣“forums”␣OR␣“comments”OR␣“virtual␣networks”␣OR␣“virtual +communities”␣OR␣“online␣communities”␣OR␣“posts”␣OR␣“tweets”␣OR␣“blogs”) +3.5 +Inclusion and Exclusion Criteria +After attaining our 251 studies from our search strings, we read the journal metadata, title and +abstract to screen studies. Our ranked criteria requires that all studies must: +(1) Originate from a peer-reviewed journal, conference proceeding, reports, or book chapters. +(2) Be written in English. +(3) Involve a computational model (network relationship, textual and/or visual machine learning +model) for identifying and classifying radicalisation, hate speech or extreme affiliation. +(4) Utilise social media platform(s) for generating their model. +(5) Computationally identify ERH via binary, multiclass, clustering or score-based algorithms. +(6) Focus on politicised discourse to exclude cyber-bullying or irrelevant benign discussions. +(7) Published after the 1st January 2015—until the 1st July 2021. +(8) Utilise English social media data if evaluating semantics and grammatical structure. +In addition to those that do not abide to any of the above, we exclude studies that: +(1) Are duplicates of existing studies. +(2) Do not specify their target affiliation to exclude broad observational studies. +We outline our further in-depth critical Quality Assessment (QA) criteria to filter irrelevant or +ambiguous studies in our supplementary material’s Quality Assessment Criteria subsection. +After the Title and Abstract Screen, we read the full text of the 57 studies for the screening stages +displayed in Table 1. With 42 studies passing the Full Text Screen, we then randomly selected studies +from the bibliographies from this ’snowball sample’ of the 42 studies until 5 studies fail QA. +Table 1. Studies found and filtered +Screen Type +Study Count +Search Strings +251 +Title and Abstract Screen +57 +Full Text Screening +42 +After Snowball Sampling +51 +3.5.1 +Threats to Validity. +While we consider a concerted range of search strings, we recognise that ERH concepts is a wide +spectrum. To focus on manifestly hateful, politicised, and violent datasets/studies, we excluded +cyber-bullying or emotion-detection studies. The potential overlap and alternate terms for ERH +(e.g., sexism as ‘misogyny classification’ [30]) could evade our search strings. Our pilot study, +subsequent tweaks to our search method, and snowball sampling minimise this lost paper dilemma. +This study does not involve external funding, and all researchers declare no conflicts of interest. +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech +000:9 +4 +KEY RESEARCH QUESTION (RQ) FINDINGS +Across the 51 studies between 2015-2021, ERH research is gaining popularity—with 4 studies from +2015-2016 increasing to 25 between 2019-2020 (and 4 studies from January to July 2021). We present +our SLR’s core findings in this section, with in-depth RQ analysis in Sections 5, 6, 7, and 8. +4.1 +Summary of the Social ERH Definitions Used by Researchers +RQ1: What are the working definitions for computationally classifying online Extremism, Radicalisation, +and Hate Speech? +Across the 51 studies, there are seldom delineations between the researchers choice of Extremism, +Radicalisation, and Hate Speech as the study’s focus––with the consensus that hate speech is +equivalent to extremist or radical views. Hence, researchers approach extremism and radicalisation +as an organisationally affiliated form of hate speech. +The consensus on hate speech’s working definition is any form of subjective and derogat- +ory speech towards protected characteristics expressed directly or indirectly in textual form—- +predominantly via racism or sexism. Benchmark datasets utilise human-annotated labels on single- +post instances of racially or gender-motivated straw man arguments, stereotyping, or post causing +offensive towards the majority of annotators (via inter-annotator agreement). Only 20% of studies +consider explicit rules or legal frameworks for defining hate speech [15, 18, 47, 53, 55, 71, 77, 84, +123, 128], with others relying on either an implicit ‘consensus’ on hate speech or utilise benchmark +datasets. Benchmark datasets typically consider categorising their data into explicit categories of +racism [31, 32, 122, 123], sexism [13, 122, 123], aggression [13, 61], or offensiveness [13, 31, 128]; +including hate categorisation via visual memes and textual captions [3, 57, 99, 111]. +Extremism and radicalisation are equivalent terms in existing academia. Islamic extremism is +the target group in 77% of US-originating extremism studies. ‘Far-right extremism’ and ‘white +supremacy’ are used interchangeably, a form of cultural bias given the variety of right-wing politics +worldwide. Only one study considered radicalisation as an ideological movement over time [12]. +4.2 +Summary of the Data Collection, Processing, and Annotation Processes +RQ2: What are the methodologies for collecting, processing and annotating datasets? +Collecting non-hateful and hateful ERH instances varies between supervised and unsupervised +(clustering) tasks. Supervised learning typically utilises manual human annotation of textual posts +extracted via tools presented in Figure 1. Semi or unsupervised data collection can include grouping +ideologies by platform, thread, or relation to a suspended extremist account. Islamic extremism +studies frequently used manifestos and official Islamic State magazines as a ‘ground truth’ for +textual similarity-based approaches for extremism detection. We found a direct correlation between +the availability of open and official research tools, and the platform of choice by researchers. Biases +extend geographically, with no studies utilising data or groups from Oceanic countries. +Figure 2 displays the skew for Twitter as the dominant platform for hate speech research. Despite +the nuance of conversations, 69% of studies classify hate on a single post per Figure 3. +Data processing often utilises extracting statistically significant ERH features—such as hateful +lexicons, emotional sentiment, psychological signals, ‘us vs them’ mentality (higher occurrence of +first and third-person plural words [46]), and references to political entities. +We categorise and frame the two approaches for dataset annotation: organisational or experience- +driven annotation. Organisational annotation utilises non-governmental anti-hate organisations [105] +or ‘expert’ annotator panels—determined via custom tests or by tertiary degree. Organisational an- +notation relies on crowdsourced annotators, balanced by self-reported political affiliation. Inter-rater +agreement or Kappa coefficient are the sole metrics for measuring annotator agreement. +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +000:10 +Govers et al. +Fig. 1. Method to collect data. +Fig. 2. Number of studies per social media platform studied. +4.3 +Summary of the State-of-the-art Computational ERH Detection Models +RQ3: What are the computational classification methods for ERH detection? +ERH detection includes text-only Natural Language Processing (NLP), network-related community +detection, and hybrid image-text models. Between 2015-2021, there was a notable shift from +traditional Machine Learning Algorithms (MLAs) towards contextual Deep Learning Algorithms +(DLAs) due to higher classification performance—typically measured by macro F1-score. +Notably, only 3 of the 21 community detection studies utilised Deep Learning Algorithms +(DLAs) [77, 84, 99]. Instead, community detection researchers tended to opt for graph-based models +such as heterogeneous graph models converting follower/following, reply/mention, and URL +networks with numeric representations for logistic regression or decision trees [12, 16, 24, 48, 80, 84]. +Community detection Machine Learning Algorithms (MLAs) performance varied by ~0.3 F1-score +(mean between studies) dependent on the selection of features. Statistically significant features +for performant MLA models include gender, topics extracted from a post’s URL(s), location, and +emotion via separate sentiment algorithms such as ExtremeSentiLex [89] and SentiStrength [112]. +For textual non-deep NLP studies, researchers classified text via converting the input into +word embeddings via Word2Vec, GloVe, or frequent words via Term Frequency-Inverse Docu- +ment Frequency (TF-IDF), and parsing it into Support Vector Machines, decision trees, or logistic +regression models. As these embeddings do not account for word order, context and nuance is +often lost—leading to higher false positives on controversial political threads. Conversely, DLAs +utilise positional and contextual word embeddings for context-sensitivity using Long Short Term +Memory (LSTM) Convolutional Neural Networks and Bidirectional Encoder Representations from +Transformers (BERT) leading to their higher performance as outlined in RQ4. +4.4 +Summary of ERH Models’ Classification Performance +RQ4: What are the highest performing models, and what challenges exist in cross-examining them? +By 2021, Support Vector Machines on emotional, sentiment, and derogatory lexicon features were +the last non-deep MLA to attain competitive F1-scores for NLP tasks compared to DLAs such as +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +3%6% +29% +32% +22% +5% +3% +Twitter APi +Custom web-crawler + Pushshift.io APl (Reddit) +Magazine or Newspaper Archives +Pre-existing Dataset or Library +Facebook Graph APl +Other APlIslamicExtremist +Sources, 9 +Stormfront, 7 +Other Far- +Blogs and +right +Apolitical +Extremist +Sources, 4 +Forums, 4 +News +Websites, 6 +Far-Left +Forums, +Facebook, +Wikipedia +2 +3 +Twitter, 36 +Vkont +akte, +Religious +Other, 5 +Reddit, 2 +Texts, 2Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech +000:11 +Fig. 3. Type of data used for an ERH classification. +Fig. 4. Distribution of target groups. +Convolutional Neural Networks (CNNs) and neural language transformers. As of 2021, BERT-base +attained the highest macro F1-score average across the seven benchmark datasets. However, cross- +examining models between datasets present various challenges due to varying criteria, social media +domain, and choice of metrics. Likewise, non-textual community detection and traditional MLA +studies resulted in lower classification F1-score by ~0.15 and ~0.2 respectively. While BERT, attention- +layered Long Short Term Memory (BiLSTM), and other ensemble DLAs attain the highest F1-scores, +no studies consider their performance trade-offs with their high computational complexity. Our +recommendations propose further research in prompt engineering, distilled models, and hybrid +multimedia-text studies—as we only identified one hybrid image-text study. +While textual DLAs outperform community detection models, grouping unknown instances +enable network models to identify bot networks and emergent terror cells. Hence, there is a growing +area of research for hybrid semi-supervised NLP and community detection models to identify new +groups and radical individuals in a domain we frame as meso-level and micro-level classification. +4.5 +Geographic Trends—Islamophobia and Exclusion in the Academic Community +To identify ERH hot spots in research, we present the first cross-researcher examination of their +institution’s location compared to their dataset(s) geolocational origin in Figure 5. For clarity, we +filter out the 29 indiscriminate global studies. +Despite the decline of the Islamic State as a conventional state-actor post-2016, western academic +research remains skewed towards researching Islamic extremist organisations operating from the +Middle East. 24% of US-originating ERH studies targeted Islamic extremism, compared to 19% +focusing on violent far-right groups and 19% for left vs right polarised speech (in discussions +containing hate speech). Despite more Islamic extremist studies from US-oriented research, over +90% of terrorist attacks and plots in the US were from far-right extremists in 2020 [54]. +European-origin studies have a reduced bias, where 25% target far-right white supremacy and +29% on Islamic extremism. Islamic extremism’s popularity is a global trend for 20% of all studies, +shown in Figure 4. Hence, there is a clear Islamophobic trend in academia—given the aversion of +far-right groups, and the lack of a change in the distribution of targeted groups between 2015-2021. +Researcher ethics and socio-legal considerations present a critical international research gap, as +only 13% of US and 28% of European studies included discussions on annotation ethics, expression +laws, or regulation. This US vs. Europe discrepancy likely emanates from the data collection and +autonomous decision-making rights guaranteed under the EU’s GDPR [36]. +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +4% +Text. Single post only +8% +I Text. Grouped by user +19% +No text. Metadata +69% +Relations (Community +Detection), or Images +Text and Graphing +(NLP and Community +Detection).lslamic +20% +20% +IFar-right White +Supremacy) +IMainstream Politics +10% +Hate Speech +9% +41% +1Other000:12 +Govers et al. +Fig. 5. Relations between researchers country of origin and their dataset’s country of focus (global/indiscrim- +inate studies excluded). Created via Flowmap.blue [21] with Mapbox [74] and OpenStreetMap [85] +4.5.1 +The Case for Oceania and the Global South. +Despite developments post-2015, such as two New Zealand terrorist attacks [94, 106] and five in +Australia [8], the rise of racial violence in South Africa [28], and the 2019-ongoing COVID-related +radicalisation [120]; no studies considered targeting Oceania or English-recognised countries in the +global south. Likewise, applying these datasets across intersectional ethnic, sexual, and cultural +domains presents a threat to validity as terms considered mundane or inoffensive to one group may +be considered inflammatory to another. Datasets are also biased towards racism towards a minority +group [31, 123], which may bias English hate speech in a white minority country such as South +Africa. Investigating language trends and model performance on Mela-, Micro- and Polynesian +groups could also offer insights in the role of religion, values (such as tikanga values in New +Zealand’s M¯aori population), taboos, lexicons, and social structures unique to indigenous cultures. +5 +SOCIO-TECHNICAL CONTEXT IN RESEARCH—CONSENSUS-DRIVEN +DEFINITIONS +What are the Working Definitions for Classifying Online Extremism, Radicalisation, and Hate Speech? +Empirically, studies often provide a generalised social definition in their introduction or back- +ground and utilise technical criteria to annotate instances for (semi)supervised learning tasks. +Hence, this research question consists of two parts: the socio-legal ERH definitions, and the +technical implementation and classification thereof outlined in the existing literature. +We identified an unexpected overlap between the definitions and models between extremism and +radicalisation studies, whereby researchers frame these concepts as synonymous with hate speech +with a political/organisational affiliation. Hate speech studies focus on protected groups as binary +‘hate or not’ [3, 4, 15, 27, 32, 57, 77, 84, 101, 109, 111, 123], or multiclass ‘racism, sexism, offensive, +or neither’ text [10, 13, 31, 45, 69, 81, 83, 90, 91, 123, 126], with a consensus that ‘Extremism = +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +Iceland +Sweden +Finland +Norya +Russia +Denmark +Belarus: +Germany- +Ukraine +Era +Kazakhstan +Romania +Mongolia +Atiantic +ain +Kyrgyzstan +Unit +Qcean +Turkey +Turkmenistan +Afghanistan +China +Morocco +Iran +‘Nepal-K +Mexico +Bangladesh +Cuba +Mauritania +Oman +India +Laos +Niger +Senega +Chad +Sudan +Yemen +Cambodia +Philipp +Venezuela +Ethiopia +SriLanka +Colombia +Maldives +Malaysia +Kenya +Ecuador +Indonesia +Tanzania +Perum +Brazil +Angola +Zambia +Bolivia +Mozambique +Namibia +Indiar +Paraguay +South +Chile +Ocear +Locationtotals +Atlantic +SouthAfrica +Ocean +outgoing 兰 incoming +ruguay +Argoitina +more outgoing +more incomingDown the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech +000:13 +Radicalisation = Hate speech with an affiliation’. Alternatively, we propose a novel computationally +grounded framework and definitions to seperate and expand ERH in Section 9 to underline the +holistic stages of extremists temporal radicalisation through disseminating hateful media. +5.0.1 +Socio-legal Context Provided in Existing Literature. +The largest discrepancy were between studies that discussed legal or ethical context to ERH, which +constituted only 20% of studies [15, 18, 47, 53, 55, 71, 77, 84, 123, 128]. The remaining 80% relied on +an implicit consensus of ‘hate speech’ (often synonymous with toxic, threatening, and targeted +speech), or ’extremism’ (often UN designated terrorist organisations like ISIS [118]). +Waseem and Hovy [123] outlined a unique eleven-piece criteria to identify offensive ‘hate +speech’ including considerations for politicised extremist speech via tweets that “promotes, but +does not directly use, hate speech or violent crime” and “shows support of problematic hash tags" +(although "problematic" was not defined). Hate speech as a supervised learning task resulted in two +categories—sexism and racism. A sexist post requires gender-oriented slurs, stereotypes, promoting +gender-based violence, or straw man arguments with gender as a focus (defined as a logical fallacy +aimed at grossly oversimplifying/altering an argument in bad faith [123]). The ambiguity for sexism +classification by human annotators was responsible for 85% of inter-annotator disagreements [123]. +5.1 +Researchers’ Consensus-driven Definitions for ERH Concepts +We aggregate the trends in ERH based on the definitions used throughout the 51 studies, and +observe that ERH concepts reflect their computational approach more than their social definitions. +Despite radicalisation being a social process of ideological movement, existing work considers the +term as synonymous to political hate speech/extremism. +Definition 1: Hate speech (researchers’ consensus) +Hate speech is the subjective and derogatory speech towards protected characteristics expressed +directly or indirectly to such groups in textual form.* +*N.B: there is a significant bias in hate speech categorical classification in practice, whereby no +studies considered categories outside of sexism (including gender-identity) or racism. +Definition 2: Extremism (researchers’ consensus) +Organisational affiliation to an ideology that discriminates against protected inalienable +characteristics or a violent political organisation. Affiliation does not always include manifestly +hateful text and may include tacit or explicit organisational support. Extremist studies often +classify organisational affiliation based on text (NLP) and community networks (follower, +following, or friend relationships). +The current academic consensus among researchers demonstrates a considerable overlap between +‘extremism’ and ‘hate speech’ definitions. In practice, extremism exclusively focused on racism +detection, or in the specific context of Jihadism [4, 15, 84], white supremacy [53, 86, 99, 109], +Ukrainian separatists [16], anti-fascism (Antifa) [53], and the sovereign citizen movement [53]. Of +the 13 studies targeting extremism, only one considered extremism by the ADL’s politically-fringe- +but-not-violent definition [1]. Tying extremism to the study of mainstream terrorist-affiliated groups +neglects rising movements, ethical movements using unethical terror-tactics, and non-violent +fragments of other terrorist groups, such as a reversion to ‘fringe’ activism. Hence, extremism’s +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +000:14 +Govers et al. +working definition is similar to terrorism when considering group affiliation detection. If investigating +extremist ideologies, then the definition is synonymous with those in hate speech studies. +Extremism’s working definition is exceptionally biased towards support for Islamic extremist +movements (10 out of 13 studies [4, 16, 24, 53, 77, 80, 84, 86, 89, 109]), with far-right ideologies a +distant second (5 out of 13 studies [16, 53, 86, 99, 109]). These organisational and ideological biases +is potentially a result of US security discourse and national interests (via the ‘War on Terror’). White +supremacy and far-right ideologies are separate terms used interchangeably without distinction. +Definition 3: Radicalisation (researchers’ consensus) +No discernible difference between extremism’s definition with both terms used interchangeably. +Radicalisation = extremism = politically affiliated hate invoking or supporting speech. +Definitions and algorithmic approaches on radicalisation detection relied on political hate speech, +or extremism via ideological affiliation—with 5 of the 8 radicalisation studies targeting textual or +network affiliation to the Islamic State (IS) [7, 47, 80, 95, 101], and 2 on white supremacy [46, 103]. +The only other notable deviations from this extremism = radicalisation dilemma is Bartal et +al.’s [12] focus on radicalisation as a temporal movement with apolitical roles. Their study investig- +ated the temporal movement from a ‘Novice’ (new poster) classification towards an ‘influencer’ +role based on their network relations and reply/response networks. Chandrasekharan et al. defined +‘radicalisation’ as the process of an entire subreddit’s patterns up to and including the time of its +ban to map subreddit-wide radicalisation [27]. Only two studies are the exception to the extremism += radicalisation = politicised hate speech consensus per the remainder of the 51 studies [12, 27]. +Fig. 6. ERH Definition Tree—visualising how ERH definitions deviate based on their algorithmic approach. +5.2 +Correlation Between Definitions and Algorithmic Approach +Uniquely, 66% of publications in the field of social-science or security studies utilised network- +driven community detection models, with extremism defined in a law enforcement context by +emphasising a user’s network-of-interactions between known annotated extremists. Hung et al. +defined extremism in a semi-supervised OSINT and HUMINT surveillance manner—requiring +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +ERH +Semantic/Topic-based (Aggregate) Hate +Radical or Extremist Network Detection +Speech Detection +Generally +Generally semi or fully +Generally +unsupervised +supervised. +unsupervised +'Big Data': +Unless cluster-based +'Big Data': +Graphs (non-textual): +Emotional Sentiment +Topics and Political Affilation +Hate Speech +Crime and Social Science Focused +E.g. SentiStrength, SIRA, LIWC +Law Enforcement Pivoted Data: +E.g. FBI 'Tripwire', Explicit Terror Afillation, TSA's +Automated Targeting System, etc. +Multiclass Sentiment +Binary Affiliation +Influencer +Racism +Terror Network +Sexism +Aggression +Offense +or ProlAnti/Neutral +Person-of +Classification +Clustering +Classificatior +Classification +Classification +Classification +Interest Detection +Classification +F +H +J +K +c +D +A +Multiclass +Ideological or +Community +Radical Scoring +Organisation or +Detection +Topic Classification +Regression +B +E +GDown the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech +000:15 +links between an extremist virtual and a physical/offline presence to extremist stimuli through +incorporating the FBI’s tripwire program [48]. Approaching extremism using relational properties +via interactions, geographic proximity, profile metadata, and semantic or network similarity raises +ethical dilemmas vis-à-vis individuals who have/had a solely virtual presence or those interested in +opposing opinions [24, 29, 76]. The relationship between definitions and algorithmic approaches +indicate that radicalisation studies skew towards community detection models, extremism towards +hybrid NLP and community detection models, and hate speech to a text-only NLP endeavour. +Table 2. Table of references for studies in each category (A, B...K) in the above ERH definition tree diagram. +Label +Studies +Label +Studies +Label +Studies +A +[12, 15, 48, 80] +B +[15, 16, 80, 89, 99, 108] +C +[48, 108] +D +[3, 4, 15, 27, 32, 57, 77, 84, 101, 109, 111, +123] +E +[14, 53, 80, 89, 93, 99, 99] +F +[1, 4, 7, 13, 52, 55, 86, 89, 90, 95, 100, 107, +124, 128, 129] +G +[15, 18, 46, 47, 103, 104, 113] +H +[10, 13, 45, 69, 81, 83, 90, 91, 123, 126] +I +[10, 13, 45, 81, 83, 90, 91, 123, 126] +J +[13, 56, 83, 90, 126, 128, 129] +K +[3, 56, 57, 67, 71, 81, 83, 90, 99, 111, 126, +128–130] +- +- +5.2.1 +Privacy and Ethics-driven Regulation. +No studies integrated or mentioned existing AI ethics regulation or standards, such as those +emerging from the EU [50], or private-sector self-regulation such as the IEEE P700x Series of +Ethical AI Design [60]. Researchers should consider the application and use cases for their proposed +models—as autonomous legal decision making, injurious use of data (outside of a reasonable +purpose), or erasure (a challenge for persistent open-source datasets), may violate regulations such +as the EU’s General Data Protection Regulation (GDPR)’s Article 22, 4, and 17 respectively [36]. +Prominently, Mozafari et al. evaluated hate speech with a ethno-linguistic context, recognising +that certain racist slurs were dependent on the culture and demographic using them [81]. +6 +BUILDING ERH DATASETS—COLLECTION, PROCESSING, AND ANNOTATION +What are the methodologies for collecting, processing and annotating datasets? +This RQ outlines the dominant platforms of choice for ERH research, the APIs and methods +for pulling data and its underlying ethical considerations. Geographic mapping demonstrates the +marginalisation of Oceania and the global south in academia. We critically evaluate the sentimental, +relationship, and contextual feature extraction and filtering techniques in community detection +and NLP studies. We conclude with the key recommendations for future data collection research. +6.1 +Prominent Platforms, Pulling, and Populations +This subsection outlines the common social media platforms, the method for sampling and extracting +(‘pulling’) textual and network/relationship data, and the type of data used in ERH datasets. +40% of studies relied on Twitter tweets for ERH detection, with Twitter being the dominant +platform for research per Figure 2. Twitter’s mainstream global reach paired with its data-scraping +Twitter API enabled researchers to target specific hashtags (topics or groups), real-time tweet +streams and reach of tweets and their community networks. Hence, the Twitter API is also the most +used method for scraping data, with other methods outlined in Figure 1. Unfortunately, revised +2021 Twitter Academic API regulations removed access to tweets from suspended accounts [17], +limiting datasets to those pre-archived. Currently, the Waseem and Hovy datasets are not available +due to relying on the Twitter API and suspended tweets [122, 123]. +For far-right ERH detection, researchers used custom web-scrapers to pull from the global +white supremacist forum Stormfront—containing topics ranging from political discussions, radic- +alising "Strategy and Tactics", and "Ideology and Philosophy" sections, and regional multilingual +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +000:16 +Govers et al. +chapters [55, 71, 81, 103, 109, 124, 126]. As a supplement or alternate to searching and collecting +hateful posts en masse, five studies considered extremist ‘ground truth’ instances by comparing tex- +tual similarity from Tor-based terror-supporting anonymous forums [4] and websites [124], radical +Islamist magazines and manifestos [7, 84, 95]. Interestingly, no studies considered extracting ground +truths from far-right manifestos or media. Likewise, no studies considered recent low-moderation +anonymised forums such as 8Chan (now 8kun) or Kiwifarms, which were extensive hubs for pro- +paganda dissemination from the Christchurch shooter [106]; Parler, notable for its organisational +influence during the 2021 Capitol Hill riots [72]; Telegram, TikTok, or Discord, despite reports on +its use for sharing suicides, mass shootings, and group-lead harassment of minority groups [44, 87]). +Hence, there is a prevalent and concerning trend towards NLP studies on mainstream platforms, +which may overlook the role of emergent, pseudo-anonymised or multimedia-oriented platforms. +6.1.1 +Data Collected. +62% of all studies evaluate ERH on a single post-by-post basis, with NLP the dominant approach per +Figure 3. Conversely, grouping posts on a per user basis frequently included annotations from cyber- +vigilante groups such as the Anonymous affiliated OSINT ‘Controlling Section’ (CtrlSec) group’s +annotations of ISIS-supporting accounts [47]. However, Twitter claims that CntrlSec annotations +were "wildly inaccurate and full of academics and journalists" [26]. Hence, researchers should avoid +unvetted cyber-vigilante data, and consider anonymising datasets to further benefit user privacy, +researcher ethics, and model performance by reducing false positives (i.e., censorship). +While NLP text detection is the dominant detection approach, 23 of the 51 studies investigated +data sources outside of textual posts per Figure 10. Research gaps include the lack of multimedia +and law enforcement studies, with only three hybrid text-image detection [57, 99, 111] and one +study utilising FBI anonymous tips, Automated Targeting System and Secure Flight data [48, 119]. +6.1.2 +Data Collection and Annotation Bias. +Due to the varying fiscal costs, biases, and time trade-offs, there is no consensus for selecting or +excluding annotators for supervised learning datasets. Hence, we frame that annotator selection +falls within two varying groups: experience-driven selection and organisation-driven selection. +For the former, experience-driven selection includes studies that utilised self-proclaimed ‘expert’ +panels as determined by their location and relevant degrees [123], are anti-racism and feminist +activists [122], or work on behalf of a private anti-discrimination organisation [105]. However, +assembling annotators by specific characteristics may be time-consuming or costly, such as crowd- +sourcing tertiary annotators via Amazon Mechanical Turk, or Figure Eight [10, 45, 71, 81, 93]. +Conversely, an organisation-driven selection approach focuses on agreement by a crowdsourced +consensus. Instead of relying on specific experience, researchers utilised custom-made tests for +knowledge of hate speech criteria based on the researchers own labelled subset [122, 128]. Likewise, +organising annotator pools can also include balancing annotators self-reported political affiliation to +reduce political bias [93]. Researchers use Inter-rater agreement, and Kappa Coefficient to determine +a post’s ERH classification. For racism, sexism, and neither classifications, annotation Fleiss’ Kappa +values ranged between 0.607 [32] to 0.83 [128], indicating moderate to strong agreement [125]. +Thirdly, unsupervised clustering enables mass data collection without time-consuming annota- +tion via Louvain grouping (to automatically group text/networks to identify emergent groups) [15, +16, 108], or grouping based on a thread’s affiliation (e.g., the now-banned r/altright [46] and v/[n- +word] [27]). Although not all posts from an extremist platform may be manifestly hateful, as evident +in the 9507 post ‘non-hate’ class in the Stormfront benchmark dataset from de Gibert et al. [32]. +Research continues to skew towards radical Islamic extremism per Figure 4, while the plurality +(41%) target generic ‘hate speech’ in ‘hate or not’, or delineations for racism, sexism, and/or offence. +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech +000:17 +6.1.3 +Benchmark Datasets. +We define a benchmark dataset as any dataset evaluated by three or more studies. The majority of +studies used custom web-scrapped datasets or Tweets pulled via the Twitter API per Figure 1. +Table 3. Datasets used by three or more studies. +Dataset +Year +Categories +Platform +of origin +Collection strategy +Used By +Waseem +and +Hovy [123] +2016 +16914 tweets: 3383 Sexist , +1972 Racist, 11559 Neutral +Twitter +11-point Hate Speech Cri- +teria +[10, 52, 55, 56, +71, 81, 83, 91, +123, 126] +FifthTribe +[39] +2016 +17350 pro-ISIS accounts +Twitter +Annotated pro-ISIS ac- +counts +[7, 84, 95, 102] +de Gibert +[32] +2018 +1196 Hate, 9507 Non-hate, +74 Skip (other) posts +Stormfront +3 annotators considering +prior posts as context +[32, +55, +71, +126] +OffenseEval +(OLID) [128] +2019 +14100 tweets. (30%) Of- +fensive or Not; Targeted +or +Untargeted +insult; +towards an Individual, +Group, or Other +Twitter +Three-level hierarchical +schema, by 6 annotators +[55, 67, 128, +130] +HatEval [13] 2019 +10000 tweets distributed +with Hateful or Not, Ag- +gressive or Not, Individual +targeted or Generic +Twitter +Crowdsourced via Figure +Eight, with 3 judgements +per Tweet +[13, 55, 71, 90, +126] +Hatebase- +Twitter [31] +2019 +25000 tweets: Hate speech, +Offensive, Neither +Twitter +3 or more CrowdFlower +annotators per tweet +[31, 52, 55, 71, +81, 83, 126] +TRAC [61] +2018 +15000 English and Hindi +posts; Overtly, Covertly, +or Not Aggressive +Facebook +Kumar et al. [62] subset, 3 +annotators per post, com- +ment or unit of discourse +[55, 56, 71] +6.2 +Feature Extraction Techniques +Figure 7 outlines the three types of feature extraction techniques. Non-contextual lexicon approaches +relate to word embeddings for entities, slurs, and emotional features. However, non-contextual +blacklists and Bag of Words (BoW) lexicon approaches cannot identify context, concepts, emergent, +or dual-use words (see the Supplementary Material’s Algorithm Handbook section for comprehensive +definitions) [32, 81, 83, 90, 122]. Contextual sentiment embeddings expand on lexicons by embedding +a form of context via positional tokens to establish an order to sentences. +We group unsupervised term clustering and dimensionality reduction methods under the +Probability-Occurrence Models category. The two dominant approaches include weighted ANOVA- +based BoW approaches and Term Frequency-Inverse Document Frequency (defined in the Supple- +mentary Material), which weigh the importance of each word in the overall document and class +corpus. Contextual sentiment embeddings result in higher F1-scoring models (per RQ4) due to their +context-sensitivity and compatibility as input embeddings for deep learning models [55, 71, 83]. +Community detection features require mapping following, friend, followee, and mention dy- +namics. Furthermore, other statistically significant metadata includes profile name, geolocation (to +investigate ERH as a disease), gender, and URLs [16, 81, 84, 123, 126]. URLs can identify rabbit holes +for misinformation or alternate forums via PageRank [88] and Hyperlink-induced Topic Search +(HITS) [59]—which extracts keywords, themes and topical relations across the web [77, 88]. +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +000:18 +Govers et al. +Fig. 7. Types of feature extraction techniques. +6.3 +Data Filtering +For context-insensitive BoW and non-deep models, stop words (e.g. the, a, etc.), misspellings, and web +data are often filtered out via regular expressions and parsing libraries [4, 27, 46, 47, 52, 69, 71, 81, 84]. +Compared to semantic or reply networks, community detection models tend to extract metadata +for separate clustering for entity and concept relationships. All data filtering techniques are thereby +aggregated and branched in Figure 8. +No studies considered satire, comedy, or irony to delineate genuine extremism and online culture. +Researchers’ implicit consensus is to treat all posts as part of the ERH category if it violates their +criteria, regardless of intent. Conversely, Figure 9 displays the 14% of the studies filtered bots +by removing but not classifying bot accounts from the ERH datasets [12, 15, 16, 46, 69, 93, 109]. +Strategies include removing duplicate spam text, filtering Reddit bots by name, and setting minimum +account statistics for verification—such as accounts with that share hashtags to at least five other +users to combat spam [16]. Likewise, Lozano et al. limited eligible users for their dataset to have at +least 100 followers, with more than 200 tweets, and at least 20 favourites to other tweets [69]. This +operates on the assumption that bots are short-lived, experience high in-degree between similar +(bot) accounts, and seldom have real-world friends or followers—as discovered by Bartal et al. [12] +Outside of removing suspicious bot accounts via human annotation in dataset generation, com- +putational means to explicitly categorise bots or trolls remains an area for future research. +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +SenticNet +LIWC +('Bag of Concepts/Narratives') +Hatebase +Word2Vec +Doc2Vec +Blacklists +Semanticizer +Non-contextualLexicon +SentiWordNet +Contextual Sentiment +SentiStrength +Approaches +Embeddings +SIMON +Entity Detection +Transfer Learning +TransferLearningDatasets +Tasks (pre-training) +(Stanford Sentiment) +EmoFeat +GPT +Dynamic +Dependency +FastText +few shot +Clustering +Parse Trees +learning +WordPiece +Theme and Narrative +Bag of +Tokenization +Clustering +Words +Entity Clustering +PCA +Probability-Occurrence Models +TF-IDF +DBSCAN +ANOVA F-value weighted BoWDown the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech +000:19 +Fig. 8. Types of data filtering techniques across NLP and community detection studies. +Fig. 9. Studies with Bot or Troll Filtering. +Fig. 10. Special Type of Data Used. +6.4 +Key Takeaways for Dataset Domain, Pre-processing, and Annotation +Twitter’s accessible API, popularity and potential for relationship modelling via reply and hashtag +networks, makes it the platform of choice for research (Figure 1). Despite the rise of far-right +extremism post-2015, Islamic extremism in the US and Europe remains the target group for the +majority of organisation-based studies, with no studies considering far-right/left manifestos. The +marginalisation of Oceania and the global south by datasets predominantly containing US white +hetero males indicates a structural bias in academia. For feature extraction, we recommend using: +(1) Contextual sentimental embeddings—due to their compatibility with deep learning models +and highest performance, per Table 4. +(2) Pre-defined lexicons—assuming they remain up-to-date with online culture. +(3) Probability-occurrence models—ideal for large-scale clustering of emergent groups [27, 99, 126]. +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +RELATIONS BETWEENPOSTS +6 +SEMANTIC FEATURE DATA AND DICTIONARIES +5 +NETWORKS +METADATA (PROFILE DATA, LOCATION) +SIMILARITY TO CORPUS DATA +3 +TEMPORAL DATA (TIME ZONE, TOPICAL) +3 +LAW ENFORCEMENT DATA +AUDIO-VISUAL DATA +0 +1 +2 +3 +4 +5 +6 +7Transliteration +correction or +removal +133tsp3akto +leetspeak +Spelling +(alphabetic +correction +representation) +Regular +Lowercase +Hashtag +Spelling +expressions +conversion +expansion +URL Topic +("#HelloWorld" to +Extraction +"Hello""World") +Hashtag +Clustering +Filtering +Word Stemming +Grammar +Techniques +Metadata +Relations +(Community +Detection) +Variant word +substitution +(alternate spellings) +LangDetect +Location +Short posts +(to remove non +(min word count) +Removal +English posts) +Stop words +URLS +Identity pseudo- +(for context +anonymisation +insensitive MLAs +(usenames, names, +mentions etc.)6% +14% +Yes +■No +Unspecified or +Irrelevant to Study +80%000:20 +Govers et al. +We do not recommend pre-defined lexicons on non-English text, new groups or ideologies—as +these lexicons may not translate to different concepts and slurs. We recommend adaptive semi or +unsupervised learning via contextual embeddings and entity/concept clustering for edge cases. +Research currently lacks multidomain datasets, pseudo-anonymous platforms, multimedia (i.e., +images, videos, audio and livestreams), and extraction of comedy, satire, or bot features. +7 +COMMUNITY DETECTION, TEXT, AND IMAGE ERH DETECTION ALGORITHMS +What are the computational classification methods for ERH detection? +Between 2015-2021, non-deep Machine Learning Algorithms (MLAs) shifted towards Deep +Learning Algorithms (DLAs) due to their superior performance and context-sensitivity (Table 4). +Support Vector Machines (SVM) and a case of a Random Forest (RF) model were the last remaining +non-deep MLAs post-2018 to outperform DLAs. Studies seldom hybridise relationship network +modelling and semantic textual analysis. Ongoing areas of research in MLAs consist of identifying +psychological signals to compete with DLAs such as Bidirectional Encoder Representations from +Transformers (BERT), Convolutional Neural Networks (CNN) and Bidirectional Long Short-Term +Memory (BiLSTM) models (defined further in the Supplementary Material’s Algorithm Handbook +section). DLAs are best oriented for text-only tasks and for hybrid image-caption models [3, 57, +99, 111]. Future NLP studies should consider higher-performing neural languages models over +BERT-base—such as RoBERTa [68], Sentence-BERT [96], or multi-billion parameter transformers +such as GPT-3 [23], GPT-Neo [19], or Jurassic-1 [65]. +7.1 +Observed Non-deep Machine Learning Algorithms (MLAs) +Studies investigating non-deep MLAs tend to test multiple models, typically Support Vector Ma- +chines (SVMs), Random Forest (RF), and Logistic Regression. Figure 11 outlines the distribution of +both deep and non-deep approaches, with SVM again the most popular MLA in 15 of the 51 studies. +Non-deep MLAs consistently under-performed for multiclass classification, whereby Ahmad et al. +identified that a prior Naïve Bayes model could not distinguish between ‘Racism’ and ‘Extremism’ +classes due to a low F1-score of 69%; while their LSTM and GRU model could detect such nuance +with a 84% F1-score [4]. Likewise, application-specific sentimental algorithms paired with MLAs +resulted in lower F1-scores compared to context-sensitive BERT models—which do not require +manual feature extraction [55, 71, 83, 126]. Sharma et al. claimed that SentiStrength was "...not +robust to various sentence structures and hence fails to capture important semantics like sarcasm, +negation, indirect words, etc. at the phrase level" [107, pg. 5]—a critique shared in six other non-deep +sentiment scoring studies [47, 69, 84, 89, 103, 130]. Consider the hypothetical case of "I am not happy +with those people", whereby context-insensitive (orderless) embeddings will not detect the negation +of happy nor the implicit euphemism for ‘those people’. +Hence, researchers have three options when designing ERH models: +(1) Avoid complex textual feature extraction and filtering by prioritising DLA development, or +(2) Prioritise manual textual and metadata feature extraction, such as psychological signals, +emotions, sarcasm, irony, temporal data, and/or +(3) Consider community detection (relationship network or topic modelling) features. +7.1.1 +Non-deep Machine Learning Algorithms in Community Detection Studies. +There is a discrepancy in the choice of algorithmic approach compared to NLP-oriented models +where less than a third of the community detection studies considered Deep Learning (DL) mod- +els [77, 84, 99], while NLP-only studies were majority DL (15 of 29). A reason for this discrepancy +would be the limited research in social media network analysis without investigating textual data, +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech +000:21 +Fig. 11. Number of instances of Machine Learning Algorithms (MLAs) used for ERH detection. +instead opting to cluster group affiliation via K-means [12, 16, 80], NbClust [12], weighted bipartite +graphing into Louvain groups [15, 16], and fast greedy clustering algorithms [80]. +We observed that graphing relationship networks result in two types of classification categories: +(1) Meso-level affiliation—semi or unsupervised affiliation of a user to an extremist group or +organisation, with a bias towards Islamic extremist groups [15, 16, 80, 84, 99, 101, 108]. +(2) Micro-level affiliation—(semi)supervised person-to-person affiliation to an annotated extremist, +such as radicalising influencers [12, 24, 77], and legal person-of-interest models [48]. +For organisational affiliation, information for clustering included the use of hashtags shared by +suspended extremist Twitter users and unknown (test) users [7, 15, 84, 95]. +For identifying a user’s affiliation to other individuals, researchers preferred non-textual graph- +based algorithms as they reduce memory complexity and avoid the perils in classifying ambiguous +text [16, 80]. Furthermore, 2016-2019 demonstrated a move from investigative graph search and +dynamic heterogeneous graphs via queries in SPARQL [48, 80] towards Louvain grouping on +bipartite graphs as a higher-performing (by F1-score) classification method [15, 16, 108]. +For hybrid NLP-community detection models, researchers mapped text and friend, follower/ing, +and mention networks via decision trees and kNN [77, 84], or used Principal Component Analysis +on extracted Wikipedia articles to map the relationships between discussed events and entities [99]. +An emerging field of community detection for extremism consists of knowledge graphs. Know- +ledge graphs represents a network of real-world entities, such as events, people, ideologies, situ- +ations, or concepts [125]. Such network representations can be stored within graph databases, +word-embeddings, or link-state models [48, 125, 127]. Link-state knowledge models consist of +undirected graphs where nodes represent entities and edges represent links between entities, such +as linking Wikipedia article titles with related articles based on those referenced in the article, as +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +SUPPORTVECTOR MACHINE (SVM) +15 +SENTIMENTAL ALGORITHMS +11 +LOGISTIC REGRESSION +10 +OTHER DECISION TREES +OTHER (NOVEL) NEURAL NETWORK +9 +GRAPHING ALGORITHMS +7 +NOVEL APPROACHES +6 +BERT +6 +CONVOLUTIONAL NEURAL NETWORK (CNN) +6 +LSTM-ONLY +6 +RANDOM FOREST +5 +ENSEMBLE NEURAL NETWORKS +5 +KNN CLASSIFICATION +NAIVE BAYES +GRU NEURAL NETWORK +PRINCIPAL COMPONENT ANALYSIS (PCA) +2 +K-MEANS CLUSTERING +2 +LDA (LINEAR DISCRIMINANT ANALYSIS) +2 +LINEAR REGRESSION +1 +LSA (LATENT SEMANTIC ANALYSIS) +1 +0 +2 +4 +6 +8 +10 +12 +14 +16000:22 +Govers et al. +used in Wikipedia2Vec [127]. Hung et al. consider a novel hybrid OSINT and law-enforcement +database graph model-—which unifies textual n-grams from social media to shared relationships +between other individuals and law enforcement events over time [48]. +Four studies consider model relationships to individual extremist affiliates [12, 24, 48, 77]. In a +direct comparison between text and relationship detection models, Saif et al. observed that text-only +semantic analysis outperformed their graph-based network model by a +6% higher F1-score [101]. +7.2 +Deep Learning Algorithms (DLAs) +DL studies are rising, with less than a third of studies including DLAs pre-2019 [10, 18, 27, 32, 45, +91, 99]. The percentage of all studies which included a DLA per year was 0% in 2016, 27.3% in +2017 [10, 27, 45, 99], and 33.3% in 2018 [18, 32, 91], compared to being the majority post-2018 (81.8% +in 2019 [1, 4, 53, 67, 71, 77, 83, 84, 90, 128–130], 54.5% in 2020 [14, 55, 57, 81, 107, 111] and 80% in +2021 [3, 52, 126])—with Figure 12 displaying the shift towards DLAs since 2015. +Fig. 12. Patterns of adoption for ERH detection algorithms over time. Colour change ordered by F1-score +trend (low to high). Brown = ~0.75 F1-score on benchmark datasets, Red = ~0.9 F1-score, Grey = No Data. +Between 2017-2018 Convolutional Neural Networks (CNN) using Long-Short Term Memory +(LSTM), GRU, Recurrent Neural Networks (RNN), or graph-based layers were the sole DLAs [10, 18, +32, 45, 91, 99]. From 2019-2021, various new approaches such as SenticNet 5 [89], ElMo (Embeddings +from Language Model) [90], custom neural networks such as an Iterative Opinion Mining using +Neural Networks (IOM-NN) model [14], and attention-based models such as BiLSTM [81, 83, 90, +128, 129]. Since 2019, there is an emerging consensus towards BERT [67, 71, 81, 107, 126, 129, 130] +due to its easy open-source models on the Hugging Face platform and high performance per Table 4. +7.2.1 +Deep Learning for Community Detection. +Only 3 of the 21 DL studies considered relationship network models [77, 84, 99]. Whereby, Mashech- +kin et al. grouped self-proclaimed Jihadist forums and VK users with Jihadist keywords as a +"suspicious users" category [77]. Uniquely, the researchers implemented a Hyperlink-induced Topic +Search (HITS) approach to calculate spatial network proximity between annotated extremists and +unknown instances. HITS identifies hubs, which are influential web pages as they link to numerous +other information sources/pages known as authorities [59]. The influence of an authority depends +on the number of hubs that redirect to the authority. An example of HITS in-action would be an ex- +tremist KavazChat forum (a hub) with numerous links to extremist manifestos (authorities) [59, 77]. +Evaluating influence in these graph networks requires measuring spatial proximity via betweenness +centrality [41] and depth-first search shortest paths where proximity to a known extremist via +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +Prior SLR trends. +Algorithmic and Feature Extraction Trends found in our SLR +Community +Expansion of Sentimental +Year-of-Hybrid-Models +Detection +algorithms. First +CNN+LSTM, BiLSTM. Use +BERT & SVM as +Blacklist +with Semantic +application of CNN +of language models +the dominant +Words. +Clustering. +LSTM, & Word2Vec. +(BERT), + approaches. +1 +2000-2010 +2010-2015 +2015 +2016 +2017 +2018 +2019 +2020 +2021 +1 +Bag of Words +Generic ML (sentimental +Year-of-Extremism- +Context-sensitive models +Approaches. +feature extraction in +detection via SVM, RNN +& features - phrase, +MLA) Bipartite Graphing. +kMeans. Emergence of +ExtremSentiLex, opinion +Waseem & Hovy dataset. +affiliation-based datasets +& propaganda mining.Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech +000:23 +following/reposting them constitutes an extremist classification. However, such relations do not +accommodate for replies to deescalate, deradicalise, or oppose extremist speech. +7.2.2 +Visual-detection Models for ERH Detection. +Despite the emergence of multimedia sources for radicalisation and ideological dissemination, +only three studies considered multimodal image and image-text sources—utilising image memes +with superimposed text from the Facebook hateful meme dataset [57] and the MultiOFF meme +dataset [111]. Only one study considered the post’s text (i.e., text not displayed on the image itself) +as context via the multimedia Stormfront post and image data from Rudinac et al. [99]. +For the Facebook hateful meme and MultiOFF datasets include images with superimposed cap- +tions [57, 111]. Both Kiela et al. [57] and Aggarwal et al. [3] extract caption text via Optical Character +Recognition (OCR) models—-a computer vision technique to convert images with printed/visual text +into machine-encoded text [125]. The three hateful meme studies utilised either both (multimodal) +or one (unimodal) of the image and its caption [3, 57, 111]. The multimodal Visual BERT-COCO +model attained the highest accuracy of 69.47%, compared to 62.8% for a caption text-only classifier +or 52.73% for image only, 64% for the ResNet152 model [3]; and 84.70% for the baseline (human) [57]. +Fig. 13. Deep learning pipeline for visual-text ERH detection based on the hateful meme studies [3, 57, 111]. +The highest performing multimodal model relied on Visual BERT [57]. Visual BERT extends +textual BERT by merging BERT’s self-attention mechanism to align input text with visual regions +of an image [64]. VisualBERT models are typically pretrained for object detection, segmentation, +and captioning using the generic Common Objects in COntext (COCO) dataset [66], such that +the model can segment and create a textual description of the objects behind an image such as +“two terrorists posing with a bomb” (Figure 13). Training otherwise acts the same as BERT-—which +involves masking certain words/tokens from a textual description of the image of what the image +depicts, and VisualBERT predicting the masked token(s) based on the image regions. We aggregate +and generalise all visual ERH detection studies architectural pipelines in Figure 13 [3, 57, 99, 111]. +No hateful meme dataset studies consider accompanying text from the original post. This raises +concerns regarding posts satirising, reporting, or providing counter-speech on hateful memes. +Only one study investigated a contextual textual post and accompanying images through a +proposed Graph Convolutional Neural Network (GCNN) model [99]. This GCNN approach extracted +semantic concepts extracted from Wikipedia, such as identifying that an image was a KKK rally— +attaining a 0.2421 F1-score for detecting forum thread affiliation across 40 Stormfront threads [99]. +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +Key: +Dotted lines = Optional path. Full line = necessary. +Red outline = Computer Vision (lmage Embeddings) +ResNet152 +Extract 1000- +Computer Vision +dimension +Model +embeddings +Extract Regions of +Create Textual Representation +Entity and Object +the Images (Visual +VisualBERT +("Two individuals with a +Detection +BERT tokens) +bomb") +Separate contextual +NLP model (e.g., +Top Text +BERT) +Textual Optical +Textual Feature +Concatenate +Hate Classifier +Character +Extraction +Embeddings +Neural Network +Recognition +"Top +"Bottom +Bottom Text +Text" +Text"000:24 +Govers et al. +8 +MODEL PERFORMANCE EVALUATION, VALIDATION, AND CHALLENGES +What are the highest performing models, and what challenges exist in cross-examining them? +Evaluating model performance presents three core challenges for future researchers: +(1) Dataset domain differences—which may include or exclude relevant features (e.g., gender, +location, or sentiment) and may involve numerous languages or groups (e.g., Islamic extrem- +ists vs. white supremacists) who will express themselves with different lexicons [20, 53]. +(2) Criteria differences—different standards for ERH definitions, criteria, filtering, and annota- +tion threaten cross-dataset analysis between models [24, 122]. Binary classification can result +in higher accuracy compared to classifying nuanced and non-trivial subsets of hate such as +racism/sexism [122], overt/covert aggression [61], or hateful group affiliation [53]. +(3) Varying and non-standard choice of metrics—Figure 14 displays the 28 metrics, which +vary depending on whether the study investigates community detection via closeness, in- +betweenness, and eigenvectors; or NLP, often via accuracy, precision, and F1-scores. +Fig. 14. Distribution of metrics used across the 51 studies—demonstrating a lack of standardisation. +Table 4. Models ranked by macro F1-score for the benchmark datasets across studies (inter-study evaluation). +Dataset +1st Highest +2nd Highest +3rd Highest +Waseem +and +Hovy [123] +0.966 (BERT with GPT-2 +fine-tuned dataset [126]) +0.932 (Ensemble RNN [91]) +0.930 (LSTM + Random Em- +bedding + GBDT [10]) +FifthTribe [39] +1.0 (RF [84]) +0.991-0.862 (SVM [7]) +0.87 (SVM [95]) +de Gibert [32] +0.859 (SP-MTL LSTM, CNN +and GRU Ensemble [55]) +0.82 (BERT [71]) +0.73 (LSTM baseline met- +ric [32]) +TRAC FB [61] +0.695 (CNN + GRU [55]) +0.64 (LSTM [61]) +0.548 (FEDA SVM [56]) +Hatebase Twit- +ter [31] +0.923 (BiLSTM with Atten- +tion modeling [83]) +0.92 (BERTbase+CNN / BiL- +STM [81], 0.86 (with racial/- +sexual debiasing module) +0.912 +(Neural +En- +semble [71]) +HatEval [13] +0.7481 +(Neural +En- +semble [71]) +0.738 +(LSTM- +ELMo+BoW) [90] +0.695 (BERT with GPT-2 +fine-tuned dataset [126]) +OffensEval [128] +0.924 (SP-MTL CNN [55]) +0.839 (BERT [130]) +0.829 (BERT 3-epochs [67]) +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +F1-SCORE +35 +ACCURACY +25 +PRECISION +21 +RECALL +19 +P-VALUE +3 +CORRELATION COEFFICIENT +NETWORKCENTRALITIES +2 +MEAN ABSOLUTE ERROR +STANDARDIZED CO-INCIDENCE RATIO +LOUVAIN (Q) MODULARITY +PATIAL PROXIMITY +1 +GINI COEFFICIENT +1 +INFORMATION GAIN +PARTITION SENSITIVITY +MEANLOG ACCURACY +1 +RACIST SCORE (CUSTOM METRIC) +1 +WILCOXON SIGNED-RANK TEST +CHI-SQUARE INDEPENDENCE +PEARSON CORRELATION COEFFICIENT +1 +PAGERANK SCORES +RADICAL SCORE (SENTISTRENGTH) +1 +HYBRID STRUCTURAL-SEMANTIC SIMILARITY +1 +POSSIBILISTIC SIMILARITY MEASURE +ROC AREA +SIRA RADICAL SCORE +1 +R^2 VALUE +0 +10 +15 +20 +25 +30 +5 +35 +40Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech +000:25 +8.1 +Benchmark Dataset Performance (Inter-study Evaluation) +We use macro F1-score as the target metric as it balances true and false positives among all classes, +and is a shared metric across the benchmark datasets [13, 31, 32, 39, 123, 128]. Table 4 outlines that +the highest F1-scoring models reflect the move towards context-sensitive DLAs like BERT, as also +displayed in Figure 12. SVMs and a single instance of a Random Forest classifier on sentimental +features were the last standing non-deep MLAs [7, 56, 84, 95]. Given the variety of MLA and DLAs +(Figure 11), approaches that frequently underperformed included Word2Vec, non-ensemble neural +networks such as CNN-only models, and baseline models [31, 46]. These baseline models include +the HateSonar model by Davidson et al. [31], Waseem and Hovy’s n-grams and gender-based +approach [123], LSTM model by de Gibert et al. [32], and C-Support Vector Classification by Basile +et al. [13]. No studies discuss memory or computational complexity, an area worthy of future +research as expanded in our Supplementary Material’s Section 3. +8.2 +Community Detection Performance +While community detection models tend to produce F1-scores ~0.15 lower than DLAs [12, 15, 16, 48, +80, 108], these comparisons rely on different datasets/metrics. Shi and Macy recommended using +Standardised Cosine Ratio as the standardised metric for structural similarity in network analysis, as +it is not biased towards the majority class, unlike Jaccard or cosine similarity [108]. For community +detection models on the same pro/anti-ISIS dataset [39], F1-scores ranged from 0.74-0.93 [7, 15, 95]. +Only one study cross-examined text and network features [101], with a hybrid dataset consisting +of annotated anti/pro-ISIS users’ posts and number of followers/ing, hashtags, mentions, and +location. Text-only semantic analysis outperformed their network-only model (0.923 F1 vs. 0.866 +respectively) [101]. However, topic (hashtag) clustering and lexicon-based sentiment detection +via SentiStrength underperformed compared to the network-only approach by a 0.07-0.1 lower +F1 [101]. Thus, unsupervised clustering models are ideal for temporal radicalisation detection and +identification of emergent or unknown groups or ideologies. There is insufficient evidence to conclude +whether community detection is superior to NLP due to the lack of shared NLP-network datasets. +For supervised community detection tasks, researchers [15, 80, 101] used network features via +Naïve Bayes [80], k-means [12, 16, 80], SVM [84], and decision trees [77, 84]. The highest F1-score +community detection model was a hybrid NLP and community detection model using network +features, keywords and metadata (i.e., language, time, location, tweet/retweet status, and whether +the post contained links or media) with a Naïve Bayes classifier—attaining a 0.89 F1-score [80]. +9 +FUTURE RESEARCH DIRECTIONS +In this section, we offer an alternate to the radicalisation = extremism = political hate speech consensus +from RQ1 and models observed in RQ3/4 to present a new framework for delineating and expanding +ERH for future work. Overall, we propose an uptake roadmap for ERH context mining to expand the +field into new research domains, deployments for industries, and elicit governance requirements. +9.1 +Ideological Isomorphism—a Novel Framework for Radicalisation Detection +Definition 4: Ideological Isomorphism (Computational Definition for Radicalisation) +The temporal movement of one’s belief space and network of interactions from a point of +normalcy towards an extremist belief space. It is an approach to detecting radicalisation with +an emphasis on non-hateful sentiment as ringleaders and/or influencers pull and absorb others +towards their hateful group’s identity, relationships, and beliefs. +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +000:26 +Govers et al. +As outlined in our novel tree-diagram dissection of ERH definitions to their computational +approach in Figure 6, there is considerable overlap in approaches between the otherwise unique +fields of extremism, radicalisation and politicised hate speech. Radicalisation’s working definition +suffers from ambiguity in the majority of studies due to its interchangeability towards extremist +affiliation and no considerations for temporal changes. Radicalisation’s computational definition +should reflect a behavioural, psychological, and ideological move towards extremism over time. +While extremist ideologies and outwards discourse towards victim groups may be manifestly +hateful, radicalisation towards target audiences may involve non-hateful uniting and persuasive +speech [58]. Hence, we propose that radicalisation detection should not be a single-post classification. +Rather, models should consider micro (individual), meso (group dynamics), and macro (global +events and trends) relations. The roots for radicalisation result from an individual’s perceptions of +injustice, threat, and self-affecting fears on a micro-level. On a meso-level, this can include the rise +of community clusters based on topics and relationships. Socially, a radicalised user draws on an +extremist group’s legitimacy, connections and group identity, trends, culture and memes [58, 70, 79]. +Hence mapping ideological isomorphism requires temporal modelling to: +(1) Detect the role of users or groups polarising or pulling others towards extreme belief spaces +(i.e., ideological isomorphism), akin to detecting online influencers [12, 46, 80, 121]. Studies +should also consider the role of alienation as a radicalising factor via farthest-first clustering. +(2) Further research into the role of friendship and persuasion by adapting sentimental ap- +proaches to consider positive reinforcement towards hateful ideologies akin to existing +research in detecting psycho-behavioural signals [84]. Furthermore, there lacks research in +computationally detecting social factors such as suicidal ideation or mental health. +(3) Investigate the interactions between groups across social media platforms as radicalisers +themselves, such as the promotion of extremist content by recommendation algorithms. +(4) Utilise community detection metrics such as centrality, Jaccard similarity, and semantic +similarity over time as measurements for classifying radicalisation for meso-level NLP (topic) +and graph-based (relational) clustering, leaving content moderation as a separate task. +(5) Consider the role of satire, journalism, and martyrs as areas for radicalisation clustering. +9.2 +Morphological Mapping and Consensus-building—a novel +computationally-grounded framework for extremism detection +Definition 5: Morphological Mapping and Consensus-building (Extremism) +The congregation of users into collective identities (‘in-groups’) in support of manifestly +unlawful actions or ideas. +While ideological isomorphism focuses on micro-level inter-personal relations, morphological +mapping pertains to clustering meso-level beliefs and community networks to extremist ideologies. +While we discovered various affiliation-based clustering approaches, no studies identified novel or +emergent movements. Establishing a ground truth for a novel extremist organisation is challenging +if such groups are decentralised or volatile. Hence, we recommend using manifestos, particu- +larly unconsidered far-right sources, and influential offline and online extremists as a benchmark +for identifying martyrdom networks and new organisations. Areas for future research include +investigating the role of trolls, physical world attacks, or misinformation in narrative-building. +Our morphological mapping framework proposes to delineate Extremism by considering the role +of group identity and ideological themes behind hate speech by considering affiliation across users +and posts. When targeting extremism, pledging ‘support’ to a terrorist organisation may not violate +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech +000:27 +context-insensitive BoW hate speech classifiers—-hence it is not appropriate to categorise extremist +affiliation under the same guise as post-by-post hate speech. Currently, extremism detection +constitutes a binary ‘pro vs anti group’ classification, which fails to capture the inner trends of +radicalisation from peaceful, to fringe beliefs, to committing to violent-inducing beliefs online, and +potentially to offline extremism. Investigating semi or unsupervised clustering (mapping) of groups +will also aid Facebook’s commitment to moderating militarised social movements, violence-inducing +conspiracy networks, terrorist organisations, or hate speech inducing communities [37]. +Thus, we propose four prerequisites for studies to fall under the extremism detection category: +(1) Investigate the interactions and similarities between groups on mainstream and anonymous +platforms to map group dynamics and extremist networks. For privacy, we recommend +group-level (non-individualistic) network and semantic clustering. +(2) Map affiliation and group dynamics. Given the lack of definitions for extreme affiliation, we +recommend using Facebook’s definition of affiliation as a basis—being the positive praise of +a designated entity or event, substantive (financial) support, or representation on behalf of a +group (i.e., membership/pledges) [37]. +(3) Investigate hateful and non-hateful community interactions, memes and trends, that reinforce +group cohesion. +(4) Map affiliation as a clustering task, akin to our proposed radicalisation framework but without +the temporal component. +9.3 +Outwards Dissemination––‘traditional’ hate speech detection updated +Definition 6: Outwards Dissemination (Hate Speech) +Targeted, harassing, or violence-inducing speech towards other members or groups based on +protected characteristics. +Hence, the projection and mainstreaming of hateful ideologies through speech, text, images, +and videos requires an outwards dissemination of views shared by extremists, such as racism. The +outwards dissemination of hate is a strictly NLP (text) and computer vision (entity and object) +classification problem. We delineate hate speech with affiliation to violent extremist groups as such +misappropriation could have devastating effects on one’s image, well-being, and safety [9, 24, 29]. +All researchers should be aware that malicious actors may exploit existing ERH models for +injurious surveillance and censorship. Future work should also consider the impacts of labels +on society at large, whereby terms such as ‘far-right’ as an alias for white supremacy is both +misleading, infers a ‘right vs wrong’ left-to-right spectrum, and ambiguous. We recommended +decoupling religious contexts in favour of technical terms such as ‘radical Islamic extremism’ or +‘terror-supporting martyrdom’ to avoid grouping religiosity to a political ideology and terrorism. +Thus, we propose three key prerequisites for a study to be in the hate speech category: +(1) Investigates textual or multimedia interactions only, whereby detecting cyber-bullying or +extremist community networks should be separate tasks. +(2) Decouple affiliation where possible. For instance, white supremacy instead of far-right (an +ambiguous term) or organisational affiliation. +(3) Consider models which include latent information, such as news, entities, or implied hate. +Datasets should explain each classification with categories for disinformation and fallacies. +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +000:28 +Govers et al. +Future work in outlining hate speech would be a systematic socio-legal cross-examination of hate +speech laws from governments and policies from social media platforms—including the emerging +consensus vis-à-vis the harmonised EU Code of Conduct for countering illegal hate speech [34]. +9.4 +Uptake Roadmap for Researchers, Industry, and Government +We present a pipeline for researchers, industries, and government analysts to approach ERH +context mining per Figure 15. In addition to this summary visualisation of our key dataset and +model recommendations, we expand on our actionable recommendations for immediate next steps +and long-term software requirements for ERH detection in our supplementary material. +Future SLRs should consider a mixture of academic studies, grey material, and technical reports +to further encompass our proposed ERH context mining field’s socio-legal component and explore +industry approaches. We recommend transforming our ethical recommendations for responsible +research outlined in our SLR design into formalised interdisciplinary guidelines to protect privacy +and researcher safety. ERH is never a singular end-goal, post, or unexpected event. Hence, detecting +erroneous behaviour emanating from mental health crises can both avoid ERH online and offline, +and present avenues for cooperation with third-parties such as suicide prevention and counselling +groups. Finally, we recommend searching for multimedia-only studies including for livestreams. +10 +CONCLUSION +ERH context mining is a novel and wide field that funnels to one fundamental aim—the pursuit +to computationally identify hateful content to enact accurate content moderation. In our work, +we harmonised Extremist affiliation, Radicalisation, and Hate Speech in politicised discussions from +2015-2021 in a socio-technical context to deconstruct and decouple all three fields of our proposed +ERH context mining framework. Hence, we propose a novel framework consisting of ideological +isomorphism (radicalisation), morphological mapping (extremism), and outwards dissemination +(politicised hate speech) based on our findings in RQ1. While hate speech included racism and +sexism, other forms of discrimination were seldom considered. Extremism and radicalisation +frequently targeted Islamic groups, particularly from US and European researchers. Binary post-by- +post classification remain the dominant approach despite the complexity of online discourse. +There is a clear and present danger in current academia emanating from the unresolved biases in +dataset collection, annotation, and algorithmic approaches per RQ2. We observed a recurring lack +of consideration for satire/comedic posts, misinformation, or multimedia sources. Likewise, data +lacked nuance without contextual replies or conversational dynamics, and were skewed towards +the US and Europe—with the global south, indigenous peoples, and Oceania all marginalised. +Computationally, we identified that deep learning algorithms result in higher F1-scores at the +expense of algorithmic complexity via RQ3/4. Context-sensitive neural language DLAs and SVM +with sentimental, semantic, and network-based features outperformed models found in prior +SLRs. However, state-of-the-art models still lack a contextual understanding of emergent entities, +conversational dynamics, events, entities and ethno-linguistic differences. To combat injurious +censorship and vigilantism, we recommended several areas for future work in context-sensitive +models, researcher ethics, and a novel approach to framing ERH in SLRs and computational studies. +The poor design and abuse of social media threatens the fabric of society and democracy. Re- +searchers, industries, and governments must consider the full start-to-finish ecosystem to ERH +context mining to understand the data, their criteria, and model performance. Without a holistic +approach to delineating and evaluating Extremism, Radicalisation, and Hate Speech, threat actors +(extremists, bots, trolls, (non-)state actors) will continue to exploit and undermine content modera- +tion systems. Hence, informed, accurate and ethical content moderation are core to responsible +platform governance while averting injurious censorship from biased models. +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech +000:29 +Fig. 15. ERH Context Mining pipeline—with key identified research gaps. +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +Sociolinguistics +Red = Computer- +science oriented. +Computational +Blue = social- +Research +Mixed = +science +Gaps: +Social Media +Criteria-building +misinformation. +Satire & +sexism. +beyond racism & +Hate speech +affiliation. +non-violent +Extremism as a +temporal process. +a friendly +Radicalisation as +Privacy & Ethics +Consider: +ERH +- +(RQ1, RQ2, RQ3) +Dataset Creation +Data Selection & +- + Collection +movements +Conspiracy +COVID-19 posts +extremist +Non-lslamic +anonymisation) +(data +Researcher safety +manifestos. +New APls +datasets +Multiple platform +platforms +/anonymous +Controversial +Consider: +Data Filtering +Extraction +Topic mapping to +Belief space +entities/events) +(Wikipedia/news +sources +informative +mapping +emotion detection +Context-sensitive +features +Suicidal ideation +signals +Psychological +Consider: +80 +Deployment (RQ3/4) +Model Choice & + Model Creation & + Performance +Privacy Paradigm. +Image detection +Hybrid NLP- +Meme detection +text generation. +conversational +Synthetic +learning. +Few-shot +(GPT) +Pretrained Models +Generative +Differential +design. +Privacy-by- +Consider: +Model Governance +benchmarks +metrics & +Industry standard +Meta-analysis +Interdisciplinary +Self-regulation +recognition +context +New event +updating) +(frequently +models +Dynamic 'online' +Consider:000:30 +Govers et al. +REFERENCES +[1] Umar Abubakar, Sulaimon A. 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Publication date: December 2021. + +Supplementary Material for: +Down the Rabbit Hole: Detecting Online Extremism, +Radicalisation, and Politicised Hate Speech +JAROD GOVERS, ORKA Lab, Department of Software Engineering, University of Waikato, NZ +PHILIP FELDMAN, ASRC Federal, US +AARON DANT, ASRC Federal, US +PANOS PATROS, ORKA Lab, Department of Software Engineering, University of Waikato, NZ +Contents +Contents +1 +1 +Definitions—The Algorithm Handbook +1 +1.1 +Definitions for Traditional (non-deep) Machine Learning Algorithms +2 +1.2 +Definitions for Deep Learning Approaches +5 +1.3 +Language Transformer Models +6 +1.4 +Definitions for Prominent Feature Extraction Techniques +7 +2 +SLR Design Considerations +8 +2.1 +Quality Assessment Criteria +8 +3 +The Case for Performance Engineering when Evaluating Models +10 +4 +Uptake Roadmap Expanded +10 +4.1 +Model Recommendations +10 +4.2 +Dataset Recommendations +11 +References +12 +1 +DEFINITIONS—THE ALGORITHM HANDBOOK +This supplementary material document includes the supplementary material referenced in the +main Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech +Systematic Literature Review. This document offers a ‘dictionary/look-up table’ for the core al- +gorithmic architectures for the non-deep machine learning and deep learning models mentioned +throughout the SLR, alongside other side findings and design considerations. We contextualise the +Authors’ addresses: Jarod Govers, jg199@students.waikato.ac.nz, ORKA Lab, Department of Software Engineering, University +of Waikato, Gate 1, Knighton Road, Hamilton, Waikato, NZ, 3216; Philip Feldman, philip.feldman@asrcfederal.com, ASRC +Federal, Beltsville, Maryland, US; Aaron Dant, aaron.dant@asrcfederal.com, ASRC Federal, Beltsville, Maryland, US; Panos +Patros, panos.patros@waikato.ac.nz, ORKA Lab, Department of Software Engineering, University of Waikato, Gate 1, +Knighton Road, Hamilton, Waikato, NZ, 3216. +Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee +provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and +the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. +Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires +prior specific permission and/or a fee. Request permissions from permissions@acm.org. +© 2021 Association for Computing Machinery. +0360-0300/2021/12-ART000 $15.00 +https://doi.org/tobereplacedwhenassignedDOI +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. +arXiv:2301.11579v1 [cs.SI] 27 Jan 2023 + +000:2 +Govers et al. +relevant strengths and weaknesses of the various algorithmic approaches for text and visual models +for ERH detection. No new findings are in this handbook/‘look-up table’. Hence, those familiar +with the models listed in the contents above need not read this section. +1.1 +Definitions for Traditional (non-deep) Machine Learning Algorithms +We aggregate common and historic non-deep machine learning algorithms into the ‘traditional’ +MLA category. Hence, this section defines each of the baseline models used for textual or community +detection models—consisting of: +(1) Sentimental Bag of Words approaches, +(2) Naïve Bayes, +(3) Decision Trees, +(4) Support Vector Machines, +(5) Clustering Models. +1.1.1 +Bag of Words (BoW). +BoW approaches simplify complex contextual sentences into a multiset (‘bag’) of individual words +by assigning a value or probability to each word in its relation to a specific document class. For +instance, a BoW approach would deconstruct the contiguous sentence, “The Eldian people are +the spawn of the devil” (where Eldian is a fictitious race), into an unordered bag of individual +words. While ‘are’, ‘the’ are unlikely to have a considerable influence on whether a sentence is hate +speech or not, the use of ‘devil’ and ‘Eldian [race]’ is more frequently paired in hate speech than +for non-hateful/off-topic text. The disregard of word order and the relationship of BoW approaches, +and MLA models at large, constitute context-insensitive models. For instance, a BoW model does +not know that ‘I love the Eldian people but hate their food’ is paring love -> Eldian, and hate -> food, +and thus would consider ‘I hate the Eldian people but love their food’ as identical. Likewise, BoW +approaches do not consider alternate word meanings/uses (e.g., ‘I ran for government’ vs. ‘I ran +away’). Nonetheless, BoW approaches are core to word-specific ‘blacklists’ in content moderation, +such as banning users who use slurs in a post. However, for nuanced and often politicised discussions +on controversial topics, simple blacklists can lead to injurious censorship—due to the context and +use of such words. +Sentimental algorithms, such as SentiStrength [42] aggregate individual words into individual +emotions—whereby ‘love’ indicates a positive sentiment, while ‘hate’ generally appears in vitriolic +speech. Figure 1 outlines an abstracted representation of the sentiment classification based on the +average sentiment score of a sentence. However, the context-insensitive BoW models again fails for +nuanced cases, whereby Sharma et al. [41] identified that SentiStrength cannot detect negations +(e.g., “I am NOT happy” where happy skews the final sentiment scores). +1.1.2 +Naïve Bayes. +Naïve Bayes classifiers represent types of probabilistic classifiers utilising Bayes theorem with +the assumption that the influence of each variable for classification is independent of each other +(i.e., naïve) [44]. For document classification, notable features are assigned a probability for their +occurrence given a specific class. For instance, a hate speech post that has an angry sentiment may +have a P(0.8) (Probability of 80%) of being hateful, given that a test hate speech dataset may be 80% +angry speech. Bayes rule represents these chains of (assumed) independent/unrelated probabilities +to form a final probability for a test instance. +Notable features for probability models include: +• Textual features—(e.g., sentimental scores, appearance of certain slurs/terms), +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +Supplementary Material for: +Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech +000:3 +Fig. 1. An abstracted example of Bag of Words approach within a Naïve Bayes classifier—demonstrating its +lack of context sensitivity and the focus on key ‘racist’ words for ERH detection tasks. +• Network data—(e.g., probability that someone who is friends with a supremacist is also a +supremacist, retweet relationships), +• Metadata—(e.g., length of a post, readability via a Flesch Reading Ease score, number of posts). +In the example of Figure 1, the probability that the tweet is racist depends on the probability that +the racist tweet is angry, contains racial terms (‘Eldian’), the semantic similarity between known +hate speech posts, and the appearance of a negative lexicon. Naïve Bayes can be a final classifier +for aggregating context-sensitive embeddings (e.g., deep learning models) and multiple ‘ensembles’ +of approaches/models—via chaining their probabilities together with this Bayes rule. +1.1.3 +Decision Trees. +Fig. 2. An example of a decision tree, with the leaf nodes constituting the classification. In the Eldian +hate speech example, this would require traversing the left branches recursively for the final ‘Hate Speech’ +classification leaf (shown via the red arrows). +Chaining the correlations between features and their class likelihood can also span a tree of +scenarios. If an annotated dataset indicates that a post is 80% likely to be racist if a sentiment-scoring +algorithm detects anger, then a binary decision emerges—if post contains angry words, then likely +hate speech; if not, then not hate speech. These rules construct decision trees, where the root +constitutes the instance (text, network, metadata, or image), and each node is a decision, with the +leaves (final node) being the expected class value (i.e., the classification) [44]. Hence, decision trees +are not naïve as they rely on specific values of other features when traversing a tree’s branches for +a prediction. +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +Example: Naive Bayes (simplified): +"The Eldian people are the spawn of the devil" +Sentiment +Target Entity +Semantic Similarity to +Lexicon also affiliated +Annotated Posts +with extremists: +Angry +“hate' +Eldian +0.9 +1 (devil) +P(Racist I Tweet) = +“Eldian' +'devil' +P(Rac.IAngry) * P(Rac.I Eldian) * P(Rac.I O.9) * P(Rac.I 1 lexicon) +“scum" +P(Racist)Post to Classify +Sentiment = Not +Sentiment = +Angry +Angry +Contains slurs +>=0.5 semantic +<0.5 semantic +No slurs +similarity +similarity += Yes (Devil) +Not Hate +Not Hate +Hate Speech +Hate Speech +Speech +Speech000:4 +Govers et al. +Creating an optimal tree that maximises accuracy and precision is not trivial due to the feature +explosion of possible rules and tree nodes. Hence, Random Forest classifiers rely on a divide- +and-conquer algorithm for generalising feature pairings into class classifications with a random +initialisation [18]. This recursive process requires finding optimal splits to maximise the separation +of classes for a final leaf, with an example tree presented in Figure 2—where a random forest would +consists of multiple trees as a forest. Ideally, a leaf node should encapsulate instances of one class. +Random forests generate multiple decision trees and select the final prediction based on the +predictions from the majority of decision trees. Utilising multiple trees with a random initial tree +state increases the range of features and values selected during the training step. Utilising multiple +trees and testing the models on untrained ‘test’ data minimises the risk of over-fitting to the training +(i.e., a classifier which performs reliably on the training dataset but not on real-world data). +Random forests strengths include its ability to tie dependent and complex features while reducing +over-fitting through pruning (i.e., reducing tree size to generalise the model). Hence, decision trees +capture related concepts in hate speech where naïve BoW approaches do not—such as the appearance +of anger/negative sentiment invoking the use of charged terms (e.g., racism as an emotional outlet) +or frequency of posts and sentiment. +1.1.4 +Support Vector Machines (SVM). +Fig. 3. Support Vector Machine where instances beyond the boundaries (support vectors) are automatically +assigned to the class. +SVMs are another supervised learning model for classification and regression tasks, seeking to +map instances in vector spaces to maximise the distance between classes [14], visualised in Figure 3. +Mapping features to multidimensional vectors can exponentially increase dimensions (an issue +shared in deep-learning models). Thus, SVMs reduce irrelevant features through specific kernels— +typically a linear, polynomial, Gaussian or sigmoid function. These kernels reduce the feature +set to draw boundaries between two classes, similar to logistic regression. These boundaries are +either hard (i.e., a binary classification) or soft—allowing outliers near the boundary for edge cases, +like niche controversial and offensive, but not ostensibly targeting protected characteristics. SVM +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +Support Vector +(with cost hyperparameter) +Neutral +Speech. +Y-Axis +Hate +Speech +Optimal hyperplane +to maximise +distance (y=mx+c) +X-AxisSupplementary Material for: +Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech +000:5 +models are computationally faster and reduce memory compared to deep learning models [3, 45], +while achieving comparative performance outlined in RQ4. Dimensionality reduction techniques +can also reduce runtime by reducing the complexity of large feature spaces from textual or network +data, such as via Principle Component Analysis [27]. +SVMs are the consistently highest performing MLAs per RQ4, while lowest complexity, with +𝑂(𝑚∗𝑛) complexity for a Linear Kernel SVC—where m = feature count, and n = number of instances. +1.1.5 +Clustering and Nearest Neighbour Classifiers. +Instead of annotated hate speech datasets, clustering methods group by textual similarity via Natural +Language Processing (NLP), and network relations via Community detection. Hence, clustering can +work in cases of fully annotated datasets as supervised learning, semi-annotated datasets as semi- +supervised learning, or unlabelled raw web scrapped data for unsupervised learning. +For supervised learning, K-Nearest Neighbour (KNN) classifiers work via evaluating the nearest +neighbours’ likeliness when projecting the textual, network, or metadata features onto a multidi- +mensional space [2]. The ‘distance’ between feature spaces typically rely on Euclidean, Manhattan, +or Minkowski distance—where the latter two are suited for non-linear feature spaces. Non-euclidean +distances are ideal where dimensions are not comparable, as Manhatten distance reduces noise/er- +rors from outliers since the gradient has a constant magnitude. +Clustering examples for hate speech detection includes K-Means, which partitions n observations +into k clusters [27]. K-Means automatically generates clusters, thus does not require annotated +datasets. Hence, K-Means can detect novel groups, including emergent extremist organisations, +or influential individuals [4]. Unsupervised clustering’s strength for ERH detection is how it +circumvents the definition issues for annotating data and can cluster large movements without +costly annotation. However, K-Means may not identify manifestly hateful posts, as it does not abide +by any standard imbued within strict annotation criteria. Evidently, in the cross examination of +a naïve approach vs their proposed K-Means derived model by Moussaouri et al. [30], the naïve +approach outperformed the possibilistic clustering by 0.07-0.14 for accuracy 0.04-0.05 for precision. +1.2 +Definitions for Deep Learning Approaches +Deep learning represents a family of machine learning algorithms with multiple layers and com- +plexity, typically via neural network architectures. Neural networks rely on training a network with +a set of weights at each layer, known as neurons. The first layer of a neural network utilises numeric +representation of an instance (e.g., hateful text) in numeric ‘tokenised’ form, which is adjusted +throughout the hidden lower layers towards a final output (typically) classification layer. Each +downwards training step results in readjusting the weights of the upper layers for the neurons— +known as backpropagation [38]. Figure 4 displays this architecture for neural networks per our +example. The benefit of DLAs in ERH detection is the preservation of word order and meaning (e.g., +“I ran” vs “I ran for president”), thus displaying context-sensitivity. Given dual-use words such as +‘queer’, or racially motivated slurs, understanding the surrounding contextual words is essential to +reduce bias via misclassifications [31]. DLAs dominant the benchmark dataset leader-board in RQ4. +1.2.1 +Convolutional Neural Networks (CNN). +Convolutional Neural Networks (CNNs) expand on the neural network model through a convolu- +tional layer—which acts as a learnable filter for textual or image embeddings [44]. Moreover, CNNs +include a pooling layer(s) to reduce the spatial complexity of the network’s features. Reducing +spatial size helps reduce the number of parameters and thus training time and memory footprint, +while reducing over-fitting by generalising patterns in the training data. +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +000:6 +Govers et al. +1.2.2 +Long Short-Term Memory (LSTM) and Gated Recurrent Unit (GRU). +LSTM and GRU aim to increase contextual awareness to process data sequences with long-term +gradients to retain information on prior tokens [12, 19]. LSTM and GRU seek to reduce the vanishing +gradients caused during backpropagation steps, which reduces classification performance as older +trained instances are effectively ‘forgotten’ due to later weight changes. Similarly, GRU’s are a +gating approach with fewer parameters and thus higher runtime, enabling larger neural networks +overall. CNN models with LSTM and GRU connections outperform CNNs on their own for hate +speech detection [22, 23, 32]. The highest performing BiLSTM model expands LSTM for bidirectional +input, via two LSTMs—where tokens in the network utilise information from past (backwards) +tokens/data and future (forwards) data [32]. The ability to uphold the temporal memory of prior +tokens (attention) constitutes a Recurrent Neural Network (RNN). +1.3 +Language Transformer Models +The state-of-the-art transformer architecture relies on self-attention—the memory retention of +neural networks where each token of a sequence is differentially weighted [8, 16]. Unlike Recurrent +Neural Networks (i.e., neural networks where nodes follow a temporal sequence), a transformer’s +attention mechanism utilises context for any position for the token sequence. Hence, transformers +can handle words out of order to increase understanding. Transformers offer greater classification +performance (see RQ4) at the expense of memory and computational overhead. A considerable +ethical threat of transformer models is their capability to predict future tokens (i.e., text generation). +For instance, a malicious actor could create realistic automated trolls or radicalising synthetic +agents as bots. Language models also risk data leakage of their trained data through predicting +tokens found in the original trained dataset, such as names or addresses [8]. +Fig. 4. An abstracted example of a neural network for text. The top text represents its raw syntactic form, +with its converted numeric embedding representation. These embeddings are responsible for altering the +weights to increase token prediction or generation (for transformers) via backpropagation. The final output +layer for this example would offer the probability that the given text is racist, sexist, or benign. +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +topic +relation +relation +entity +The +Eldian +people +are +the +spawn +of +the +devil +0.125 +0.553 +0.814 +0.125 +0.25 +0.5 +0.19 +0.25 +0.666 +Segment and Positional tokens +(i.e. computational embeddings for position/order of words). +KKKKKKKK +Contextual interconnected layers (e.g. neural network) +Racism +Sexism +Neither +Probability +Probability +ProbabilitySupplementary Material for: +Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech +000:7 +1.3.1 +Cross-encoders (e.g., BERT). +Bidirectional Encoding Representations from Transformers (BERT) is the most common cross- +encoder observed for ERH detection [16], with the highest performance of all NLP models. Cross- +encoders offer higher performance for classification tasks, through retaining information over a +given sequence with a label (i.e., self-attention). BERT’s core strength is its memory retention of all +tokens in a sentence, thus upholding full context-sensitivity of every word in the post it seeks to +classify. However, cross-encoders are computationally expensive due to high parameter counts +(110 million parameters for BERT-base, 365 million for BERT-large), an issue further outlined in +RQ4. Hence, an area of ongoing research includes model distillation (optimising and reducing +parameter count to reduce memory requirements and training time), specialised training datasets, +and alternate layers [26, 37, 46]. BERT is pre-trained on entries from English Wikipedia (2.5 million +words) and the English BookCorpus (800 million words) [16]. Hence, such pre-trained models are +then fine-tuned on a smaller dataset (typically 1000+ instances, per RQ2’s benchmark datasets) to +optimise the BERT weights to detect hate speech with the context of its pre-trained corpus. +1.3.2 +Generative Pre-trained Models (GPT). +Similarly, the state-of-the-art GPT transformer architecture expands on the encoder blocks (shared +with BERT) to include decoder blocks [8]. Hence, GPT works on a single token (i.e., word vector) +and produces estimates for the sequence’s next token—ideal for tasks such as text generation, +summarising, question answering, and information retrieval. +GPT models differ from BERT-based models via masked self-attention—an alternate form of +context-sensitivity where the model only knows the context of the prior words in the sentence. +GPT-2/3 [8], GPT-Neo [7], and Jurassic-1 [25], are notable 2019-2021 era multi-billion parameter +models—where larger datasets and parameter count result in more human-like text generation and +higher performance in information retrieval tasks [8]. +GPT’s core strength in ERH detection synthetic hate speech generation via a GPT model fine- +tuned on a hateful corpus—as investigated by Wullah et al. (see RQ3) [45]. However, state-of-the-art +GPT models utilise up to 178B parameters, whereby memory and computational requirements +scale linearly. Hence, future GPT work in synthetic text generation should consider inference tasks +over fine-tuning. Specifically, inference utilises a pre-trained model’s on-demand text generation +capability through prompts rather than altering each of the billions of weights. Using the auto- +complete-like inference capabilities for generating realistic synthetic hate speech posts constitutes +a novel case of prompt engineering in ERH detection and thus is a potential future research project. +1.4 +Definitions for Prominent Feature Extraction Techniques +This subsection outlines the three most common feature extraction techniques used for textual ERH +detection—as outlined in RQ2 in the SLR. These models seek to identify hateful lexicons from text, +or create numerical representations for word or sentence meaning via embeddings. We deconstruct +the six most common feature extraction techniques observed in our SLR. +1.4.1 +Word2Vec. +Word2Vec is a model to convert words into vector embeddings, which compares synonymous words +(e.g., ‘hate’ and ‘disgust’) via numerical vectors [29]. Word2Vec compares these word-to-vector +embeddings via semantic similarity by evaluating their cosine similarity between their vectors +(e.g., comparing word vectors of an unknown class instance to words from a known ‘hate speech’ +instance(s) to make a ‘hate or not’ classification). On a word-level basis, the vector value for ‘king’ +- value for man + value for woman would result in a vector similar to queen [29]. In our case, a +‘Islamist extremist’ and ‘ISIS’ are semantically similar akin to ‘White Supremacy’ and ‘Nazism’. +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +000:8 +Govers et al. +1.4.2 +Doc2Vec. +Similar to Word2Vec, Doc2Vec aggregates vector embeddings for paragraphs in addition to indi- +vidual words [24]. Thereby offering memory of the current context and paragraph’s topic—useful +for understanding a whole post’s sentiment and meaning. +1.4.3 +N-grams. +N-grams represent contiguous sequences of n-number of characters for frequency analysis given +their non-linear distribution in English, as well as when comparing a radical vs non-radical cor- +pus [44]. This linguistic model is often paired with methods such as TF-IDF or BoW. +1.4.4 +Term Frequency-Inverse Document Frequency (TF-IDF). +TF-IDF determines the relevance of a word in a document by comparing its frequency in the +document compared to its inverse number for the frequency of that word across all documents [44]. +Thereby, assigning each word a weight to signify its semantic importance compared to the wider +corpus. For instance, radical Islamist dog-whistle terms (i.e., coded or suggestive political messages +intended to support a group) appeared disproportionately in extremist text compared to a neutral +religious corpus [36]. +1.4.5 +SenticNet. +SenticNet embeds pattern matching, parser trees, and LSTM-CNN models for sentiment analysis, +with the aim to replace a naïve BoW approach within a proclaimed bag of concepts and narrat- +ives [10]. Specifically, it includes feature extraction methods of concept parsing (i.e., understanding +linguistic patterns in natural language into conceptual pairs), subjectivity and polarity inference, +alongside personality and emotion extraction. +1.4.6 +Global Vectors for Word Representation (GloVe). +GloVe offers an unsupervised learning algorithm for context-independent word-to-vector embed- +dings [34]. While similar in creating vectors akin to Word2Vec, GloVe instead establishes word +co-occurrences using matrix factorization (i.e., co-occurrence matrix of word [row] and context +[usage of the word in the document]) and dimensionality reduction techniques. +2 +SLR DESIGN CONSIDERATIONS +This supplementary material section outlines the additional criteria and considerations for selecting +papers and ensuring privacy-protections for users, groups and collected data. In essence, this section +offers a meta-analysis of the ethics and selection process used throughout the SLR. +2.1 +Quality Assessment Criteria +The following includes our paper inclusion quality check criteria—with a score of 13 or higher +required for inclusion in the final paper selection (i.e., final 51 papers included). +We propose a critical criteria for quality assessment to filter irrelevant or ambiguous studies. +Specifically, for a study that passed a title and abstract screen, we assess the study’s clarity for +ERH definitions and annotations (for objective and legible classifications), methodical clarity (i.e. +outlining each study’s algorithmic model, methods, data collection processes, and statistical analys- +is/evaluation methods), and socio-technical considerations. We weighted each quality assessment +section to prioritise their research methodology and clarity in their technical methods over their +Conceptual Quality for studies encompassing broader socio-technical issues such as ethics, legality, +or ERH clarity. After a ten paper pilot study, we selected a score threshold of 65% to exclude +irrelevant or ambiguous studies. Our supplementary material document includes the criteria and +scoring for our quality assessment rubric. +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +Supplementary Material for: +Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech +000:9 +2.1.1 +Computational Quality (0 = None, 1 = Partial, 2 = Full). +(1) Is the radicalisation/affiliation detection model clearly defined? +(2) Is the radicalisation/affiliation detection model’s algorithm clearly defined? +(3) Is the training data reputable? +(4) Are the models results compared to similar state-of-the-art methods? +(5) Is the methodology for designing and conducting their experiment clearly defined? +(6) Are patterns and trends discussed and presented clearly? +2.1.2 +Epistemological Quality (0 = None, 1 = Partial, 2 = Full). +(1) Does the source(s) (data or researchers) avoid any conflict of interests or expressed biases? +(i.e., explicit support/funding from a political think tank or state agency). +(2) Does the study provide a cited or evidence-based definition for “radicalisation”, “hate speech” +or "extremist" affiliation? +(3) Are the dataset annotations vetted by more than one annotator to reduce bias? +2.1.3 +Conceptual Quality (0 or 0.5 value, as not critical but useful). +(1) Does the study discuss social or ethical issues in ERH detection (e.g. censorship)? +(2) Do the authors discuss the legality of their model or definitions? +(3) Does the study evaluate its model across multiple social media platforms? +(4) Does the study discuss regulatory frameworks or recommendations for social media platforms +based on their findings? +2.1.4 +Researcher Ethics. +We focus on key terms and compositions of ERH examples to protect the privacy of the individuals +exposed, as recommended by meta-studies on extremism research ethics [9, 13, 28]. When linking +ERH detection to real-world groups and events, we solely focus on events and organisations which +resulted in media attention or criminal convictions. In no part during this SLR did we attempt to +track users, groups, or correlate online users to any personally identifiable information (name, +location, username etc.) given the ease of composing online data into a traceable online fingerprint. +Similar to the social norms in New Zealand in the aftermath of the Christchurch shooting, no +extremists, terrorists, and/or criminals are referred by name to minimise publicity. We recognise +the potential for political or cultural bias in this charged field by citing international non-partisan +Non-governmental Organisations when framing ERH concepts, and avoid searching any party or +ideology in our search strategy. Moreover, we encourage that our findings and recommendations +invoke an open debate among social media platforms, governments, and the wider public. However, +we do not condone the use of ERH detection in social media as a form of autonomous law. We +recommend human-in-the-loop processes when handling or classifying data via independent +reviews, privacy protections, and complaint and redress mechanisms for deployed models. +Our recommendations thereby focus on Open Source Intelligence (OSINT) oriented studies +that do not consider governmental or private-conversation surveillance (with the exception of +one hybridised study that appeared in our search [20]). We thereby consider ERH detection as +a computational method aimed at garnering community-insights, trends, and flagging for social +media platforms themselves to use. Whether ERH detection policies should encourage deplatform- +ing, deranking, demonetisation, fact-checking, or targeted counter-speech/prevention programs +require further research. We encourage open interdisciplinary research in public and private- +communications—particularly ethical and legal discussions. +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +000:10 +Govers et al. +3 +THE CASE FOR PERFORMANCE ENGINEERING WHEN EVALUATING MODELS +While high F1-scores help enforce community guidelines via accurate predictions and reduce +injurious censorship from false positives, runtime performance trade-offs are seldom discussed. +DLAs may perform within 1% (F1-score) of their MLA counterparts in NLP studies but require +significantly higher computational resources. For instance, fine-tuning a BERT-large model for NLP +tasks requires Graphics or Tensor Processing Units (GPU or TPU), restricting researchers from +testing large language models [45, 47]. For community detection, uncompressed network models +can include up to 27.4 million links [6], which significantly increases computational and memory +requirements for a minimal 1-5% performance gain. Specifically, using a Possibilistic Approach (PA) +with dimensionality reduction reduced subgraph mining runtime by up to 67% (1500 seconds to +500 seconds on an 8-core 3.2GHz system), while reducing accuracy by only 4% [30]. Furthermore, +community-level insights on topics with millions of tweets, relations, and discussions can lead +to a network explosion with a non-deterministic polynomial runtime [5, 30]. In graph-detection +approaches, performance engineering and optimisation for mining frequent subgraphs and graph- +traversal is an active area of research [30]). No NLP studies consider performance engineering for +DLAs despite developments in model distillation and sentence-level embeddings [37]. +Thus, we recommend that researchers consider performance trade-offs in future work and +investigate a possible standardised performance-complexity metric (e.g. parameter count vs. F1- +score ratio) to build scalable, energy-efficient and fiscally-viable models. Moreover, fine-tuning +or retraining DLAs, or regenerating frequent subgraphs for community detection, should be a +frequent endeavour to adapt to the rapidly evolving topics, entities, and events throughout online +discourse. Due to the computational costs of fine-tuning or training multi-billion parameter models, +we recommend approaches that do not require expensive training, such as few-shot learning (i.e., +giving several known instances of ERH and a unseen ‘test’ instance) and prompt engineering [8]. +4 +UPTAKE ROADMAP EXPANDED +This supplementary section expands on the dataset and model research gaps highlighted in Figure +16 of the main Down the Rabbit Hole SLR document. We categorise these research recommendations +into eight core components for our proposed ERH Context Mining research field. +4.1 +Model Recommendations +The two predominant recommendations for future work are investigating the role of changes in +hateful affiliation or speech over time to satisfy the temporal requirement for Radicalisation detection, +and to train models on multiclass datasets from multiple platforms. We note that only one study +considered temporal data on both meso and macro (changes within and between groups), and micro +(individual) levels, although recommended as future work within four other studies [3, 11, 20, 40]. +Moreover, we recommend expanding on DLAs as the target for future research based on their +leading performance in RQ4. Neural language models offer a macro-level societal understanding +due to their pre-trained corpus on academic sources, OpenWebText2 Reddit discussions, and +Wikipedia [8]. Furthermore, transformer models beyond 764 million parameters are untested. +Bot, troll, meme, entity, dis/misinformation and satire detection remain underdeveloped—-which +could lead to censorship or undermine democratic institutions. Five studies recommended multi- +media detection as future work [1, 5, 11, 17, 35]. +To protect user privacy from recreating user content from neural language models, we encourage +privacy-by-design software engineering through machine learning paradigms such as Differential +Privacy (DP). DP-paradigm models and datasets reduce the potential for self-identification from +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +Supplementary Material for: +Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech +000:11 +trained models (i.e. data leakage, such as names or usernames in open-source datasets), as DP- +paradigm models use pseudo-anonymised patterns of groups and hate. +4.2 +Dataset Recommendations +To investigate the roles of radicalisation, we recommend expanding on the dataset annotation +approach by de Gibert et al. [15] by creating a conversation-level dataset with public non-hateful +replies to a post for context. Moreover, future benchmark datasets should consider pulling data across +platforms to investigate macro-level radicalisation trends between platforms. We note that only +two studies considered anti-Asian sentiment in COVID-related tweets, targeting a seldom explored +topic and demographic [21, 39] worthy of expansion given the ongoing COVID-19 pandemic. +Likewise, future datasets should consider the role of indigenous discussions and potential re- +searcher biases given the Anglo-dominant field of ERH research. Given the rise of COVID extrem- +ism [43], far-right movements, and xenophobia in Oceania. Hence, we recommend geotargeted +datasets to consider the differences for investigating ERH topics, which would demonstrate NZ’s +commitment to our Christchurch Call to Action Summit. Investigating unexplored and minority +groups could also provide imperative insights for social scientists regarding the conversational +dynamics, morphological mapping, and ideological isomorphism from radical minority groups +towards the majority. Likewise, research on vulnerable communities (youth, gender and sexual +minorities, religious, racial, and geographically distant peoples) would aid social media platforms +in both identifying unique radicalising risks, as well as avenues for support and de-escalation. In +the mental health end, we recommend building on Nouh et al.’s proposed approach of extracting +Fig. 5. ERH Context Mining (ERH-CM) eight core components for Research, Industry, and Government. +ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021. + +Collective +Researcher +Researcher, +Tasks +Industry, +Govt. +1. Harm +Reduction +Tasks +Introspection & +Reflection +** +Community +2. +Detection +7. Privacy +副 +ERH-CM +3 +Increased +& Data +Context +Sovereignty +6. Self-regulation +Diverse +& Ethics +Datasets +5. Quality Assessment +Industry +Collective +& Control +Tasks +(human-in-the-loop) +Researcher, +Industry +Tasks000:12 +Govers et al. +textual, psychological and behavioural features [33], both due to its performance, as well as its +potential for analysing societal factors and ERH roots such as correlations between mental health +issues (isolation, depression etc.) and vulnerability to radicalisation towards violent extremism. +For any counter-extremism or de-radicalisation studies, we recommend work in ethical and legal +guidelines to protect privacy, avoid backlash or inadvertent algorithmic amplification. +Investigating posts from periods of political, or social crisis (e.g., COVID health measures, post- +terror attack discourse etc.) could also help identify cases of ERH on mainstream platforms before +they are deplatformed/removed. 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Publication date: December 2021. + diff --git a/5NFJT4oBgHgl3EQfkiwH/content/tmp_files/load_file.txt b/5NFJT4oBgHgl3EQfkiwH/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..41ab458628e00a28d9f167256812d55ffb30eb6c --- /dev/null +++ b/5NFJT4oBgHgl3EQfkiwH/content/tmp_files/load_file.txt @@ -0,0 +1,3154 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf,len=3153 +page_content='Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech JAROD GOVERS, ORKA Lab, Department of Software Engineering, University of Waikato, NZ PHILIP FELDMAN, ASRC Federal, US AARON DANT, ASRC Federal, US PANOS PATROS, ORKA Lab, Department of Software Engineering, University of Waikato, NZ Social media is a modern person’s digital voice to project and engage with new ideas and mobilise communities— a power shared with extremists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Given the societal risks of unvetted content-moderating algorithms for Extremism, Radicalisation, and Hate speech (ERH) detection, responsible software engineering must understand the who, what, when, where, and why such models are necessary to protect user safety and free expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, we propose and examine the unique research field of ERH context mining to unify disjoint studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Specifically, we evaluate the start-to-finish design process from socio-technical definition-building and dataset collection strategies to technical algorithm design and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Our 2015-2021 51-study Systematic Literature Review (SLR) provides the first cross-examination of textual, network, and visual approaches to detecting extremist affiliation, hateful content, and radicalisation towards groups and movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We identify consensus-driven ERH definitions and propose solutions to existing ideological and geographic biases, particularly due to the lack of research in Oceania/Australasia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Our hybridised investigation on Natural Language Processing, Community Detection, and visual-text models demonstrates the dominating performance of textual transformer-based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We conclude with vital recommendations for ERH context mining researchers and propose an uptake roadmap with guidelines for researchers, industries, and governments to enable a safer cyberspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' CCS Concepts: • Computing methodologies → Discourse, dialogue and pragmatics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Lexical semantics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Information extraction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Machine learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Natural language processing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Knowledge representa- tion and reasoning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Image and video acquisition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' • Applied computing → Sociology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' • Social and profes- sional topics → Hate speech;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Technology and censorship;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Political speech;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Governmental regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Additional Key Words and Phrases: extremism, radicalisation, machine learning, community detection, natural language processing, neural networks, hate speech, sociolinguistics ACM Reference Format: Jarod Govers, Philip Feldman, Aaron Dant, and Panos Patros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, 0, Article 000 (December 2021), 35 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='org/tobereplacedwhenassignedDOI Authors’ addresses: Jarod Govers, jg199@students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='waikato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='nz, ORKA Lab, Department of Software Engineering, University of Waikato, Gate 1, Knighton Road, Hamilton, Waikato, NZ, 3216;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Philip Feldman, philip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='feldman@asrcfederal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='com, ASRC Federal, Beltsville, Maryland, US;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Aaron Dant, aaron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='dant@asrcfederal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='com, ASRC Federal, Beltsville, Maryland, US;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Panos Patros, panos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='patros@waikato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='nz, ORKA Lab, Department of Software Engineering, University of Waikato, Gate 1, Knighton Road, Hamilton, Waikato, NZ, 3216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Copyrights for components of this work owned by others than ACM must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='00 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='org/tobereplacedwhenassignedDOI ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='11579v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='SI] 27 Jan 2023 000:2 Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Contents Abstract 1 Contents 2 1 Introduction 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Motivation and Contributions 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 Structure 4 2 Social Context to Social Network Analysis 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Extremism and Radicalisation Decoupled 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 Hate Speech Decoupled 5 3 Systematic Literature Review Design and Protocol 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Trends and Shortfalls in Prior SLRs 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 Research Questions 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='3 Databases 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='4 Search Strings 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='5 Inclusion and Exclusion Criteria 8 4 Key Research Question (RQ) Findings 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Summary of the Social ERH Definitions Used by Researchers 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 Summary of the Data Collection, Processing, and Annotation Processes 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='3 Summary of the State-of-the-art Computational ERH Detection Models 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='4 Summary of ERH Models’ Classification Performance 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='5 Geographic Trends—Islamophobia and Exclusion in the Academic Community 11 5 Socio-technical Context in Research—Consensus-driven Definitions 12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Researchers’ Consensus-driven Definitions for ERH Concepts 13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 Correlation Between Definitions and Algorithmic Approach 14 6 Building ERH Datasets—Collection, Processing, and Annotation 15 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Prominent Platforms, Pulling, and Populations 15 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 Feature Extraction Techniques 17 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='3 Data Filtering 18 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='4 Key Takeaways for Dataset Domain, Pre-processing, and Annotation 19 7 Community Detection, Text, and Image ERH Detection Algorithms 20 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Observed Non-deep Machine Learning Algorithms (MLAs) 20 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 Deep Learning Algorithms (DLAs) 22 8 Model Performance Evaluation, Validation, and Challenges 24 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Benchmark Dataset Performance (Inter-study Evaluation) 25 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 Community Detection Performance 25 9 Future Research Directions 25 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Ideological Isomorphism—a Novel Framework for Radicalisation Detection 25 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 Morphological Mapping and Consensus-building—a novel computationally- grounded framework for extremism detection 26 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='3 Outwards Dissemination––‘traditional’ hate speech detection updated 27 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='4 Uptake Roadmap for Researchers, Industry, and Government 28 10 Conclusion 28 References 30 ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech 000:3 1 INTRODUCTION Online social media empowers users to communicate with friends and the wider world, organise events and movements, and engage with communities all at the palm of our hands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Social media platforms are a frequent aid for modern political exchanges and organisation [115], with extremes amplified by algorithmic recommendation systems, such as on Twitter [49], TikTok [87], and YouTube [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Furthermore, the semantic expression of ideas differs between social media platforms, with Twitter’s short 280 character limit resulting in more narcissistic and aggressive content compared to Facebook [75], and anonymous platforms such as 4Chan instilling a vitriolic ‘group think’ in political threads/‘boards’ [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hateful, emotive, and ‘click-worthy’ content permeates virtual discourse, which can radicalise users down an ideological rabbit hole towards real-world violent action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The individuals behind the 2014 Isla Vista and 2019 Christchurch shootings appeared as individual ‘lone-wolf attacks’ without an allegiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' However, investigations found a deep network of perverse and violent communities across social media [40, 106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Likewise, exploiting social media to plan politically motivated attacks towards the civilian population to coerce political change (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', terrorism) delegitimises democracies, social cohesion, and physical/mental health [28, 58, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' As a response, social media platforms employ text and visual content-moderation systems to detect and remove hate speech, extremism, and radicalising content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' This paper offers a state-of-the- art Systematic Literature Review (SLR) on the definitions, data collection, annotation, processing, and model considerations for Extremism, Radicalisation, and Hate speech (ERH) detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Motivation and Contributions Existing ERH literature reviews exist as independent microcosms, often focusing on specific types of models, typically text-only Natural Language Processing (NLP) models or non-textual network analysis via community detection models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Studies seldom cross-examine models and evaluate the performance between non-textual network analysis (a ‘who-knows-who’ approach), textual, and/or multimedia approaches for ERH detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' While we identified ten prior literature reviews for hateful content detection, none consider the similarities and definitional nuance between ERH concepts and what Extremism, Hate Speech, or Radicalisation means in practice by researchers [2, 5, 20, 42, 43, 51, 82, 97, 110, 114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Evaluating the consensus for ERH definitions, dataset collection and extraction techniques, model choice and performance are all essential to create ethical models without injurious censorship or blowback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Through understanding the groups, beliefs, data, and algorithms behind existing content- moderation models—we can reliably critique often overlooked social concepts, such as algorithmic bias, and ensure compliance between social definitions and computational practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, this new field of ERH context mining extracts the context to classifications—enabling researchers, industries, and governments to assess the state of social discourse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Given the rise of state-sponsored disinformation campaigns to undermine democratic institutions and social media campaigns, the time is now for ERH research within politicised discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The three core contributions for this paper are: (1) The establishment of consensus-driven working definitions for Extremism, Radicalisation, and Hate Speech within the novel field of ERH context mining—and a proposed frame- work/roadmap for future researchers, social media platforms, and government advisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (2) The critical examination of existing textual (NLP), network (community detection), and hybrid text-image datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (3) The identification and cross-examination of the state-of-the-art models’ performance on benchmark datasets and relevant challenges with the current ERH detection metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 000:4 Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 Structure For a high-level summary of this SLR’s findings, refer to Section 4 on Key Research Question Findings, and Section 9 for Recommendations for Future Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' These key summaries condense and contextual- ise the 51 studies observed between 2015-2021, which we use to build our proposed computational ERH definitions and technological roadmap for researchers, industry, and government in Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' For a holistic understanding, we present a social context to our motivations in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Related work and areas our SLR improves on are outlined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Section 3 outlines the systematic protocol used to collect the 51 studies between 2015-2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Further to the summaries presented in Section 4, we present an in-depth analysis and cross-examination of studies definitions of ERH concepts in Section 5, approaches for collecting and processing data in Section 6, algorithmic approaches for classification in Section 7, and their performance in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We conclude with recommendations for future SLRs, and studies in Section 9, and conclusions in Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 2 SOCIAL CONTEXT TO SOCIAL NETWORK ANALYSIS Analysing social media requires the socio-technical considerations of what constitutes hate speech, extremism, and radicalisation (ERH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' To detect such concepts, computational models can investigate multimodal sources—including textual meaning and intent through Natural Language Processing (NLP), computer vision for images, and evaluating user relationships through community detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, this section decouples and analyses ERH’s social background and definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Extremism and Radicalisation Decoupled Extremism’s definition appears in two main flavours: politically fringe belief systems outside the social contract or violence supporting organisational affiliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The Anti-Defamation League (ADL) frames extremism as a concept "used to describe religious, social or political belief systems that exist substantially outside of belief systems more broadly accepted in society" [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' For instance, under the ADL’s definition, extremism can be a peaceful positive force for mainstreaming subjugated beliefs, such as for civil rights movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' This construct of a socially mainstream belief constitutes the Overton window [73]—and is not the target for content moderation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Conceptually, extremism typically involves hostility towards an apparent ‘foreign’ group based on an opposing characteristic or ideology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Core tenants of extremism can stem from political trauma, power vacuums and opportunity, alongside societal detachment and exclusion [58, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, extremism often relies on defending and congregating people(s) around a central ideology, whose followers and devotees are considered ‘in-group’ [116].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Extremists unify through hostility and a perceived injustice from an ‘out-group’ of people(s) that do not conform to the extremist narrative—typically in a ‘us vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' them’ manner [28, 78, 116].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, extremism detection algorithms can use non-textual relationships as an identifying factor via clustering users into communities (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', community detection) [20, 110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Thus, extremism can simply reduce to any form of a fringe group whose identity represents the vocal antithesis of another group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' There is no one conceptual factor to make an extremist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Extremism can also emanate from political securitisation—whereby state actors transform a specific referent object (such as Buzan’s five dimensions of society: societal, military, political, economic, and environmental security [25];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' or individuals and groups [28]) towards matters of national security, requiring extraordinary political measures [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' As the state normalises policies into matters of existential national security, society can adapt and ideate decisions to ones of existential ‘life or death’ nature [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' For example, the ‘Great Replacement’ conspiracy theory claims that non-European immigrants and children are “colonizers" or "occupiers”, and an “internal enemy”—-with the intent to securitise migration, race, religion, and culture into wars with wording to invoke fears of a fifth column or ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech 000:5 racial invasion/replacement [22, 106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The Christchurch Shooter took direct interest in securitising migrants as an extreme military threat, as far to name his manifesto after the conspiracy [106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Extremism is not a strictly demographic ‘majority vs minority’ concern, as it encapsulates movements demanding radical change and earmarked by a sense of rewarding personal and social relationships, self-esteem, and belief of a wider purpose against a perceived adversarial force [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Exploiting desires for vengeance and hostility are also key recruitment strategies [20, 58, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Outside of political, cultural, and socio-economic factors, mental health and media are intrinsically inalienable contributing factors [28, 58, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Likewise, repeated media reports of footage and body counts can gamify and normalise extremism as a macabre sport for notoriety [20, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Within industry, Facebook, Twitter, YouTube, and the European Union frame extremism as a form of indirect or direct support for civilian-oriented and politically motivated violence for coercive change [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Facebook expands this industry-wide consensus to include Militarised Social Movements and "violence-inducing conspiracy networks such as QAnon" [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Radicalisation focuses on the process of ideological movement towards a different belief, which the EU frames as a "phased and complex process in which an individual or a group embraces a radical ideology or belief that accepts, uses or condones violence" [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Terrorism consists of politically motivated violence towards the civilian population to coerce, intimidate, or force specific political objectives, as an end-point for violent radicalisation to project extremism [25, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Borum delineates the passive ideological movement of radicalisation from active decisions to engage in ‘action pathways’ consisting of physical terrorism, or hate crimes [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Political radicalisation towards increasingly aggrandising groups can also manifest in Roe’s two sides of nationalism: positive socio-cultural and negative ethnic/racial nationalism [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' These balancing forces create a form of societal security dilemma whereby the actions of one society to strengthen its identity can cause a reaction in another societal group, weakening security between all groups-—a radicalising spiral which can manifest into a polarised ‘culture war’ [70, 98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' However, integration over assimilation can inversely undermining culture, self-expression and group cohesion, leading to alienation and oppression by the dominant political or normative force [28, 98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Nonetheless, ERH detection does not offer a panacea to combating global terrorism, nor does surveillance offer a ‘catch-all’ solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' In the case of the livestreamed Christchurch shooter, the New Zealand Security Intelligence Service concluded that “the most likely (if not only) way NZSIS could have discovered [the individual]’s plans to conduct what is now known of [the individual]’s plans to conduct his terrorist attacks would have been via his manifesto.” [106, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' However, the individual did not disseminate this until immediately before the attack, and his 8Chan posts did not pass the criteria to garner a search warrant [106, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, extremism detection is an evolutionary arms race between effective and ethical defences vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' new tactics to evade detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 Hate Speech Decoupled Obtaining viewpoint neutrality to categorise hate speech is challenging due to human biases and the risk of hate speech undermining liberties through mainstreaming intolerance—the paradox of tolerance where a society tolerant without limit may have their rights seized by those projected intolerance [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Popper encapsulates this challenge by formulating that "if we are not prepared to defend a tolerant society against the onslaught of the intolerant, then the tolerant will be destroyed, and tolerance with them" [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Defining clear hate speech restrictions are needed to protect expression rights and victim groups rights and safety [11, 117].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The European Union defines hate speech as "all conduct publicly inciting to violence or hatred directed against a group of persons or a member of such a group defined by reference to race, colour, religion, descent or national or ethnic origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='" [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Whereas, the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Department of Justice frames that: "A hate crime is a traditional offence like murder, arson, or vandalism with an added element ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 000:6 Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' of bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' [consisting of a] criminal offence against a person or property motivated in whole or in part by an offender’s bias against a race, religion, disability, sexual orientation, ethnicity, gender, or gender identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='" [38] Notably, governmental laws may differ from industry content moderation policies via the omission of sexual, gender, religious or disability protections, and may include threats of violence and non-violent but insulting speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The United Nations outlines the international consensus on hate speech as "any kind of commu- nication in speech, writing or behaviour, that attacks or uses pejorative or discriminatory language with reference to a person or a group on the basis of who they are, in other words, based on their religion, ethnicity, nationality, race, colour, descent, gender or other identity factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='" [117, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 2] What all these definitions have in common is that they all involve speech directed at a portion of the population based on a protected class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 3 SYSTEMATIC LITERATURE REVIEW DESIGN AND PROTOCOL This SLR investigates the state-of-the-art approaches, datasets, ethical, socio-legal, and technical implementations used for extremism, radicalisation, and politicised hate speech detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We conduct a preliminary review of prior ERH-related SLRs to establish the trends and research gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' For the purposes of our SLR’s design, and to embed Open-Source Intelligence (OSINT) and Social Media Intelligence (SOCMINT) principles, we define social media data as any online medium where users can interactively communicate, exchange or influence others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We accept external data sources, such as manifesto or news sites if interactive—such as via comment sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Furthermore, we propose and use a novel quality assessment criteria to filter irrelevant or ambiguous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Trends and Shortfalls in Prior SLRs Searching for Extremism, Radicalisation, Hate speech (ERH) and related terms, resulted in ten literature reviews ranging from January 2011 to April 2021 [2, 5, 20, 42, 43, 51, 82, 97, 110, 114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Aldera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' observed only one survey before 2011 (covering 2003-2011) and another in 2013, indicating the limited, exclusionary, but developing nature of reviews in this ERH detection area [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Prior SLRs seldom delineated or elaborated on Extremism, Hate Speech and Radicalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Neither “extremism”, "radicalism" [2, 5, 20, 42, 43, 110, 114] or “hate speech” oriented SLRs [51, 82, 97] cross-reference each other despite 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='3% of the data reviewed in the “hate speech” oriented review by Adek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' encompassing hate speech in a political context [114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' This lack of overlap presents an industry-wide challenge for social media companies who may oversee developments in ‘hate speech’ detection which could transfer to a ‘extremism/radicalisation detection’ model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' SLRs prior to 2015 found that deep learning approaches (DLAs), such as Convolutional Neural Networks (CNN) and Long Short-Term Memory (LSTM), resulted in 5-20% lower F1-scores than non-deep approaches (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Naïve Bayes, Support Vector Machines, and Random Forest classifiers) [2, 5, 42, 51, 97, 114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' DLAs post-2015 indicated a pivotal change towards higher-performing language transformers such as Bidirectional Encoder Representations from Transformers (BERT) models [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Domains and Criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' No review delineated or removed studies that did not use English social media data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' This presents three areas of concern for researchers when attempting to compare the performance of models: (1) Results may not be comparable, if they use culture-specific lexical datasets, or language models trained on other languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (2) Linguistic differences and conveyance in language—as what may be culturally appro- priate for the majority class may appear offensive to minority groups and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (3) The choice of language(s) influences the distribution of target groups—with a bias towards Islamic extremism given its global reach in both Western (predominantly ISIS) and ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech 000:7 Eastern countries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', with studied online movements in the Russian Caucasus Region [77]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' It is worth investigating whether Gaikwad et al.’s finding that 64% of studies solely target ‘Jihadism’ corroborates with our study, which targets only English data [42, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Our SLR incorporates the key approaches of dataset evaluation (including their accessibility, labels, source and target group(s)), data collection and scraping approaches, Machine Learning and Deep Learning algorithms, pre-processing techniques, research biases, and socio-legal contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Unlike prior SLRs, our SLR conceptualises all elements for ERH context mining—consisting of a user’s ideological radicalisation to an extremist position, and then projected via hate speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 Research Questions Our Research Questions (RQ) investigate the full process of ERH Context Mining—incl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' data collec- tion, annotation, pre-processing, model generation and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' These RQs consist of: (1) What are the working definitions for classifying online Extremism, Radicalisation, and Hate Speech?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (2) What are the methodologies for collecting, processing and annotating datasets?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (3) What are the computational classification methods for ERH detection?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (4) What are the highest performing models, and what challenges exist in cross-examining them?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Given the overlap of studies across prior SLRs targeting extremism or radicalisation or hate speech, RQ1 addresses the similarities and differences between researchers’ definitions of ERH concepts and their computational classification approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We dissect the ERH component of ERH context mining and propose consensus-driven working definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' RQ2 addresses the vital context for ERH models—the data used and features extracted or filtered out from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Furthermore, identifying frequently used benchmark datasets provides the basis for critical appraisal of the state-of-the-art algorithmic approaches in the community detection, multi- media, and NLP spheres in RQ3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Covering algorithmic approaches is not in itself novel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' However, we consider novel, niche, and overlooked features relevant for an ERH model to make accurate classifications—namely, bot/troll detection, transfer learning, the role of bias, and a hybridised eval- uation of NLP and non-textual community detection models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We also consider critical challenges, choice of metrics, and performance considerations not observed in prior SLRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='3 Databases Given the cross-disciplinary, global and socio-technical concepts for ERH detection, we queried the following range of software engineering, computer science, crime and security studies databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ProQuest (with the “peer-reviewed” filter, including queries to the below databases) Association for Computing Machinery (ACM) Digital Library SpringerLink ResearchGate Wiley Institute of Electrical and Electronics Engineers (IEEE) Xplore Association for Computational Linguistics Portal Public Library of Science (PLOS) ONE Database Google Scholar—as a last line to capture other journals missed in the above searched databases 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='4 Search Strings The first round of study collection included automated database search strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' A second round included a targeted manual search strategy with dissected keyword combinations to expand ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 000:8 Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' All results were added to the Title and Abstract Screening list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The following database ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='search query also included time filters (2015-2021) and peer-review-only filters: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='(“artificial␣intelligence”␣OR␣“machine␣learning”␣OR␣“data␣mining”␣OR␣“natural ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='language␣processing”␣OR␣“multiclass␣classification”␣OR␣“model”␣OR␣“analysis”␣OR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='“intelligence”␣OR␣“modelling”␣OR␣“detection”) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='AND␣(“hate␣speech”␣OR␣“radicalisation”␣OR␣“radicalization”␣OR␣“extremism”) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='AND␣(“social␣media”␣OR␣“forums”␣OR␣“comments”OR␣“virtual␣networks”␣OR␣“virtual ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='communities”␣OR␣“online␣communities”␣OR␣“posts”␣OR␣“tweets”␣OR␣“blogs”) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='5 Inclusion and Exclusion Criteria After attaining our 251 studies from our search strings, we read the journal metadata, title and abstract to screen studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Our ranked criteria requires that all studies must: (1) Originate from a peer-reviewed journal, conference proceeding, reports, or book chapters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (2) Be written in English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (3) Involve a computational model (network relationship, textual and/or visual machine learning model) for identifying and classifying radicalisation, hate speech or extreme affiliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (4) Utilise social media platform(s) for generating their model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (5) Computationally identify ERH via binary, multiclass, clustering or score-based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (6) Focus on politicised discourse to exclude cyber-bullying or irrelevant benign discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (7) Published after the 1st January 2015—until the 1st July 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (8) Utilise English social media data if evaluating semantics and grammatical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' In addition to those that do not abide to any of the above, we exclude studies that: (1) Are duplicates of existing studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (2) Do not specify their target affiliation to exclude broad observational studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We outline our further in-depth critical Quality Assessment (QA) criteria to filter irrelevant or ambiguous studies in our supplementary material’s Quality Assessment Criteria subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' After the Title and Abstract Screen, we read the full text of the 57 studies for the screening stages displayed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' With 42 studies passing the Full Text Screen, we then randomly selected studies from the bibliographies from this ’snowball sample’ of the 42 studies until 5 studies fail QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Studies found and filtered Screen Type Study Count Search Strings 251 Title and Abstract Screen 57 Full Text Screening 42 After Snowball Sampling 51 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Threats to Validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' While we consider a concerted range of search strings, we recognise that ERH concepts is a wide spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' To focus on manifestly hateful, politicised, and violent datasets/studies, we excluded cyber-bullying or emotion-detection studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The potential overlap and alternate terms for ERH (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', sexism as ‘misogyny classification’ [30]) could evade our search strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Our pilot study, subsequent tweaks to our search method, and snowball sampling minimise this lost paper dilemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' This study does not involve external funding, and all researchers declare no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech 000:9 4 KEY RESEARCH QUESTION (RQ) FINDINGS Across the 51 studies between 2015-2021, ERH research is gaining popularity—with 4 studies from 2015-2016 increasing to 25 between 2019-2020 (and 4 studies from January to July 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We present our SLR’s core findings in this section, with in-depth RQ analysis in Sections 5, 6, 7, and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Summary of the Social ERH Definitions Used by Researchers RQ1: What are the working definitions for computationally classifying online Extremism, Radicalisation, and Hate Speech?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Across the 51 studies, there are seldom delineations between the researchers choice of Extremism, Radicalisation, and Hate Speech as the study’s focus––with the consensus that hate speech is equivalent to extremist or radical views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, researchers approach extremism and radicalisation as an organisationally affiliated form of hate speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The consensus on hate speech’s working definition is any form of subjective and derogat- ory speech towards protected characteristics expressed directly or indirectly in textual form—- predominantly via racism or sexism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Benchmark datasets utilise human-annotated labels on single- post instances of racially or gender-motivated straw man arguments, stereotyping, or post causing offensive towards the majority of annotators (via inter-annotator agreement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Only 20% of studies consider explicit rules or legal frameworks for defining hate speech [15, 18, 47, 53, 55, 71, 77, 84, 123, 128], with others relying on either an implicit ‘consensus’ on hate speech or utilise benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Benchmark datasets typically consider categorising their data into explicit categories of racism [31, 32, 122, 123], sexism [13, 122, 123], aggression [13, 61], or offensiveness [13, 31, 128];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' including hate categorisation via visual memes and textual captions [3, 57, 99, 111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Extremism and radicalisation are equivalent terms in existing academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Islamic extremism is the target group in 77% of US-originating extremism studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ‘Far-right extremism’ and ‘white supremacy’ are used interchangeably, a form of cultural bias given the variety of right-wing politics worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Only one study considered radicalisation as an ideological movement over time [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 Summary of the Data Collection, Processing, and Annotation Processes RQ2: What are the methodologies for collecting, processing and annotating datasets?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Collecting non-hateful and hateful ERH instances varies between supervised and unsupervised (clustering) tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Supervised learning typically utilises manual human annotation of textual posts extracted via tools presented in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Semi or unsupervised data collection can include grouping ideologies by platform, thread, or relation to a suspended extremist account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Islamic extremism studies frequently used manifestos and official Islamic State magazines as a ‘ground truth’ for textual similarity-based approaches for extremism detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We found a direct correlation between the availability of open and official research tools, and the platform of choice by researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Biases extend geographically, with no studies utilising data or groups from Oceanic countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Figure 2 displays the skew for Twitter as the dominant platform for hate speech research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Despite the nuance of conversations, 69% of studies classify hate on a single post per Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Data processing often utilises extracting statistically significant ERH features—such as hateful lexicons, emotional sentiment, psychological signals, ‘us vs them’ mentality (higher occurrence of first and third-person plural words [46]), and references to political entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We categorise and frame the two approaches for dataset annotation: organisational or experience- driven annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Organisational annotation utilises non-governmental anti-hate organisations [105] or ‘expert’ annotator panels—determined via custom tests or by tertiary degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Organisational an- notation relies on crowdsourced annotators, balanced by self-reported political affiliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Inter-rater agreement or Kappa coefficient are the sole metrics for measuring annotator agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 000:10 Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Method to collect data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Number of studies per social media platform studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='3 Summary of the State-of-the-art Computational ERH Detection Models RQ3: What are the computational classification methods for ERH detection?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ERH detection includes text-only Natural Language Processing (NLP), network-related community detection, and hybrid image-text models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Between 2015-2021, there was a notable shift from traditional Machine Learning Algorithms (MLAs) towards contextual Deep Learning Algorithms (DLAs) due to higher classification performance—typically measured by macro F1-score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Notably, only 3 of the 21 community detection studies utilised Deep Learning Algorithms (DLAs) [77, 84, 99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Instead, community detection researchers tended to opt for graph-based models such as heterogeneous graph models converting follower/following, reply/mention, and URL networks with numeric representations for logistic regression or decision trees [12, 16, 24, 48, 80, 84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Community detection Machine Learning Algorithms (MLAs) performance varied by ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='3 F1-score (mean between studies) dependent on the selection of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Statistically significant features for performant MLA models include gender, topics extracted from a post’s URL(s), location, and emotion via separate sentiment algorithms such as ExtremeSentiLex [89] and SentiStrength [112].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' For textual non-deep NLP studies, researchers classified text via converting the input into word embeddings via Word2Vec, GloVe, or frequent words via Term Frequency-Inverse Docu- ment Frequency (TF-IDF), and parsing it into Support Vector Machines, decision trees, or logistic regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' As these embeddings do not account for word order, context and nuance is often lost—leading to higher false positives on controversial political threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Conversely, DLAs utilise positional and contextual word embeddings for context-sensitivity using Long Short Term Memory (LSTM) Convolutional Neural Networks and Bidirectional Encoder Representations from Transformers (BERT) leading to their higher performance as outlined in RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='4 Summary of ERH Models’ Classification Performance RQ4: What are the highest performing models, and what challenges exist in cross-examining them?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' By 2021, Support Vector Machines on emotional, sentiment, and derogatory lexicon features were the last non-deep MLA to attain competitive F1-scores for NLP tasks compared to DLAs such as ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 3%6% 29% 32% 22% 5% 3% Twitter APi Custom web-crawler Pushshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='io APl (Reddit) Magazine or Newspaper Archives Pre-existing Dataset or Library Facebook Graph APl Other APlIslamicExtremist Sources, 9 Stormfront, 7 Other Far- Blogs and right Apolitical Extremist Sources, 4 Forums, 4 News Websites, 6 Far-Left Forums, Facebook, Wikipedia 2 3 Twitter, 36 Vkont akte, Religious Other, 5 Reddit, 2 Texts, 2Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech 000:11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Type of data used for an ERH classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Distribution of target groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Convolutional Neural Networks (CNNs) and neural language transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' As of 2021, BERT-base attained the highest macro F1-score average across the seven benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' However, cross- examining models between datasets present various challenges due to varying criteria, social media domain, and choice of metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Likewise, non-textual community detection and traditional MLA studies resulted in lower classification F1-score by ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='15 and ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' While BERT, attention- layered Long Short Term Memory (BiLSTM), and other ensemble DLAs attain the highest F1-scores, no studies consider their performance trade-offs with their high computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Our recommendations propose further research in prompt engineering, distilled models, and hybrid multimedia-text studies—as we only identified one hybrid image-text study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' While textual DLAs outperform community detection models, grouping unknown instances enable network models to identify bot networks and emergent terror cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, there is a growing area of research for hybrid semi-supervised NLP and community detection models to identify new groups and radical individuals in a domain we frame as meso-level and micro-level classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='5 Geographic Trends—Islamophobia and Exclusion in the Academic Community To identify ERH hot spots in research, we present the first cross-researcher examination of their institution’s location compared to their dataset(s) geolocational origin in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' For clarity, we filter out the 29 indiscriminate global studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Despite the decline of the Islamic State as a conventional state-actor post-2016, western academic research remains skewed towards researching Islamic extremist organisations operating from the Middle East.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 24% of US-originating ERH studies targeted Islamic extremism, compared to 19% focusing on violent far-right groups and 19% for left vs right polarised speech (in discussions containing hate speech).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Despite more Islamic extremist studies from US-oriented research, over 90% of terrorist attacks and plots in the US were from far-right extremists in 2020 [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' European-origin studies have a reduced bias, where 25% target far-right white supremacy and 29% on Islamic extremism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Islamic extremism’s popularity is a global trend for 20% of all studies, shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, there is a clear Islamophobic trend in academia—given the aversion of far-right groups, and the lack of a change in the distribution of targeted groups between 2015-2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Researcher ethics and socio-legal considerations present a critical international research gap, as only 13% of US and 28% of European studies included discussions on annotation ethics, expression laws, or regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' This US vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Europe discrepancy likely emanates from the data collection and autonomous decision-making rights guaranteed under the EU’s GDPR [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 4% Text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Single post only 8% I Text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Grouped by user 19% No text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Metadata 69% Relations (Community Detection), or Images Text and Graphing (NLP and Community Detection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='lslamic 20% 20% IFar-right White Supremacy) IMainstream Politics 10% Hate Speech 9% 41% 1Other000:12 Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Relations between researchers country of origin and their dataset’s country of focus (global/indiscrim- inate studies excluded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Created via Flowmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='blue [21] with Mapbox [74] and OpenStreetMap [85] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 The Case for Oceania and the Global South.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Despite developments post-2015, such as two New Zealand terrorist attacks [94, 106] and five in Australia [8], the rise of racial violence in South Africa [28], and the 2019-ongoing COVID-related radicalisation [120];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' no studies considered targeting Oceania or English-recognised countries in the global south.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Likewise, applying these datasets across intersectional ethnic, sexual, and cultural domains presents a threat to validity as terms considered mundane or inoffensive to one group may be considered inflammatory to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Datasets are also biased towards racism towards a minority group [31, 123], which may bias English hate speech in a white minority country such as South Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Investigating language trends and model performance on Mela-, Micro- and Polynesian groups could also offer insights in the role of religion, values (such as tikanga values in New Zealand’s M¯aori population), taboos, lexicons, and social structures unique to indigenous cultures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 5 SOCIO-TECHNICAL CONTEXT IN RESEARCH—CONSENSUS-DRIVEN DEFINITIONS What are the Working Definitions for Classifying Online Extremism, Radicalisation, and Hate Speech?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Empirically, studies often provide a generalised social definition in their introduction or back- ground and utilise technical criteria to annotate instances for (semi)supervised learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, this research question consists of two parts: the socio-legal ERH definitions, and the technical implementation and classification thereof outlined in the existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We identified an unexpected overlap between the definitions and models between extremism and radicalisation studies, whereby researchers frame these concepts as synonymous with hate speech with a political/organisational affiliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hate speech studies focus on protected groups as binary ‘hate or not’ [3, 4, 15, 27, 32, 57, 77, 84, 101, 109, 111, 123], or multiclass ‘racism, sexism, offensive, or neither’ text [10, 13, 31, 45, 69, 81, 83, 90, 91, 123, 126], with a consensus that ‘Extremism = ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='Argoitina ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='more outgoing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='more incomingDown the Rabbit Hole: Detecting Online Extremism,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Radicalisation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' and Politicised Hate Speech 000:13 Radicalisation = Hate speech with an affiliation’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Alternatively, we propose a novel computationally grounded framework and definitions to seperate and expand ERH in Section 9 to underline the holistic stages of extremists temporal radicalisation through disseminating hateful media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Socio-legal Context Provided in Existing Literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The largest discrepancy were between studies that discussed legal or ethical context to ERH, which constituted only 20% of studies [15, 18, 47, 53, 55, 71, 77, 84, 123, 128].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The remaining 80% relied on an implicit consensus of ‘hate speech’ (often synonymous with toxic, threatening, and targeted speech), or ’extremism’ (often UN designated terrorist organisations like ISIS [118]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Waseem and Hovy [123] outlined a unique eleven-piece criteria to identify offensive ‘hate speech’ including considerations for politicised extremist speech via tweets that “promotes, but does not directly use, hate speech or violent crime” and “shows support of problematic hash tags" (although "problematic" was not defined).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hate speech as a supervised learning task resulted in two categories—sexism and racism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' A sexist post requires gender-oriented slurs, stereotypes, promoting gender-based violence, or straw man arguments with gender as a focus (defined as a logical fallacy aimed at grossly oversimplifying/altering an argument in bad faith [123]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The ambiguity for sexism classification by human annotators was responsible for 85% of inter-annotator disagreements [123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Researchers’ Consensus-driven Definitions for ERH Concepts We aggregate the trends in ERH based on the definitions used throughout the 51 studies, and observe that ERH concepts reflect their computational approach more than their social definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Despite radicalisation being a social process of ideological movement, existing work considers the term as synonymous to political hate speech/extremism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Definition 1: Hate speech (researchers’ consensus) Hate speech is the subjective and derogatory speech towards protected characteristics expressed directly or indirectly to such groups in textual form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' * N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='B: there is a significant bias in hate speech categorical classification in practice, whereby no studies considered categories outside of sexism (including gender-identity) or racism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Definition 2: Extremism (researchers’ consensus) Organisational affiliation to an ideology that discriminates against protected inalienable characteristics or a violent political organisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Affiliation does not always include manifestly hateful text and may include tacit or explicit organisational support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Extremist studies often classify organisational affiliation based on text (NLP) and community networks (follower, following, or friend relationships).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The current academic consensus among researchers demonstrates a considerable overlap between ‘extremism’ and ‘hate speech’ definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' In practice, extremism exclusively focused on racism detection, or in the specific context of Jihadism [4, 15, 84], white supremacy [53, 86, 99, 109], Ukrainian separatists [16], anti-fascism (Antifa) [53], and the sovereign citizen movement [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Of the 13 studies targeting extremism, only one considered extremism by the ADL’s politically-fringe- but-not-violent definition [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Tying extremism to the study of mainstream terrorist-affiliated groups neglects rising movements, ethical movements using unethical terror-tactics, and non-violent fragments of other terrorist groups, such as a reversion to ‘fringe’ activism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, extremism’s ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 000:14 Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' working definition is similar to terrorism when considering group affiliation detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' If investigating extremist ideologies, then the definition is synonymous with those in hate speech studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Extremism’s working definition is exceptionally biased towards support for Islamic extremist movements (10 out of 13 studies [4, 16, 24, 53, 77, 80, 84, 86, 89, 109]), with far-right ideologies a distant second (5 out of 13 studies [16, 53, 86, 99, 109]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' These organisational and ideological biases is potentially a result of US security discourse and national interests (via the ‘War on Terror’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' White supremacy and far-right ideologies are separate terms used interchangeably without distinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Definition 3: Radicalisation (researchers’ consensus) No discernible difference between extremism’s definition with both terms used interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Radicalisation = extremism = politically affiliated hate invoking or supporting speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Definitions and algorithmic approaches on radicalisation detection relied on political hate speech, or extremism via ideological affiliation—with 5 of the 8 radicalisation studies targeting textual or network affiliation to the Islamic State (IS) [7, 47, 80, 95, 101], and 2 on white supremacy [46, 103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The only other notable deviations from this extremism = radicalisation dilemma is Bartal et al.’s [12] focus on radicalisation as a temporal movement with apolitical roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Their study investig- ated the temporal movement from a ‘Novice’ (new poster) classification towards an ‘influencer’ role based on their network relations and reply/response networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Chandrasekharan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' defined ‘radicalisation’ as the process of an entire subreddit’s patterns up to and including the time of its ban to map subreddit-wide radicalisation [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Only two studies are the exception to the extremism = radicalisation = politicised hate speech consensus per the remainder of the 51 studies [12, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ERH Definition Tree—visualising how ERH definitions deviate based on their algorithmic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 Correlation Between Definitions and Algorithmic Approach Uniquely, 66% of publications in the field of social-science or security studies utilised network- driven community detection models, with extremism defined in a law enforcement context by emphasising a user’s network-of-interactions between known annotated extremists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' defined extremism in a semi-supervised OSINT and HUMINT surveillance manner—requiring ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ERH Semantic/Topic-based (Aggregate) Hate Radical or Extremist Network Detection Speech Detection Generally Generally semi or fully Generally unsupervised supervised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=" unsupervised 'Big Data': Unless cluster-based 'Big Data': Graphs (non-textual): Emotional Sentiment Topics and Political Affilation Hate Speech Crime and Social Science Focused E." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' SentiStrength, SIRA, LIWC Law Enforcement Pivoted Data: E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=" FBI 'Tripwire', Explicit Terror Afillation, TSA's Automated Targeting System, etc." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Multiclass Sentiment Binary Affiliation Influencer Racism Terror Network Sexism Aggression Offense or ProlAnti/Neutral Person-of Classification Clustering Classificatior Classification Classification Classification Interest Detection Classification F H J K c D A Multiclass Ideological or Community Radical Scoring Organisation or Detection Topic Classification Regression B E GDown the Rabbit Hole: Detecting Online Extremism,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Radicalisation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' and Politicised Hate Speech 000:15 links between an extremist virtual and a physical/offline presence to extremist stimuli through incorporating the FBI’s tripwire program [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Approaching extremism using relational properties via interactions, geographic proximity, profile metadata, and semantic or network similarity raises ethical dilemmas vis-à-vis individuals who have/had a solely virtual presence or those interested in opposing opinions [24, 29, 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The relationship between definitions and algorithmic approaches indicate that radicalisation studies skew towards community detection models, extremism towards hybrid NLP and community detection models, and hate speech to a text-only NLP endeavour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Table of references for studies in each category (A, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='K) in the above ERH definition tree diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Label Studies Label Studies Label Studies A [12, 15, 48, 80] B [15, 16, 80, 89, 99, 108] C [48, 108] D [3, 4, 15, 27, 32, 57, 77, 84, 101, 109, 111, 123] E [14, 53, 80, 89, 93, 99, 99] F [1, 4, 7, 13, 52, 55, 86, 89, 90, 95, 100, 107, 124, 128, 129] G [15, 18, 46, 47, 103, 104, 113] H [10, 13, 45, 69, 81, 83, 90, 91, 123, 126] I [10, 13, 45, 81, 83, 90, 91, 123, 126] J [13, 56, 83, 90, 126, 128, 129] K [3, 56, 57, 67, 71, 81, 83, 90, 99, 111, 126, 128–130] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Privacy and Ethics-driven Regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' No studies integrated or mentioned existing AI ethics regulation or standards, such as those emerging from the EU [50], or private-sector self-regulation such as the IEEE P700x Series of Ethical AI Design [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Researchers should consider the application and use cases for their proposed models—as autonomous legal decision making, injurious use of data (outside of a reasonable purpose), or erasure (a challenge for persistent open-source datasets), may violate regulations such as the EU’s General Data Protection Regulation (GDPR)’s Article 22, 4, and 17 respectively [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Prominently, Mozafari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' evaluated hate speech with a ethno-linguistic context, recognising that certain racist slurs were dependent on the culture and demographic using them [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 6 BUILDING ERH DATASETS—COLLECTION, PROCESSING, AND ANNOTATION What are the methodologies for collecting, processing and annotating datasets?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' This RQ outlines the dominant platforms of choice for ERH research, the APIs and methods for pulling data and its underlying ethical considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Geographic mapping demonstrates the marginalisation of Oceania and the global south in academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We critically evaluate the sentimental, relationship, and contextual feature extraction and filtering techniques in community detection and NLP studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We conclude with the key recommendations for future data collection research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Prominent Platforms, Pulling, and Populations This subsection outlines the common social media platforms, the method for sampling and extracting (‘pulling’) textual and network/relationship data, and the type of data used in ERH datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 40% of studies relied on Twitter tweets for ERH detection, with Twitter being the dominant platform for research per Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Twitter’s mainstream global reach paired with its data-scraping Twitter API enabled researchers to target specific hashtags (topics or groups), real-time tweet streams and reach of tweets and their community networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, the Twitter API is also the most used method for scraping data, with other methods outlined in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Unfortunately, revised 2021 Twitter Academic API regulations removed access to tweets from suspended accounts [17], limiting datasets to those pre-archived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Currently, the Waseem and Hovy datasets are not available due to relying on the Twitter API and suspended tweets [122, 123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' For far-right ERH detection, researchers used custom web-scrapers to pull from the global white supremacist forum Stormfront—containing topics ranging from political discussions, radic- alising "Strategy and Tactics", and "Ideology and Philosophy" sections, and regional multilingual ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 000:16 Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' chapters [55, 71, 81, 103, 109, 124, 126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' As a supplement or alternate to searching and collecting hateful posts en masse, five studies considered extremist ‘ground truth’ instances by comparing tex- tual similarity from Tor-based terror-supporting anonymous forums [4] and websites [124], radical Islamist magazines and manifestos [7, 84, 95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Interestingly, no studies considered extracting ground truths from far-right manifestos or media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Likewise, no studies considered recent low-moderation anonymised forums such as 8Chan (now 8kun) or Kiwifarms, which were extensive hubs for pro- paganda dissemination from the Christchurch shooter [106];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Parler, notable for its organisational influence during the 2021 Capitol Hill riots [72];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Telegram, TikTok, or Discord, despite reports on its use for sharing suicides, mass shootings, and group-lead harassment of minority groups [44, 87]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, there is a prevalent and concerning trend towards NLP studies on mainstream platforms, which may overlook the role of emergent, pseudo-anonymised or multimedia-oriented platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Data Collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 62% of all studies evaluate ERH on a single post-by-post basis, with NLP the dominant approach per Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Conversely, grouping posts on a per user basis frequently included annotations from cyber- vigilante groups such as the Anonymous affiliated OSINT ‘Controlling Section’ (CtrlSec) group’s annotations of ISIS-supporting accounts [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' However, Twitter claims that CntrlSec annotations were "wildly inaccurate and full of academics and journalists" [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, researchers should avoid unvetted cyber-vigilante data, and consider anonymising datasets to further benefit user privacy, researcher ethics, and model performance by reducing false positives (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', censorship).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' While NLP text detection is the dominant detection approach, 23 of the 51 studies investigated data sources outside of textual posts per Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Research gaps include the lack of multimedia and law enforcement studies, with only three hybrid text-image detection [57, 99, 111] and one study utilising FBI anonymous tips, Automated Targeting System and Secure Flight data [48, 119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 Data Collection and Annotation Bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Due to the varying fiscal costs, biases, and time trade-offs, there is no consensus for selecting or excluding annotators for supervised learning datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, we frame that annotator selection falls within two varying groups: experience-driven selection and organisation-driven selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' For the former, experience-driven selection includes studies that utilised self-proclaimed ‘expert’ panels as determined by their location and relevant degrees [123], are anti-racism and feminist activists [122], or work on behalf of a private anti-discrimination organisation [105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' However, assembling annotators by specific characteristics may be time-consuming or costly, such as crowd- sourcing tertiary annotators via Amazon Mechanical Turk, or Figure Eight [10, 45, 71, 81, 93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Conversely, an organisation-driven selection approach focuses on agreement by a crowdsourced consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Instead of relying on specific experience, researchers utilised custom-made tests for knowledge of hate speech criteria based on the researchers own labelled subset [122, 128].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Likewise, organising annotator pools can also include balancing annotators self-reported political affiliation to reduce political bias [93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Researchers use Inter-rater agreement, and Kappa Coefficient to determine a post’s ERH classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' For racism, sexism, and neither classifications, annotation Fleiss’ Kappa values ranged between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='607 [32] to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='83 [128], indicating moderate to strong agreement [125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Thirdly, unsupervised clustering enables mass data collection without time-consuming annota- tion via Louvain grouping (to automatically group text/networks to identify emergent groups) [15, 16, 108], or grouping based on a thread’s affiliation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', the now-banned r/altright [46] and v/[n- word] [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Although not all posts from an extremist platform may be manifestly hateful, as evident in the 9507 post ‘non-hate’ class in the Stormfront benchmark dataset from de Gibert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Research continues to skew towards radical Islamic extremism per Figure 4, while the plurality (41%) target generic ‘hate speech’ in ‘hate or not’, or delineations for racism, sexism, and/or offence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech 000:17 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='3 Benchmark Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We define a benchmark dataset as any dataset evaluated by three or more studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The majority of studies used custom web-scrapped datasets or Tweets pulled via the Twitter API per Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Datasets used by three or more studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Dataset Year Categories Platform of origin Collection strategy Used By Waseem and Hovy [123] 2016 16914 tweets: 3383 Sexist ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 1972 Racist,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 11559 Neutral Twitter 11-point Hate Speech Cri- teria [10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 52,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 55,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 56,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 71,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 81,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 83,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 91,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 123,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 126] FifthTribe [39] 2016 17350 pro-ISIS accounts Twitter Annotated pro-ISIS ac- counts [7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 84,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 95,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 102] de Gibert [32] 2018 1196 Hate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 9507 Non-hate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 74 Skip (other) posts Stormfront 3 annotators considering prior posts as context [32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 55,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 71,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 126] OffenseEval (OLID) [128] 2019 14100 tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (30%) Of- fensive or Not;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Targeted or Untargeted insult;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' towards an Individual,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' or Other Twitter Three-level hierarchical schema,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' by 6 annotators [55,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 67,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 130] HatEval [13] 2019 10000 tweets distributed with Hateful or Not,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Ag- gressive or Not,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Individual targeted or Generic Twitter Crowdsourced via Figure Eight,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' with 3 judgements per Tweet [13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 55,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 71,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 90,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 126] Hatebase- Twitter [31] 2019 25000 tweets: Hate speech,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Offensive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Neither Twitter 3 or more CrowdFlower annotators per tweet [31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 52,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 55,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 71,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 81,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 83,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 126] TRAC [61] 2018 15000 English and Hindi posts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Overtly, Covertly, or Not Aggressive Facebook Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' [62] subset, 3 annotators per post, com- ment or unit of discourse [55, 56, 71] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 Feature Extraction Techniques Figure 7 outlines the three types of feature extraction techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Non-contextual lexicon approaches relate to word embeddings for entities, slurs, and emotional features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' However, non-contextual blacklists and Bag of Words (BoW) lexicon approaches cannot identify context, concepts, emergent, or dual-use words (see the Supplementary Material’s Algorithm Handbook section for comprehensive definitions) [32, 81, 83, 90, 122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Contextual sentiment embeddings expand on lexicons by embedding a form of context via positional tokens to establish an order to sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We group unsupervised term clustering and dimensionality reduction methods under the Probability-Occurrence Models category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The two dominant approaches include weighted ANOVA- based BoW approaches and Term Frequency-Inverse Document Frequency (defined in the Supple- mentary Material), which weigh the importance of each word in the overall document and class corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Contextual sentiment embeddings result in higher F1-scoring models (per RQ4) due to their context-sensitivity and compatibility as input embeddings for deep learning models [55, 71, 83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Community detection features require mapping following, friend, followee, and mention dy- namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Furthermore, other statistically significant metadata includes profile name, geolocation (to investigate ERH as a disease), gender, and URLs [16, 81, 84, 123, 126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' URLs can identify rabbit holes for misinformation or alternate forums via PageRank [88] and Hyperlink-induced Topic Search (HITS) [59]—which extracts keywords, themes and topical relations across the web [77, 88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 000:18 Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Types of feature extraction techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='3 Data Filtering For context-insensitive BoW and non-deep models, stop words (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' the, a, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ), misspellings, and web data are often filtered out via regular expressions and parsing libraries [4, 27, 46, 47, 52, 69, 71, 81, 84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Compared to semantic or reply networks, community detection models tend to extract metadata for separate clustering for entity and concept relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' All data filtering techniques are thereby aggregated and branched in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' No studies considered satire, comedy, or irony to delineate genuine extremism and online culture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Researchers’ implicit consensus is to treat all posts as part of the ERH category if it violates their criteria, regardless of intent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Conversely, Figure 9 displays the 14% of the studies filtered bots by removing but not classifying bot accounts from the ERH datasets [12, 15, 16, 46, 69, 93, 109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Strategies include removing duplicate spam text, filtering Reddit bots by name, and setting minimum account statistics for verification—such as accounts with that share hashtags to at least five other users to combat spam [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Likewise, Lozano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' limited eligible users for their dataset to have at least 100 followers, with more than 200 tweets, and at least 20 favourites to other tweets [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' This operates on the assumption that bots are short-lived, experience high in-degree between similar (bot) accounts, and seldom have real-world friends or followers—as discovered by Bartal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' [12] Outside of removing suspicious bot accounts via human annotation in dataset generation, com- putational means to explicitly categorise bots or trolls remains an area for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='SenticNet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='LIWC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content="('Bag of Concepts/Narratives') " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='Hatebase ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='Word2Vec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='Doc2Vec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='Blacklists ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='Semanticizer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='Non-contextualLexicon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='SentiWordNet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='Contextual Sentiment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='SentiStrength ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='Approaches ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='Embeddings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='SIMON ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='Entity Detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='Transfer Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='TransferLearningDatasets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='Tasks (pre-training) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='(Stanford Sentiment) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='EmoFeat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='GPT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='Dynamic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='Dependency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='FastText ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='few shot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='Clustering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='Parse Trees ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='WordPiece ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='Theme and Narrative ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='Bag of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='Tokenization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='Clustering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='Words ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='Entity Clustering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='PCA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='Probability-Occurrence Models ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='TF-IDF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='DBSCAN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='ANOVA F-value weighted BoWDown the Rabbit Hole: Detecting Online Extremism,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Radicalisation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' and Politicised Hate Speech 000:19 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Types of data filtering techniques across NLP and community detection studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Studies with Bot or Troll Filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Special Type of Data Used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='4 Key Takeaways for Dataset Domain, Pre-processing, and Annotation Twitter’s accessible API, popularity and potential for relationship modelling via reply and hashtag networks, makes it the platform of choice for research (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Despite the rise of far-right extremism post-2015, Islamic extremism in the US and Europe remains the target group for the majority of organisation-based studies, with no studies considering far-right/left manifestos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The marginalisation of Oceania and the global south by datasets predominantly containing US white hetero males indicates a structural bias in academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' For feature extraction, we recommend using: (1) Contextual sentimental embeddings—due to their compatibility with deep learning models and highest performance, per Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (2) Pre-defined lexicons—assuming they remain up-to-date with online culture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (3) Probability-occurrence models—ideal for large-scale clustering of emergent groups [27, 99, 126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' RELATIONS BETWEENPOSTS 6 SEMANTIC FEATURE DATA AND DICTIONARIES 5 NETWORKS METADATA (PROFILE DATA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' LOCATION) SIMILARITY TO CORPUS DATA 3 TEMPORAL DATA (TIME ZONE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' TOPICAL) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='LAW 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+page_content='Removal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='English posts) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='Stop words ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='URLS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='Identity pseudo- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='(for context ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='anonymisation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='insensitive MLAs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='(usenames,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' names,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' mentions etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' )6% 14% Yes ■No Unspecified or Irrelevant to Study 80%000:20 Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We do not recommend pre-defined lexicons on non-English text, new groups or ideologies—as these lexicons may not translate to different concepts and slurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We recommend adaptive semi or unsupervised learning via contextual embeddings and entity/concept clustering for edge cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Research currently lacks multidomain datasets, pseudo-anonymous platforms, multimedia (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', images, videos, audio and livestreams), and extraction of comedy, satire, or bot features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 7 COMMUNITY DETECTION, TEXT, AND IMAGE ERH DETECTION ALGORITHMS What are the computational classification methods for ERH detection?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Between 2015-2021, non-deep Machine Learning Algorithms (MLAs) shifted towards Deep Learning Algorithms (DLAs) due to their superior performance and context-sensitivity (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Support Vector Machines (SVM) and a case of a Random Forest (RF) model were the last remaining non-deep MLAs post-2018 to outperform DLAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Studies seldom hybridise relationship network modelling and semantic textual analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Ongoing areas of research in MLAs consist of identifying psychological signals to compete with DLAs such as Bidirectional Encoder Representations from Transformers (BERT), Convolutional Neural Networks (CNN) and Bidirectional Long Short-Term Memory (BiLSTM) models (defined further in the Supplementary Material’s Algorithm Handbook section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' DLAs are best oriented for text-only tasks and for hybrid image-caption models [3, 57, 99, 111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Future NLP studies should consider higher-performing neural languages models over BERT-base—such as RoBERTa [68], Sentence-BERT [96], or multi-billion parameter transformers such as GPT-3 [23], GPT-Neo [19], or Jurassic-1 [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Observed Non-deep Machine Learning Algorithms (MLAs) Studies investigating non-deep MLAs tend to test multiple models, typically Support Vector Ma- chines (SVMs), Random Forest (RF), and Logistic Regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Figure 11 outlines the distribution of both deep and non-deep approaches, with SVM again the most popular MLA in 15 of the 51 studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Non-deep MLAs consistently under-performed for multiclass classification, whereby Ahmad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' identified that a prior Naïve Bayes model could not distinguish between ‘Racism’ and ‘Extremism’ classes due to a low F1-score of 69%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' while their LSTM and GRU model could detect such nuance with a 84% F1-score [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Likewise, application-specific sentimental algorithms paired with MLAs resulted in lower F1-scores compared to context-sensitive BERT models—which do not require manual feature extraction [55, 71, 83, 126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' claimed that SentiStrength was ".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='not robust to various sentence structures and hence fails to capture important semantics like sarcasm, negation, indirect words, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' at the phrase level" [107, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 5]—a critique shared in six other non-deep sentiment scoring studies [47, 69, 84, 89, 103, 130].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Consider the hypothetical case of "I am not happy with those people", whereby context-insensitive (orderless) embeddings will not detect the negation of happy nor the implicit euphemism for ‘those people’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, researchers have three options when designing ERH models: (1) Avoid complex textual feature extraction and filtering by prioritising DLA development, or (2) Prioritise manual textual and metadata feature extraction, such as psychological signals, emotions, sarcasm, irony, temporal data, and/or (3) Consider community detection (relationship network or topic modelling) features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Non-deep Machine Learning Algorithms in Community Detection Studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' There is a discrepancy in the choice of algorithmic approach compared to NLP-oriented models where less than a third of the community detection studies considered Deep Learning (DL) mod- els [77, 84, 99], while NLP-only studies were majority DL (15 of 29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' A reason for this discrepancy would be the limited research in social media network analysis without investigating textual data, ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech 000:21 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Number of instances of Machine Learning Algorithms (MLAs) used for ERH detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' instead opting to cluster group affiliation via K-means [12, 16, 80], NbClust [12], weighted bipartite graphing into Louvain groups [15, 16], and fast greedy clustering algorithms [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We observed that graphing relationship networks result in two types of classification categories: (1) Meso-level affiliation—semi or unsupervised affiliation of a user to an extremist group or organisation, with a bias towards Islamic extremist groups [15, 16, 80, 84, 99, 101, 108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (2) Micro-level affiliation—(semi)supervised person-to-person affiliation to an annotated extremist, such as radicalising influencers [12, 24, 77], and legal person-of-interest models [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' For organisational affiliation, information for clustering included the use of hashtags shared by suspended extremist Twitter users and unknown (test) users [7, 15, 84, 95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' For identifying a user’s affiliation to other individuals, researchers preferred non-textual graph- based algorithms as they reduce memory complexity and avoid the perils in classifying ambiguous text [16, 80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Furthermore, 2016-2019 demonstrated a move from investigative graph search and dynamic heterogeneous graphs via queries in SPARQL [48, 80] towards Louvain grouping on bipartite graphs as a higher-performing (by F1-score) classification method [15, 16, 108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' For hybrid NLP-community detection models, researchers mapped text and friend, follower/ing, and mention networks via decision trees and kNN [77, 84], or used Principal Component Analysis on extracted Wikipedia articles to map the relationships between discussed events and entities [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' An emerging field of community detection for extremism consists of knowledge graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Know- ledge graphs represents a network of real-world entities, such as events, people, ideologies, situ- ations, or concepts [125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Such network representations can be stored within graph databases, word-embeddings, or link-state models [48, 125, 127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Link-state knowledge models consist of undirected graphs where nodes represent entities and edges represent links between entities, such as linking Wikipedia article titles with related articles based on those referenced in the article, as ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='SUPPORTVECTOR MACHINE (SVM) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='SENTIMENTAL ALGORITHMS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='LOGISTIC REGRESSION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='OTHER DECISION TREES ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='OTHER (NOVEL) NEURAL NETWORK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='GRAPHING ALGORITHMS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='NOVEL APPROACHES ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='BERT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='CONVOLUTIONAL NEURAL NETWORK (CNN) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='LSTM-ONLY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='RANDOM FOREST ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='ENSEMBLE NEURAL NETWORKS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='KNN CLASSIFICATION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='NAIVE BAYES ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='GRU NEURAL NETWORK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='PRINCIPAL COMPONENT ANALYSIS (PCA) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='K-MEANS CLUSTERING ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='LDA (LINEAR DISCRIMINANT ANALYSIS) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='LINEAR REGRESSION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='LSA (LATENT SEMANTIC ANALYSIS) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='16000:22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' used in Wikipedia2Vec [127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' consider a novel hybrid OSINT and law-enforcement database graph model-—which unifies textual n-grams from social media to shared relationships between other individuals and law enforcement events over time [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Four studies consider model relationships to individual extremist affiliates [12, 24, 48, 77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' In a direct comparison between text and relationship detection models, Saif et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' observed that text-only semantic analysis outperformed their graph-based network model by a +6% higher F1-score [101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 Deep Learning Algorithms (DLAs) DL studies are rising, with less than a third of studies including DLAs pre-2019 [10, 18, 27, 32, 45, 91, 99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The percentage of all studies which included a DLA per year was 0% in 2016, 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='3% in 2017 [10, 27, 45, 99], and 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='3% in 2018 [18, 32, 91], compared to being the majority post-2018 (81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='8% in 2019 [1, 4, 53, 67, 71, 77, 83, 84, 90, 128–130], 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='5% in 2020 [14, 55, 57, 81, 107, 111] and 80% in 2021 [3, 52, 126])—with Figure 12 displaying the shift towards DLAs since 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Patterns of adoption for ERH detection algorithms over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Colour change ordered by F1-score trend (low to high).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Brown = ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='75 F1-score on benchmark datasets, Red = ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='9 F1-score, Grey = No Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Between 2017-2018 Convolutional Neural Networks (CNN) using Long-Short Term Memory (LSTM), GRU, Recurrent Neural Networks (RNN), or graph-based layers were the sole DLAs [10, 18, 32, 45, 91, 99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' From 2019-2021, various new approaches such as SenticNet 5 [89], ElMo (Embeddings from Language Model) [90], custom neural networks such as an Iterative Opinion Mining using Neural Networks (IOM-NN) model [14], and attention-based models such as BiLSTM [81, 83, 90, 128, 129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Since 2019, there is an emerging consensus towards BERT [67, 71, 81, 107, 126, 129, 130] due to its easy open-source models on the Hugging Face platform and high performance per Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Deep Learning for Community Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Only 3 of the 21 DL studies considered relationship network models [77, 84, 99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Whereby, Mashech- kin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' grouped self-proclaimed Jihadist forums and VK users with Jihadist keywords as a "suspicious users" category [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Uniquely, the researchers implemented a Hyperlink-induced Topic Search (HITS) approach to calculate spatial network proximity between annotated extremists and unknown instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' HITS identifies hubs, which are influential web pages as they link to numerous other information sources/pages known as authorities [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The influence of an authority depends on the number of hubs that redirect to the authority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' An example of HITS in-action would be an ex- tremist KavazChat forum (a hub) with numerous links to extremist manifestos (authorities) [59, 77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Evaluating influence in these graph networks requires measuring spatial proximity via betweenness centrality [41] and depth-first search shortest paths where proximity to a known extremist via ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Prior SLR trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Algorithmic and Feature Extraction Trends found in our SLR Community Expansion of Sentimental Year-of-Hybrid-Models Detection algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' First CNN+LSTM, BiLSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Use BERT & SVM as Blacklist with Semantic application of CNN of language models the dominant Words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' LSTM, & Word2Vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (BERT), approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 1 2000-2010 2010-2015 2015 2016 2017 2018 2019 2020 2021 1 Bag of Words Generic ML (sentimental Year-of-Extremism- Context-sensitive models Approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' feature extraction in detection via SVM, RNN & features - phrase, MLA) Bipartite Graphing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' kMeans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Emergence of ExtremSentiLex, opinion Waseem & Hovy dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' affiliation-based datasets & propaganda mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech 000:23 following/reposting them constitutes an extremist classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' However, such relations do not accommodate for replies to deescalate, deradicalise, or oppose extremist speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 Visual-detection Models for ERH Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Despite the emergence of multimedia sources for radicalisation and ideological dissemination, only three studies considered multimodal image and image-text sources—utilising image memes with superimposed text from the Facebook hateful meme dataset [57] and the MultiOFF meme dataset [111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Only one study considered the post’s text (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', text not displayed on the image itself) as context via the multimedia Stormfront post and image data from Rudinac et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' For the Facebook hateful meme and MultiOFF datasets include images with superimposed cap- tions [57, 111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Both Kiela et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' [57] and Aggarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' [3] extract caption text via Optical Character Recognition (OCR) models—-a computer vision technique to convert images with printed/visual text into machine-encoded text [125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The three hateful meme studies utilised either both (multimodal) or one (unimodal) of the image and its caption [3, 57, 111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The multimodal Visual BERT-COCO model attained the highest accuracy of 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='47%, compared to 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='8% for a caption text-only classifier or 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='73% for image only, 64% for the ResNet152 model [3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' and 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='70% for the baseline (human) [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Deep learning pipeline for visual-text ERH detection based on the hateful meme studies [3, 57, 111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The highest performing multimodal model relied on Visual BERT [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Visual BERT extends textual BERT by merging BERT’s self-attention mechanism to align input text with visual regions of an image [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' VisualBERT models are typically pretrained for object detection, segmentation, and captioning using the generic Common Objects in COntext (COCO) dataset [66], such that the model can segment and create a textual description of the objects behind an image such as “two terrorists posing with a bomb” (Figure 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Training otherwise acts the same as BERT-—which involves masking certain words/tokens from a textual description of the image of what the image depicts, and VisualBERT predicting the masked token(s) based on the image regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We aggregate and generalise all visual ERH detection studies architectural pipelines in Figure 13 [3, 57, 99, 111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' No hateful meme dataset studies consider accompanying text from the original post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' This raises concerns regarding posts satirising, reporting, or providing counter-speech on hateful memes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Only one study investigated a contextual textual post and accompanying images through a proposed Graph Convolutional Neural Network (GCNN) model [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' This GCNN approach extracted semantic concepts extracted from Wikipedia, such as identifying that an image was a KKK rally— attaining a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2421 F1-score for detecting forum thread affiliation across 40 Stormfront threads [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Key: Dotted lines = Optional path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Full line = necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Red outline = Computer Vision (lmage Embeddings) ResNet152 Extract 1000- Computer Vision dimension Model embeddings Extract Regions of Create Textual Representation Entity and Object the Images (Visual VisualBERT ("Two individuals with a Detection BERT tokens) bomb") Separate contextual NLP model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Top Text BERT) Textual Optical Textual Feature Concatenate Hate Classifier Character Extraction Embeddings Neural Network Recognition "Top "Bottom Bottom Text Text" Text"000:24 Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 8 MODEL PERFORMANCE EVALUATION, VALIDATION, AND CHALLENGES What are the highest performing models, and what challenges exist in cross-examining them?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Evaluating model performance presents three core challenges for future researchers: (1) Dataset domain differences—which may include or exclude relevant features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', gender, location, or sentiment) and may involve numerous languages or groups (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Islamic extrem- ists vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' white supremacists) who will express themselves with different lexicons [20, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (2) Criteria differences—different standards for ERH definitions, criteria, filtering, and annota- tion threaten cross-dataset analysis between models [24, 122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Binary classification can result in higher accuracy compared to classifying nuanced and non-trivial subsets of hate such as racism/sexism [122], overt/covert aggression [61], or hateful group affiliation [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (3) Varying and non-standard choice of metrics—Figure 14 displays the 28 metrics, which vary depending on whether the study investigates community detection via closeness, in- betweenness, and eigenvectors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' or NLP, often via accuracy, precision, and F1-scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Distribution of metrics used across the 51 studies—demonstrating a lack of standardisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Models ranked by macro F1-score for the benchmark datasets across studies (inter-study evaluation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Dataset 1st Highest 2nd Highest 3rd Highest Waseem and Hovy [123] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='966 (BERT with GPT-2 fine-tuned dataset [126]) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='932 (Ensemble RNN [91]) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='930 (LSTM + Random Em- bedding + GBDT [10]) FifthTribe [39] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='0 (RF [84]) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='991-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='862 (SVM [7]) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='87 (SVM [95]) de Gibert [32] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='859 (SP-MTL LSTM, CNN and GRU Ensemble [55]) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='82 (BERT [71]) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='73 (LSTM baseline met- ric [32]) TRAC FB [61] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='695 (CNN + GRU [55]) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='64 (LSTM [61]) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='548 (FEDA SVM [56]) Hatebase Twit- ter [31] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='923 (BiLSTM with Atten- tion modeling [83]) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='92 (BERTbase+CNN / BiL- STM [81], 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='86 (with racial/- sexual debiasing module) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='912 (Neural En- semble [71]) HatEval [13] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='7481 (Neural En- semble [71]) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='738 (LSTM- ELMo+BoW) [90] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='695 (BERT with GPT-2 fine-tuned dataset [126]) OffensEval [128] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='924 (SP-MTL CNN [55]) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='839 (BERT [130]) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='829 (BERT 3-epochs [67]) ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='F1-SCORE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='ACCURACY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='PRECISION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='RECALL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='P-VALUE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='CORRELATION COEFFICIENT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='NETWORKCENTRALITIES ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='MEAN ABSOLUTE ERROR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='STANDARDIZED CO-INCIDENCE RATIO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='LOUVAIN (Q) MODULARITY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='PATIAL PROXIMITY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='GINI COEFFICIENT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='INFORMATION GAIN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='PARTITION SENSITIVITY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='MEANLOG ACCURACY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='RACIST SCORE (CUSTOM METRIC) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='WILCOXON SIGNED-RANK TEST ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='CHI-SQUARE INDEPENDENCE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='PEARSON CORRELATION COEFFICIENT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='PAGERANK SCORES ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='RADICAL SCORE (SENTISTRENGTH) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='HYBRID STRUCTURAL-SEMANTIC SIMILARITY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='POSSIBILISTIC SIMILARITY MEASURE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='ROC AREA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='SIRA RADICAL SCORE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='R^2 VALUE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='40Down the Rabbit Hole: Detecting Online Extremism,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Radicalisation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' and Politicised Hate Speech 000:25 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Benchmark Dataset Performance (Inter-study Evaluation) We use macro F1-score as the target metric as it balances true and false positives among all classes, and is a shared metric across the benchmark datasets [13, 31, 32, 39, 123, 128].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Table 4 outlines that the highest F1-scoring models reflect the move towards context-sensitive DLAs like BERT, as also displayed in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' SVMs and a single instance of a Random Forest classifier on sentimental features were the last standing non-deep MLAs [7, 56, 84, 95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Given the variety of MLA and DLAs (Figure 11), approaches that frequently underperformed included Word2Vec, non-ensemble neural networks such as CNN-only models, and baseline models [31, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' These baseline models include the HateSonar model by Davidson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' [31], Waseem and Hovy’s n-grams and gender-based approach [123], LSTM model by de Gibert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' [32], and C-Support Vector Classification by Basile et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' No studies discuss memory or computational complexity, an area worthy of future research as expanded in our Supplementary Material’s Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 Community Detection Performance While community detection models tend to produce F1-scores ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='15 lower than DLAs [12, 15, 16, 48, 80, 108], these comparisons rely on different datasets/metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Shi and Macy recommended using Standardised Cosine Ratio as the standardised metric for structural similarity in network analysis, as it is not biased towards the majority class, unlike Jaccard or cosine similarity [108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' For community detection models on the same pro/anti-ISIS dataset [39], F1-scores ranged from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='74-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='93 [7, 15, 95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Only one study cross-examined text and network features [101], with a hybrid dataset consisting of annotated anti/pro-ISIS users’ posts and number of followers/ing, hashtags, mentions, and location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Text-only semantic analysis outperformed their network-only model (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='923 F1 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='866 respectively) [101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' However, topic (hashtag) clustering and lexicon-based sentiment detection via SentiStrength underperformed compared to the network-only approach by a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='07-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 lower F1 [101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Thus, unsupervised clustering models are ideal for temporal radicalisation detection and identification of emergent or unknown groups or ideologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' There is insufficient evidence to conclude whether community detection is superior to NLP due to the lack of shared NLP-network datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' For supervised community detection tasks, researchers [15, 80, 101] used network features via Naïve Bayes [80], k-means [12, 16, 80], SVM [84], and decision trees [77, 84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The highest F1-score community detection model was a hybrid NLP and community detection model using network features, keywords and metadata (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', language, time, location, tweet/retweet status, and whether the post contained links or media) with a Naïve Bayes classifier—attaining a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='89 F1-score [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 9 FUTURE RESEARCH DIRECTIONS In this section, we offer an alternate to the radicalisation = extremism = political hate speech consensus from RQ1 and models observed in RQ3/4 to present a new framework for delineating and expanding ERH for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Overall, we propose an uptake roadmap for ERH context mining to expand the field into new research domains, deployments for industries, and elicit governance requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Ideological Isomorphism—a Novel Framework for Radicalisation Detection Definition 4: Ideological Isomorphism (Computational Definition for Radicalisation) The temporal movement of one’s belief space and network of interactions from a point of normalcy towards an extremist belief space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' It is an approach to detecting radicalisation with an emphasis on non-hateful sentiment as ringleaders and/or influencers pull and absorb others towards their hateful group’s identity, relationships, and beliefs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 000:26 Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' As outlined in our novel tree-diagram dissection of ERH definitions to their computational approach in Figure 6, there is considerable overlap in approaches between the otherwise unique fields of extremism, radicalisation and politicised hate speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Radicalisation’s working definition suffers from ambiguity in the majority of studies due to its interchangeability towards extremist affiliation and no considerations for temporal changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Radicalisation’s computational definition should reflect a behavioural, psychological, and ideological move towards extremism over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' While extremist ideologies and outwards discourse towards victim groups may be manifestly hateful, radicalisation towards target audiences may involve non-hateful uniting and persuasive speech [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, we propose that radicalisation detection should not be a single-post classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Rather, models should consider micro (individual), meso (group dynamics), and macro (global events and trends) relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The roots for radicalisation result from an individual’s perceptions of injustice, threat, and self-affecting fears on a micro-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' On a meso-level, this can include the rise of community clusters based on topics and relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Socially, a radicalised user draws on an extremist group’s legitimacy, connections and group identity, trends, culture and memes [58, 70, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence mapping ideological isomorphism requires temporal modelling to: (1) Detect the role of users or groups polarising or pulling others towards extreme belief spaces (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', ideological isomorphism), akin to detecting online influencers [12, 46, 80, 121].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Studies should also consider the role of alienation as a radicalising factor via farthest-first clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (2) Further research into the role of friendship and persuasion by adapting sentimental ap- proaches to consider positive reinforcement towards hateful ideologies akin to existing research in detecting psycho-behavioural signals [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Furthermore, there lacks research in computationally detecting social factors such as suicidal ideation or mental health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (3) Investigate the interactions between groups across social media platforms as radicalisers themselves, such as the promotion of extremist content by recommendation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (4) Utilise community detection metrics such as centrality, Jaccard similarity, and semantic similarity over time as measurements for classifying radicalisation for meso-level NLP (topic) and graph-based (relational) clustering, leaving content moderation as a separate task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (5) Consider the role of satire, journalism, and martyrs as areas for radicalisation clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 Morphological Mapping and Consensus-building—a novel computationally-grounded framework for extremism detection Definition 5: Morphological Mapping and Consensus-building (Extremism) The congregation of users into collective identities (‘in-groups’) in support of manifestly unlawful actions or ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' While ideological isomorphism focuses on micro-level inter-personal relations, morphological mapping pertains to clustering meso-level beliefs and community networks to extremist ideologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' While we discovered various affiliation-based clustering approaches, no studies identified novel or emergent movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Establishing a ground truth for a novel extremist organisation is challenging if such groups are decentralised or volatile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, we recommend using manifestos, particu- larly unconsidered far-right sources, and influential offline and online extremists as a benchmark for identifying martyrdom networks and new organisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Areas for future research include investigating the role of trolls, physical world attacks, or misinformation in narrative-building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Our morphological mapping framework proposes to delineate Extremism by considering the role of group identity and ideological themes behind hate speech by considering affiliation across users and posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' When targeting extremism, pledging ‘support’ to a terrorist organisation may not violate ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech 000:27 context-insensitive BoW hate speech classifiers—-hence it is not appropriate to categorise extremist affiliation under the same guise as post-by-post hate speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Currently, extremism detection constitutes a binary ‘pro vs anti group’ classification, which fails to capture the inner trends of radicalisation from peaceful, to fringe beliefs, to committing to violent-inducing beliefs online, and potentially to offline extremism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Investigating semi or unsupervised clustering (mapping) of groups will also aid Facebook’s commitment to moderating militarised social movements, violence-inducing conspiracy networks, terrorist organisations, or hate speech inducing communities [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Thus, we propose four prerequisites for studies to fall under the extremism detection category: (1) Investigate the interactions and similarities between groups on mainstream and anonymous platforms to map group dynamics and extremist networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' For privacy, we recommend group-level (non-individualistic) network and semantic clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (2) Map affiliation and group dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Given the lack of definitions for extreme affiliation, we recommend using Facebook’s definition of affiliation as a basis—being the positive praise of a designated entity or event, substantive (financial) support, or representation on behalf of a group (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', membership/pledges) [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (3) Investigate hateful and non-hateful community interactions, memes and trends, that reinforce group cohesion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (4) Map affiliation as a clustering task, akin to our proposed radicalisation framework but without the temporal component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='3 Outwards Dissemination––‘traditional’ hate speech detection updated Definition 6: Outwards Dissemination (Hate Speech) Targeted, harassing, or violence-inducing speech towards other members or groups based on protected characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, the projection and mainstreaming of hateful ideologies through speech, text, images, and videos requires an outwards dissemination of views shared by extremists, such as racism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The outwards dissemination of hate is a strictly NLP (text) and computer vision (entity and object) classification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We delineate hate speech with affiliation to violent extremist groups as such misappropriation could have devastating effects on one’s image, well-being, and safety [9, 24, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' All researchers should be aware that malicious actors may exploit existing ERH models for injurious surveillance and censorship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Future work should also consider the impacts of labels on society at large, whereby terms such as ‘far-right’ as an alias for white supremacy is both misleading, infers a ‘right vs wrong’ left-to-right spectrum, and ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We recommended decoupling religious contexts in favour of technical terms such as ‘radical Islamic extremism’ or ‘terror-supporting martyrdom’ to avoid grouping religiosity to a political ideology and terrorism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Thus, we propose three key prerequisites for a study to be in the hate speech category: (1) Investigates textual or multimedia interactions only, whereby detecting cyber-bullying or extremist community networks should be separate tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (2) Decouple affiliation where possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' For instance, white supremacy instead of far-right (an ambiguous term) or organisational affiliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (3) Consider models which include latent information, such as news, entities, or implied hate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Datasets should explain each classification with categories for disinformation and fallacies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 000:28 Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Future work in outlining hate speech would be a systematic socio-legal cross-examination of hate speech laws from governments and policies from social media platforms—including the emerging consensus vis-à-vis the harmonised EU Code of Conduct for countering illegal hate speech [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='4 Uptake Roadmap for Researchers, Industry, and Government We present a pipeline for researchers, industries, and government analysts to approach ERH context mining per Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' In addition to this summary visualisation of our key dataset and model recommendations, we expand on our actionable recommendations for immediate next steps and long-term software requirements for ERH detection in our supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Future SLRs should consider a mixture of academic studies, grey material, and technical reports to further encompass our proposed ERH context mining field’s socio-legal component and explore industry approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We recommend transforming our ethical recommendations for responsible research outlined in our SLR design into formalised interdisciplinary guidelines to protect privacy and researcher safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ERH is never a singular end-goal, post, or unexpected event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, detecting erroneous behaviour emanating from mental health crises can both avoid ERH online and offline, and present avenues for cooperation with third-parties such as suicide prevention and counselling groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Finally, we recommend searching for multimedia-only studies including for livestreams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 10 CONCLUSION ERH context mining is a novel and wide field that funnels to one fundamental aim—the pursuit to computationally identify hateful content to enact accurate content moderation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' In our work, we harmonised Extremist affiliation, Radicalisation, and Hate Speech in politicised discussions from 2015-2021 in a socio-technical context to deconstruct and decouple all three fields of our proposed ERH context mining framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, we propose a novel framework consisting of ideological isomorphism (radicalisation), morphological mapping (extremism), and outwards dissemination (politicised hate speech) based on our findings in RQ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' While hate speech included racism and sexism, other forms of discrimination were seldom considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Extremism and radicalisation frequently targeted Islamic groups, particularly from US and European researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Binary post-by- post classification remain the dominant approach despite the complexity of online discourse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' There is a clear and present danger in current academia emanating from the unresolved biases in dataset collection, annotation, and algorithmic approaches per RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We observed a recurring lack of consideration for satire/comedic posts, misinformation, or multimedia sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Likewise, data lacked nuance without contextual replies or conversational dynamics, and were skewed towards the US and Europe—with the global south, indigenous peoples, and Oceania all marginalised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Computationally, we identified that deep learning algorithms result in higher F1-scores at the expense of algorithmic complexity via RQ3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Context-sensitive neural language DLAs and SVM with sentimental, semantic, and network-based features outperformed models found in prior SLRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' However, state-of-the-art models still lack a contextual understanding of emergent entities, conversational dynamics, events, entities and ethno-linguistic differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' To combat injurious censorship and vigilantism, we recommended several areas for future work in context-sensitive models, researcher ethics, and a novel approach to framing ERH in SLRs and computational studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The poor design and abuse of social media threatens the fabric of society and democracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Re- searchers, industries, and governments must consider the full start-to-finish ecosystem to ERH context mining to understand the data, their criteria, and model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Without a holistic approach to delineating and evaluating Extremism, Radicalisation, and Hate Speech, threat actors (extremists, bots, trolls, (non-)state actors) will continue to exploit and undermine content modera- tion systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, informed, accurate and ethical content moderation are core to responsible platform governance while averting injurious censorship from biased models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech 000:29 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ERH Context Mining pipeline—with key identified research gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Sociolinguistics Red = Computer- science oriented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Computational Blue = social- Research Mixed = science Gaps: Social Media Criteria-building misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Satire & sexism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' beyond racism & Hate speech affiliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' non-violent Extremism as a temporal process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' a friendly Radicalisation as Privacy & Ethics Consider: ERH (RQ1, RQ2, RQ3) Dataset Creation Data Selection & Collection movements Conspiracy COVID-19 posts extremist Non-lslamic anonymisation) (data Researcher safety manifestos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' New APls datasets Multiple platform platforms /anonymous Controversial Consider: Data Filtering Extraction Topic mapping to Belief space entities/events) (Wikipedia/news sources informative mapping emotion detection Context-sensitive features Suicidal ideation signals Psychological Consider: 80 Deployment (RQ3/4) Model Choice & Model Creation & Performance Privacy Paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Image detection Hybrid NLP- Meme detection text generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' conversational Synthetic learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Few-shot 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='18653/v1/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='emnlp- demos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='4 [128] Marcos Zampieri, Shervin Malmasi, Preslav Nakov, Sara Rosenthal, Noura Farra, and Ritesh Kumar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Predicting the Type and Target of Offensive Posts in Social Media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Association for Computational Linguistics, Minneapolis, Minnesota, 1415–1420.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='18653/v1/N19-1144 [129] Marcos Zampieri, Shervin Malmasi, Preslav Nakov, Sara Rosenthal, Noura Farra, and Ritesh Kumar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' SemEval- 2019 Task 6: Identifying and Categorizing Offensive Language in Social Media (OffensEval).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' In Proceedings of the 13th International Workshop on Semantic Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Association for Computational Linguistics, Minneapolis, Minnesota, USA, 75–86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='18653/v1/S19-2010 [130] Jian Zhu, Zuoyu Tian, and Sandra Kübler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' UM-IU@LING at SemEval-2019 Task 6: Identifying Offensive Tweets Using BERT and SVMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' In Proceedings of the 13th International Workshop on Semantic Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Association for Computational Linguistics, Minneapolis, Minnesota, USA, 788–795.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='18653/v1/S19-2138 ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Supplementary Material for: Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech JAROD GOVERS, ORKA Lab, Department of Software Engineering, University of Waikato, NZ PHILIP FELDMAN, ASRC Federal, US AARON DANT, ASRC Federal, US PANOS PATROS, ORKA Lab, Department of Software Engineering, University of Waikato, NZ Contents Contents 1 1 Definitions—The Algorithm Handbook 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Definitions for Traditional (non-deep) Machine Learning Algorithms 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 Definitions for Deep Learning Approaches 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='3 Language Transformer Models 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='4 Definitions for Prominent Feature Extraction Techniques 7 2 SLR Design Considerations 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Quality Assessment Criteria 8 3 The Case for Performance Engineering when Evaluating Models 10 4 Uptake Roadmap Expanded 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Model Recommendations 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 Dataset Recommendations 11 References 12 1 DEFINITIONS—THE ALGORITHM HANDBOOK This supplementary material document includes the supplementary material referenced in the main Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech Systematic Literature Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' This document offers a ‘dictionary/look-up table’ for the core al- gorithmic architectures for the non-deep machine learning and deep learning models mentioned throughout the SLR, alongside other side findings and design considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We contextualise the Authors’ addresses: Jarod Govers, jg199@students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='waikato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='nz, ORKA Lab, Department of Software Engineering, University of Waikato, Gate 1, Knighton Road, Hamilton, Waikato, NZ, 3216;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Philip Feldman, philip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='feldman@asrcfederal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='com, ASRC Federal, Beltsville, Maryland, US;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Aaron Dant, aaron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='dant@asrcfederal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='com, ASRC Federal, Beltsville, Maryland, US;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Panos Patros, panos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='patros@waikato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='nz, ORKA Lab, Department of Software Engineering, University of Waikato, Gate 1, Knighton Road, Hamilton, Waikato, NZ, 3216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='11579v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='SI] 27 Jan 2023 000:2 Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' relevant strengths and weaknesses of the various algorithmic approaches for text and visual models for ERH detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' No new findings are in this handbook/‘look-up table’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, those familiar with the models listed in the contents above need not read this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Definitions for Traditional (non-deep) Machine Learning Algorithms We aggregate common and historic non-deep machine learning algorithms into the ‘traditional’ MLA category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, this section defines each of the baseline models used for textual or community detection models—consisting of: (1) Sentimental Bag of Words approaches, (2) Naïve Bayes, (3) Decision Trees, (4) Support Vector Machines, (5) Clustering Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Bag of Words (BoW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' BoW approaches simplify complex contextual sentences into a multiset (‘bag’) of individual words by assigning a value or probability to each word in its relation to a specific document class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' For instance, a BoW approach would deconstruct the contiguous sentence, “The Eldian people are the spawn of the devil” (where Eldian is a fictitious race), into an unordered bag of individual words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' While ‘are’, ‘the’ are unlikely to have a considerable influence on whether a sentence is hate speech or not, the use of ‘devil’ and ‘Eldian [race]’ is more frequently paired in hate speech than for non-hateful/off-topic text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The disregard of word order and the relationship of BoW approaches, and MLA models at large, constitute context-insensitive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' For instance, a BoW model does not know that ‘I love the Eldian people but hate their food’ is paring love -> Eldian, and hate -> food, and thus would consider ‘I hate the Eldian people but love their food’ as identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Likewise, BoW approaches do not consider alternate word meanings/uses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', ‘I ran for government’ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ‘I ran away’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Nonetheless, BoW approaches are core to word-specific ‘blacklists’ in content moderation, such as banning users who use slurs in a post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' However, for nuanced and often politicised discussions on controversial topics, simple blacklists can lead to injurious censorship—due to the context and use of such words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Sentimental algorithms, such as SentiStrength [42] aggregate individual words into individual emotions—whereby ‘love’ indicates a positive sentiment, while ‘hate’ generally appears in vitriolic speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Figure 1 outlines an abstracted representation of the sentiment classification based on the average sentiment score of a sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' However, the context-insensitive BoW models again fails for nuanced cases, whereby Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' [41] identified that SentiStrength cannot detect negations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', “I am NOT happy” where happy skews the final sentiment scores).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 Naïve Bayes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Naïve Bayes classifiers represent types of probabilistic classifiers utilising Bayes theorem with the assumption that the influence of each variable for classification is independent of each other (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', naïve) [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' For document classification, notable features are assigned a probability for their occurrence given a specific class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' For instance, a hate speech post that has an angry sentiment may have a P(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='8) (Probability of 80%) of being hateful, given that a test hate speech dataset may be 80% angry speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Bayes rule represents these chains of (assumed) independent/unrelated probabilities to form a final probability for a test instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Notable features for probability models include: Textual features—(e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', sentimental scores, appearance of certain slurs/terms), ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Supplementary Material for: Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech 000:3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' An abstracted example of Bag of Words approach within a Naïve Bayes classifier—demonstrating its lack of context sensitivity and the focus on key ‘racist’ words for ERH detection tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Network data—(e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', probability that someone who is friends with a supremacist is also a supremacist, retweet relationships), Metadata—(e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', length of a post, readability via a Flesch Reading Ease score, number of posts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' In the example of Figure 1, the probability that the tweet is racist depends on the probability that the racist tweet is angry, contains racial terms (‘Eldian’), the semantic similarity between known hate speech posts, and the appearance of a negative lexicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Naïve Bayes can be a final classifier for aggregating context-sensitive embeddings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', deep learning models) and multiple ‘ensembles’ of approaches/models—via chaining their probabilities together with this Bayes rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='3 Decision Trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' An example of a decision tree, with the leaf nodes constituting the classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' In the Eldian hate speech example, this would require traversing the left branches recursively for the final ‘Hate Speech’ classification leaf (shown via the red arrows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Chaining the correlations between features and their class likelihood can also span a tree of scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' If an annotated dataset indicates that a post is 80% likely to be racist if a sentiment-scoring algorithm detects anger, then a binary decision emerges—if post contains angry words, then likely hate speech;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' if not, then not hate speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' These rules construct decision trees, where the root constitutes the instance (text, network, metadata, or image), and each node is a decision, with the leaves (final node) being the expected class value (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', the classification) [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, decision trees are not naïve as they rely on specific values of other features when traversing a tree’s branches for a prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Example: Naive Bayes (simplified): "The Eldian people are the spawn of the devil" Sentiment Target Entity Semantic Similarity to Lexicon also affiliated Annotated Posts with extremists: Angry “hate\' Eldian 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content="9 1 (devil) P(Racist I Tweet) = “Eldian' 'devil' P(Rac." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='IAngry) * P(Rac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='I Eldian) * P(Rac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='I O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='9) * P(Rac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='I 1 lexicon) “scum" P(Racist)Post to Classify Sentiment = Not Sentiment = Angry Angry Contains slurs >=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='5 semantic <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='5 semantic No slurs similarity similarity = Yes (Devil) Not Hate Not Hate Hate Speech Hate Speech Speech Speech000:4 Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Creating an optimal tree that maximises accuracy and precision is not trivial due to the feature explosion of possible rules and tree nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, Random Forest classifiers rely on a divide- and-conquer algorithm for generalising feature pairings into class classifications with a random initialisation [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' This recursive process requires finding optimal splits to maximise the separation of classes for a final leaf, with an example tree presented in Figure 2—where a random forest would consists of multiple trees as a forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Ideally, a leaf node should encapsulate instances of one class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Random forests generate multiple decision trees and select the final prediction based on the predictions from the majority of decision trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Utilising multiple trees with a random initial tree state increases the range of features and values selected during the training step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Utilising multiple trees and testing the models on untrained ‘test’ data minimises the risk of over-fitting to the training (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', a classifier which performs reliably on the training dataset but not on real-world data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Random forests strengths include its ability to tie dependent and complex features while reducing over-fitting through pruning (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', reducing tree size to generalise the model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, decision trees capture related concepts in hate speech where naïve BoW approaches do not—such as the appearance of anger/negative sentiment invoking the use of charged terms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', racism as an emotional outlet) or frequency of posts and sentiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='4 Support Vector Machines (SVM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Support Vector Machine where instances beyond the boundaries (support vectors) are automatically assigned to the class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' SVMs are another supervised learning model for classification and regression tasks, seeking to map instances in vector spaces to maximise the distance between classes [14], visualised in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Mapping features to multidimensional vectors can exponentially increase dimensions (an issue shared in deep-learning models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Thus, SVMs reduce irrelevant features through specific kernels— typically a linear, polynomial, Gaussian or sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' These kernels reduce the feature set to draw boundaries between two classes, similar to logistic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' These boundaries are either hard (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', a binary classification) or soft—allowing outliers near the boundary for edge cases, like niche controversial and offensive, but not ostensibly targeting protected characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' SVM ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Support Vector (with cost hyperparameter) Neutral Speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Y-Axis Hate Speech Optimal hyperplane to maximise distance (y=mx+c) X-AxisSupplementary Material for: Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech 000:5 models are computationally faster and reduce memory compared to deep learning models [3, 45], while achieving comparative performance outlined in RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Dimensionality reduction techniques can also reduce runtime by reducing the complexity of large feature spaces from textual or network data, such as via Principle Component Analysis [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' SVMs are the consistently highest performing MLAs per RQ4, while lowest complexity, with 𝑂(𝑚∗𝑛) complexity for a Linear Kernel SVC—where m = feature count, and n = number of instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='5 Clustering and Nearest Neighbour Classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Instead of annotated hate speech datasets, clustering methods group by textual similarity via Natural Language Processing (NLP), and network relations via Community detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, clustering can work in cases of fully annotated datasets as supervised learning, semi-annotated datasets as semi- supervised learning, or unlabelled raw web scrapped data for unsupervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' For supervised learning, K-Nearest Neighbour (KNN) classifiers work via evaluating the nearest neighbours’ likeliness when projecting the textual, network, or metadata features onto a multidi- mensional space [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The ‘distance’ between feature spaces typically rely on Euclidean, Manhattan, or Minkowski distance—where the latter two are suited for non-linear feature spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Non-euclidean distances are ideal where dimensions are not comparable, as Manhatten distance reduces noise/er- rors from outliers since the gradient has a constant magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Clustering examples for hate speech detection includes K-Means, which partitions n observations into k clusters [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' K-Means automatically generates clusters, thus does not require annotated datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, K-Means can detect novel groups, including emergent extremist organisations, or influential individuals [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Unsupervised clustering’s strength for ERH detection is how it circumvents the definition issues for annotating data and can cluster large movements without costly annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' However, K-Means may not identify manifestly hateful posts, as it does not abide by any standard imbued within strict annotation criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Evidently, in the cross examination of a naïve approach vs their proposed K-Means derived model by Moussaouri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' [30], the naïve approach outperformed the possibilistic clustering by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='07-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='14 for accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='04-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='05 for precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 Definitions for Deep Learning Approaches Deep learning represents a family of machine learning algorithms with multiple layers and com- plexity, typically via neural network architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Neural networks rely on training a network with a set of weights at each layer, known as neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The first layer of a neural network utilises numeric representation of an instance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', hateful text) in numeric ‘tokenised’ form, which is adjusted throughout the hidden lower layers towards a final output (typically) classification layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Each downwards training step results in readjusting the weights of the upper layers for the neurons— known as backpropagation [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Figure 4 displays this architecture for neural networks per our example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The benefit of DLAs in ERH detection is the preservation of word order and meaning (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', “I ran” vs “I ran for president”), thus displaying context-sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Given dual-use words such as ‘queer’, or racially motivated slurs, understanding the surrounding contextual words is essential to reduce bias via misclassifications [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' DLAs dominant the benchmark dataset leader-board in RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Convolutional Neural Networks (CNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Convolutional Neural Networks (CNNs) expand on the neural network model through a convolu- tional layer—which acts as a learnable filter for textual or image embeddings [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Moreover, CNNs include a pooling layer(s) to reduce the spatial complexity of the network’s features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Reducing spatial size helps reduce the number of parameters and thus training time and memory footprint, while reducing over-fitting by generalising patterns in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 000:6 Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 Long Short-Term Memory (LSTM) and Gated Recurrent Unit (GRU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' LSTM and GRU aim to increase contextual awareness to process data sequences with long-term gradients to retain information on prior tokens [12, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' LSTM and GRU seek to reduce the vanishing gradients caused during backpropagation steps, which reduces classification performance as older trained instances are effectively ‘forgotten’ due to later weight changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Similarly, GRU’s are a gating approach with fewer parameters and thus higher runtime, enabling larger neural networks overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' CNN models with LSTM and GRU connections outperform CNNs on their own for hate speech detection [22, 23, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The highest performing BiLSTM model expands LSTM for bidirectional input, via two LSTMs—where tokens in the network utilise information from past (backwards) tokens/data and future (forwards) data [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The ability to uphold the temporal memory of prior tokens (attention) constitutes a Recurrent Neural Network (RNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='3 Language Transformer Models The state-of-the-art transformer architecture relies on self-attention—the memory retention of neural networks where each token of a sequence is differentially weighted [8, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Unlike Recurrent Neural Networks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', neural networks where nodes follow a temporal sequence), a transformer’s attention mechanism utilises context for any position for the token sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, transformers can handle words out of order to increase understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Transformers offer greater classification performance (see RQ4) at the expense of memory and computational overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' A considerable ethical threat of transformer models is their capability to predict future tokens (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', text generation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' For instance, a malicious actor could create realistic automated trolls or radicalising synthetic agents as bots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Language models also risk data leakage of their trained data through predicting tokens found in the original trained dataset, such as names or addresses [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' An abstracted example of a neural network for text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The top text represents its raw syntactic form, with its converted numeric embedding representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' These embeddings are responsible for altering the weights to increase token prediction or generation (for transformers) via backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' The final output layer for this example would offer the probability that the given text is racist, sexist, or benign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' topic relation relation entity The Eldian people are the spawn of the devil 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='553 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='814 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='666 Segment and Positional tokens (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' computational embeddings for position/order of words).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' KKKKKKKK Contextual interconnected layers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' neural network) Racism Sexism Neither Probability Probability ProbabilitySupplementary Material for: Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech 000:7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Cross-encoders (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', BERT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Bidirectional Encoding Representations from Transformers (BERT) is the most common cross- encoder observed for ERH detection [16], with the highest performance of all NLP models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Cross- encoders offer higher performance for classification tasks, through retaining information over a given sequence with a label (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', self-attention).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' BERT’s core strength is its memory retention of all tokens in a sentence, thus upholding full context-sensitivity of every word in the post it seeks to classify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' However, cross-encoders are computationally expensive due to high parameter counts (110 million parameters for BERT-base, 365 million for BERT-large), an issue further outlined in RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, an area of ongoing research includes model distillation (optimising and reducing parameter count to reduce memory requirements and training time), specialised training datasets, and alternate layers [26, 37, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' BERT is pre-trained on entries from English Wikipedia (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='5 million words) and the English BookCorpus (800 million words) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, such pre-trained models are then fine-tuned on a smaller dataset (typically 1000+ instances, per RQ2’s benchmark datasets) to optimise the BERT weights to detect hate speech with the context of its pre-trained corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 Generative Pre-trained Models (GPT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Similarly, the state-of-the-art GPT transformer architecture expands on the encoder blocks (shared with BERT) to include decoder blocks [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, GPT works on a single token (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', word vector) and produces estimates for the sequence’s next token—ideal for tasks such as text generation, summarising, question answering, and information retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' GPT models differ from BERT-based models via masked self-attention—an alternate form of context-sensitivity where the model only knows the context of the prior words in the sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' GPT-2/3 [8], GPT-Neo [7], and Jurassic-1 [25], are notable 2019-2021 era multi-billion parameter models—where larger datasets and parameter count result in more human-like text generation and higher performance in information retrieval tasks [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' GPT’s core strength in ERH detection synthetic hate speech generation via a GPT model fine- tuned on a hateful corpus—as investigated by Wullah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (see RQ3) [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' However, state-of-the-art GPT models utilise up to 178B parameters, whereby memory and computational requirements scale linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, future GPT work in synthetic text generation should consider inference tasks over fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Specifically, inference utilises a pre-trained model’s on-demand text generation capability through prompts rather than altering each of the billions of weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Using the auto- complete-like inference capabilities for generating realistic synthetic hate speech posts constitutes a novel case of prompt engineering in ERH detection and thus is a potential future research project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='4 Definitions for Prominent Feature Extraction Techniques This subsection outlines the three most common feature extraction techniques used for textual ERH detection—as outlined in RQ2 in the SLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' These models seek to identify hateful lexicons from text, or create numerical representations for word or sentence meaning via embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We deconstruct the six most common feature extraction techniques observed in our SLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Word2Vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Word2Vec is a model to convert words into vector embeddings, which compares synonymous words (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', ‘hate’ and ‘disgust’) via numerical vectors [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Word2Vec compares these word-to-vector embeddings via semantic similarity by evaluating their cosine similarity between their vectors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', comparing word vectors of an unknown class instance to words from a known ‘hate speech’ instance(s) to make a ‘hate or not’ classification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' On a word-level basis, the vector value for ‘king’ value for man + value for woman would result in a vector similar to queen [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' In our case, a ‘Islamist extremist’ and ‘ISIS’ are semantically similar akin to ‘White Supremacy’ and ‘Nazism’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 000:8 Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 Doc2Vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Similar to Word2Vec, Doc2Vec aggregates vector embeddings for paragraphs in addition to indi- vidual words [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Thereby offering memory of the current context and paragraph’s topic—useful for understanding a whole post’s sentiment and meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='3 N-grams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' N-grams represent contiguous sequences of n-number of characters for frequency analysis given their non-linear distribution in English, as well as when comparing a radical vs non-radical cor- pus [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' This linguistic model is often paired with methods such as TF-IDF or BoW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='4 Term Frequency-Inverse Document Frequency (TF-IDF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' TF-IDF determines the relevance of a word in a document by comparing its frequency in the document compared to its inverse number for the frequency of that word across all documents [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Thereby, assigning each word a weight to signify its semantic importance compared to the wider corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' For instance, radical Islamist dog-whistle terms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', coded or suggestive political messages intended to support a group) appeared disproportionately in extremist text compared to a neutral religious corpus [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='5 SenticNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' SenticNet embeds pattern matching, parser trees, and LSTM-CNN models for sentiment analysis, with the aim to replace a naïve BoW approach within a proclaimed bag of concepts and narrat- ives [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Specifically, it includes feature extraction methods of concept parsing (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', understanding linguistic patterns in natural language into conceptual pairs), subjectivity and polarity inference, alongside personality and emotion extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='6 Global Vectors for Word Representation (GloVe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' GloVe offers an unsupervised learning algorithm for context-independent word-to-vector embed- dings [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' While similar in creating vectors akin to Word2Vec, GloVe instead establishes word co-occurrences using matrix factorization (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', co-occurrence matrix of word [row] and context [usage of the word in the document]) and dimensionality reduction techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 2 SLR DESIGN CONSIDERATIONS This supplementary material section outlines the additional criteria and considerations for selecting papers and ensuring privacy-protections for users, groups and collected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' In essence, this section offers a meta-analysis of the ethics and selection process used throughout the SLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Quality Assessment Criteria The following includes our paper inclusion quality check criteria—with a score of 13 or higher required for inclusion in the final paper selection (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', final 51 papers included).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We propose a critical criteria for quality assessment to filter irrelevant or ambiguous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Specifically, for a study that passed a title and abstract screen, we assess the study’s clarity for ERH definitions and annotations (for objective and legible classifications), methodical clarity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' outlining each study’s algorithmic model, methods, data collection processes, and statistical analys- is/evaluation methods), and socio-technical considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We weighted each quality assessment section to prioritise their research methodology and clarity in their technical methods over their Conceptual Quality for studies encompassing broader socio-technical issues such as ethics, legality, or ERH clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' After a ten paper pilot study, we selected a score threshold of 65% to exclude irrelevant or ambiguous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Our supplementary material document includes the criteria and scoring for our quality assessment rubric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Supplementary Material for: Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech 000:9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Computational Quality (0 = None, 1 = Partial, 2 = Full).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (1) Is the radicalisation/affiliation detection model clearly defined?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (2) Is the radicalisation/affiliation detection model’s algorithm clearly defined?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (3) Is the training data reputable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (4) Are the models results compared to similar state-of-the-art methods?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (5) Is the methodology for designing and conducting their experiment clearly defined?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (6) Are patterns and trends discussed and presented clearly?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 Epistemological Quality (0 = None, 1 = Partial, 2 = Full).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (1) Does the source(s) (data or researchers) avoid any conflict of interests or expressed biases?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', explicit support/funding from a political think tank or state agency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (2) Does the study provide a cited or evidence-based definition for “radicalisation”, “hate speech” or "extremist" affiliation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (3) Are the dataset annotations vetted by more than one annotator to reduce bias?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='3 Conceptual Quality (0 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='5 value, as not critical but useful).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (1) Does the study discuss social or ethical issues in ERH detection (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' censorship)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (2) Do the authors discuss the legality of their model or definitions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (3) Does the study evaluate its model across multiple social media platforms?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' (4) Does the study discuss regulatory frameworks or recommendations for social media platforms based on their findings?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='4 Researcher Ethics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We focus on key terms and compositions of ERH examples to protect the privacy of the individuals exposed, as recommended by meta-studies on extremism research ethics [9, 13, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' When linking ERH detection to real-world groups and events, we solely focus on events and organisations which resulted in media attention or criminal convictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' In no part during this SLR did we attempt to track users, groups, or correlate online users to any personally identifiable information (name, location, username etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=') given the ease of composing online data into a traceable online fingerprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Similar to the social norms in New Zealand in the aftermath of the Christchurch shooting, no extremists, terrorists, and/or criminals are referred by name to minimise publicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We recognise the potential for political or cultural bias in this charged field by citing international non-partisan Non-governmental Organisations when framing ERH concepts, and avoid searching any party or ideology in our search strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Moreover, we encourage that our findings and recommendations invoke an open debate among social media platforms, governments, and the wider public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' However, we do not condone the use of ERH detection in social media as a form of autonomous law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We recommend human-in-the-loop processes when handling or classifying data via independent reviews, privacy protections, and complaint and redress mechanisms for deployed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Our recommendations thereby focus on Open Source Intelligence (OSINT) oriented studies that do not consider governmental or private-conversation surveillance (with the exception of one hybridised study that appeared in our search [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We thereby consider ERH detection as a computational method aimed at garnering community-insights, trends, and flagging for social media platforms themselves to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Whether ERH detection policies should encourage deplatform- ing, deranking, demonetisation, fact-checking, or targeted counter-speech/prevention programs require further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We encourage open interdisciplinary research in public and private- communications—particularly ethical and legal discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 000:10 Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 3 THE CASE FOR PERFORMANCE ENGINEERING WHEN EVALUATING MODELS While high F1-scores help enforce community guidelines via accurate predictions and reduce injurious censorship from false positives, runtime performance trade-offs are seldom discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' DLAs may perform within 1% (F1-score) of their MLA counterparts in NLP studies but require significantly higher computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' For instance, fine-tuning a BERT-large model for NLP tasks requires Graphics or Tensor Processing Units (GPU or TPU), restricting researchers from testing large language models [45, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' For community detection, uncompressed network models can include up to 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='4 million links [6], which significantly increases computational and memory requirements for a minimal 1-5% performance gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Specifically, using a Possibilistic Approach (PA) with dimensionality reduction reduced subgraph mining runtime by up to 67% (1500 seconds to 500 seconds on an 8-core 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2GHz system), while reducing accuracy by only 4% [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Furthermore, community-level insights on topics with millions of tweets, relations, and discussions can lead to a network explosion with a non-deterministic polynomial runtime [5, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' In graph-detection approaches, performance engineering and optimisation for mining frequent subgraphs and graph- traversal is an active area of research [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' No NLP studies consider performance engineering for DLAs despite developments in model distillation and sentence-level embeddings [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Thus, we recommend that researchers consider performance trade-offs in future work and investigate a possible standardised performance-complexity metric (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' parameter count vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' F1- score ratio) to build scalable, energy-efficient and fiscally-viable models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Moreover, fine-tuning or retraining DLAs, or regenerating frequent subgraphs for community detection, should be a frequent endeavour to adapt to the rapidly evolving topics, entities, and events throughout online discourse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Due to the computational costs of fine-tuning or training multi-billion parameter models, we recommend approaches that do not require expensive training, such as few-shot learning (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', giving several known instances of ERH and a unseen ‘test’ instance) and prompt engineering [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 4 UPTAKE ROADMAP EXPANDED This supplementary section expands on the dataset and model research gaps highlighted in Figure 16 of the main Down the Rabbit Hole SLR document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We categorise these research recommendations into eight core components for our proposed ERH Context Mining research field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='1 Model Recommendations The two predominant recommendations for future work are investigating the role of changes in hateful affiliation or speech over time to satisfy the temporal requirement for Radicalisation detection, and to train models on multiclass datasets from multiple platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We note that only one study considered temporal data on both meso and macro (changes within and between groups), and micro (individual) levels, although recommended as future work within four other studies [3, 11, 20, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Moreover, we recommend expanding on DLAs as the target for future research based on their leading performance in RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Neural language models offer a macro-level societal understanding due to their pre-trained corpus on academic sources, OpenWebText2 Reddit discussions, and Wikipedia [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Furthermore, transformer models beyond 764 million parameters are untested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Bot, troll, meme, entity, dis/misinformation and satire detection remain underdeveloped—-which could lead to censorship or undermine democratic institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Five studies recommended multi- media detection as future work [1, 5, 11, 17, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' To protect user privacy from recreating user content from neural language models, we encourage privacy-by-design software engineering through machine learning paradigms such as Differential Privacy (DP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' DP-paradigm models and datasets reduce the potential for self-identification from ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Supplementary Material for: Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech 000:11 trained models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' data leakage, such as names or usernames in open-source datasets), as DP- paradigm models use pseudo-anonymised patterns of groups and hate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='2 Dataset Recommendations To investigate the roles of radicalisation, we recommend expanding on the dataset annotation approach by de Gibert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' [15] by creating a conversation-level dataset with public non-hateful replies to a post for context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Moreover, future benchmark datasets should consider pulling data across platforms to investigate macro-level radicalisation trends between platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' We note that only two studies considered anti-Asian sentiment in COVID-related tweets, targeting a seldom explored topic and demographic [21, 39] worthy of expansion given the ongoing COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Likewise, future datasets should consider the role of indigenous discussions and potential re- searcher biases given the Anglo-dominant field of ERH research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Given the rise of COVID extrem- ism [43], far-right movements, and xenophobia in Oceania.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Hence, we recommend geotargeted datasets to consider the differences for investigating ERH topics, which would demonstrate NZ’s commitment to our Christchurch Call to Action Summit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Investigating unexplored and minority groups could also provide imperative insights for social scientists regarding the conversational dynamics, morphological mapping, and ideological isomorphism from radical minority groups towards the majority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Likewise, research on vulnerable communities (youth, gender and sexual minorities, religious, racial, and geographically distant peoples) would aid social media platforms in both identifying unique radicalising risks, as well as avenues for support and de-escalation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' In the mental health end, we recommend building on Nouh et al.’s proposed approach of extracting Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ERH Context Mining (ERH-CM) eight core components for Research, Industry, and Government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Collective Researcher Researcher, Tasks Industry, Govt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Harm Reduction Tasks Introspection & Reflection ** Community 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Detection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Privacy 副 ERH-CM 3 Increased & Data Context Sovereignty 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Self-regulation Diverse & Ethics Datasets 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Quality Assessment Industry Collective & Control Tasks (human-in-the-loop) Researcher, Industry Tasks000:12 Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' textual, psychological and behavioural features [33], both due to its performance, as well as its potential for analysing societal factors and ERH roots such as correlations between mental health issues (isolation, depression etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=') and vulnerability to radicalisation towards violent extremism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' For any counter-extremism or de-radicalisation studies, we recommend work in ethical and legal guidelines to protect privacy, avoid backlash or inadvertent algorithmic amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Investigating posts from periods of political, or social crisis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=', COVID health measures, post- terror attack discourse etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=') could also help identify cases of ERH on mainstream platforms before they are deplatformed/removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' Event-based datasets would provide unique sociological insights on the role of societal stress and emergencies on the human psyche and online group dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'} +page_content=' To reduce the cost, variability in inter-annotator agreement, and psychological impact of human annotation, we recommend unsupervised clustering-based research and propose using synthetic conversational agents to simulate extremist discourse.' metadata={'source': 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100644 index 0000000000000000000000000000000000000000..f6a70f99240f2bf6932569ea0838ae20c64a8a1b --- /dev/null +++ b/9NFST4oBgHgl3EQfaziV/content/tmp_files/2301.13797v1.pdf.txt @@ -0,0 +1,1418 @@ +STOCHASTIC APPROACHES: MODELING THE PROBABILITY OF +ENCOUNTERS BETWEEN H2-MOLECULES AND METALLIC +ATOMIC CLUSTERS IN A CUBIC BOX +Maximiliano L. Riddick, Leandro Andrini +Instituto de Investigaciones Fisicoquimicas Teóricas y Aplicadas +Departamento de Química, Fac. de Ciencias Exactas (INIFTA/ UNLP-CONICET) +Departamento de Matemática, Fac. de Ciencias Exactas, UNLP +La Plata, Argentina +mriddick@mate.unlp.edu.ar +Enrique E. Álvarez +Instituto de Cálculo, Fac. Ciencias Exactas y Naturales, Ciudad Universitaria, Pabellón II, UBA (CABA) +Departamento de Fisicomatemática, Fac. de Ingeniería, UNLP +Ciudad Autónoma de Buenos Aires and La Plata, Argentina +Félix G. Requejo +Instituto de Investigaciones Fisicoquimicas Teóricas y Aplicadas +Departamento de Química, Fac. de Ciencias Exactas (INIFTA/ UNLP-CONICET) +Departamento de Física, Fac. de Ciencias Exactas, UNLP +La Plata, Argentina +ABSTRACT +In recent years the advance of chemical synthesis has made it possible to obtain “naked”clusters of +different transition metals. It is well known that cluster experiments allow studying the fundamental +reactive behavior of catalytic materials in an environment that avoids the complications present in +extended solid-phase research. In physicochemical terms, the question that arises is the chemical +reduction of metallic clusters could be affected by the presence of H2 molecules, that is, by the +probability of encounter that these small metal atomic agglomerates can have with these reducing +species. Therefore, we consider the stochastic movement of N molecules of hydrogen in a cubic +box containing M metallic atomic clusters in a confined region of the box. We use a Wiener process +to simulate the stochastic process, with σ given by the Maxwell-Boltzmann relationships, which +enabled us to obtain an analytical expression for the probability density function. This expression is +an exact expression, obtained under an original proposal outlined in this work, i.e. obtained from +considerations of mathematical rebounds. On this basis, we obtained the probability of encounter +for three different volumes, 0.1 +3, 0.2 +3 and 0.4 +3 m +3, at three different temperatures in each case, +293, 373 and 473 K, for 10 +1 ≤ N ≤ 10 +10, comparing the results with those obtained considering +the distribution of the position as a Truncated Normal Distribution. Finally, we observe that the +probability is significantly affected by the number N of molecules and by the size of the box, not by +the temperature. +Keywords Wiener Process · Probability of encounters · Molecular Collisions · Atomic-Clusters · Mathematical +Rebounds +arXiv:2301.13797v1 [cond-mat.mtrl-sci] 10 Jan 2023 + +M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv. +1 +Introduction +In the last two decades there has been an important development in clusters chemistry, and consequently new questions +arise on the basis of these developments [1, 2, 3, 4, 5, 6, 7]. This interest is due to an atomic clusters containing +up to a few dozen atoms exhibit features that are very different from the corresponding bulk properties and that can +depend very sensitively on cluster size [8]. In particular, many of these transition metal clusters are used in the field of +catalysis [1, 9, 10, 11]. One of the basic principles of catalysis is that when the smaller the metal particles, the larger the +fraction of the metal atoms that are exposed at surfaces, where they are accessible to reactant molecules and available +for catalysis [1]. It is well known in chemistry that the encounter between two molecules can give rise to a chemical +reaction, and from the mathematical aspect there are two fundamental ways to represent these types of situations as +continuous, represented by differential equations whose variables are concentrations, or as discrete, represented by +stochastic processes whose variables are the number of molecules [12]. +Without loss of generality, it can be considered that the molecular chemisorption is due to the encounter between +a molecule and a surface (or a cluster in this case) with the energy necessary for the phenomenon of adsorption to +occur [13]. Besides, the kinetics of hydrogen chemisorption by neutral gas-phase metal clusters exhibits a complex +dependence on both cluster size and metal type [14]. For different chemical purposes, for example, in the case of copper +clusters (Cun) is very important to have control of the chemisorption of hydrogen on these clusters, i.e. the formation of +Cun-H2 species [15]. +From a reductionist point of view, the molecular chemisorption is a problem of encounter between bodies: metal clusters +and reactant molecules. In our first approximation (mathematical reduction) we will consider the problem as a problem +of encounter or collisions between bodies. We are interested in proposing this strategy because we are focused to answer +what is the probability of meeting between N hydrogen molecules (N-H2) and a fixed M metallic clusters (M-Men), +for a given time t, where the H2 move freely in a bounded volume V of R3-space. Under this assumption, we are going +to consider H2-molecules and Men-clusters as rigid spheres of radii r1 and r2, respectively. Then, it is considered that +there will be a collision whenever the center-to-center distance between an H2-molecule and a Men-cluster is equal to +r12 = r1 + r2 [16]. Also, in this context we propose the H2-molecules follow a Brownian motion, namely: (a) it has +continuous trajectories (sample paths) and (b) the increments of the paths in disjoint time intervals are independent zero +mean Gaussian random variables with variance proportional to the duration of the time interval [17]. +The pioneering work of T.D. Gillespie [16, 18, 19] have given rise to a large number of works that are proposed different +algorithms for the calculation for numerically simulating the time evolution of a well-stirred chemically reacting system, +although despite recent major improvements in the efficiency of the stochastic simulation algorithm, its drawback +remains the great amount of computer time that is often required to simulate a desired amount of system time [20]. While +our method is a simple reduction to collisions of molecules, allows to calculate the probability of encounter (scheduled +in R) for a large number of molecules (≈ 106) and clusters (≈ 1020) with advantages regarding the cost of calculation, +and the effects of first approximation can provide statistical support to the design of experiments. This calculation +is possible using a stochastic model (Wiener process) in the context of considerations from the Maxwell-Boltzmann +theory. +2 +A first theoretical approaching +As we announced in the introduction, we will assume that hydrogen molecules have a random movement, whence +let H(t) = (X(t), Y (t), Z(t)) the random variable which specify the space point where the H2 hydrogen molecule +is at time t. Trivially, H(t) depends on an initially point H(0) = (x0, y0, z0). Thus, when the initial starting point is +undefined, H(t) = H(t, x0, y0, z0). Our interest is in how probably is that the distance between H(t) and a fixed point +(a, b, c) is smaller than ϵ. The fixed point (a, b, c) are the coordinates for Men. +Let’s consider the random variable D(t) as the variable that measures the distance between H(t) and the fixed point +(a, b, c). Following the classical Pythagorean relationship, D(t) = +� +(X(t) − a)2 + (Y (t) − b)2 + (Z(t) − c)2, and +in general D(t) = D(t, x0, y0, z0, a, b, c). +Now, given a time window [0, τ], let +Rτ := +� +� +� +1 +if min +t∈[0,τ) D(t) ≤ ϵ, +0 +otherwise. +(1) +So, for a fixed t0 > 0, we define G(t0) = P(D(t0) ≤ ϵ) = +� ϵ +0 D(s)ds. Then, P(Rτ = 1) = +� τ +0 G(t)dt. +2 + +M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv. +Thus, given τ > 0, Rτ depends only on the initial values (x0, y0, z0, a, b, c). Now, if we have M-Men, the probability +that the H2 molecule does not meet with any of the clusters is P(Rτ1 = 0, Rτ2 = 0, ..., RτM = 0) = pA, where +Rτi, i ∈ {1, ..., M}, follows the definition given in the eq. 1. +If N-H2 molecules are in the environment, let Aj the event “the j-th hydrogen molecule meet with a metallic cluster”. +Under random starting points, we are interested in P(AC +1 ∩ AC +2 ∩ ... ∩ AC +N) = pN +A according to the independence +among the hydrogen molecules. +2.1 +Adaptation to our context +Next, we proceed to realize the analysis according to the Brownian Motion Theory [17], in which the movement of +the particle is independent among different axis, and we are going to assume that it follows a Wiener process [21, 22]. +Then, +X(t) = x0 + WX(t) +Y (t) = y0 + WY (t) +Z(t) = z0 + WZ(t) +And we will say that WX(t), WY (t) and WZ(t) are following a Wiener processes with σ = +� +kbT +m , where kb is the +Boltzmann’s constant, T is the absolute temperature in Kelvin (K) and m is the H2’s mass in kg. That is, we are +imposing a physical behavior that obeys Maxwell-Boltzmann’s considerations. According this: +X(t) ∼ N(x0, σ2t) +Y (t) ∼ N(y0, σ2t) +Z(t) ∼ N(z0, σ2t) +With density function fX(x, t|x0), fY (y, t|y0) and fZ(z, t|z0), respectively. Under these assumptions: +fX(x, t|x0) = +1 +√ +2πσ2t +exp +� +−1 +2 +�x − x0 +σ +√ +t +�2� +fY (y, t|y0) = +1 +√ +2πσ2t +exp +� +−1 +2 +�y − y0 +σ +√ +t +�2� +fZ(z, t|z0) = +1 +√ +2πσ2t +exp +� +−1 +2 +�z − z0 +σ +√ +t +�2� +2.1.1 +Unbounded conditions +Under unbounded conditions, as it is well known, the density of the particle position in the space for a fixed t follows +the expression: +fXY Z(x, y, z, t|x0, y0, z0) = fX(x, t|x0) · fY (y, t|y0) · fZ(z, t|z0) = += +1 +( +� +2πσ2t)3 exp +� +−1 +2 +�(x − x0)2 + (y − y0)2 + (z − z0)2 +σ2t +�� +This function is continuous in the variables x, y, z, t, then is also integrable in a measurable context. Because of this +fact, Fubini’s theorem is aplicable. Now, calling ν = +� +(x−x0)2+(y−y0)2+(z−z0)2 +2σ2 +, and integrating over the variable t by +substitution, results: +3 + +M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv. +fXY Z(x, y, z, τ|x0, y0, z0) = +1 +−ν( +√ +2πσ2)3 +� τ +0 +−ν +( +√ +t)3 exp +� +− +� ν +√ +t +�2� +dt += +1 +ν( +√ +2πσ2)3 +� ∞ +√ ν +τ +e−u2du +Remembering that the erfc function [23] is defined by: +erfc(z) = +2 +√π +� ∞ +z +e−t2dt +we conclude: +fXY Z(x, y, z, τ|x0, y0, z0) = +1 +2πν( +√ +2σ2)3 erfc +��ν +τ +� +From the physical-experimental perspective that the problem is lays out, the unbounded system lacks interest, so we +will proceed to study the case of the bounded system. +2.1.2 +Bounded conditions +We assume that the experiment takes place into a cubic recipe centered at the origin. This implies that X(t), Y (t) and +Z(t) ∈ [−L; L], for a fixed volume V = L3 in R3-space. +In a similar issue the traditional way of approaching is by “truncation" [24, 25]. A drawback of this approach is the +fact that the truncation does not represent precisely the reflection on the boundaries. An illustrative and motivational +argument is given by the following example: suppose a random walk of N = 4 steps, with starting point at the origin. +Then, the walker moves 1 step at right or left (with equal probability) at each step. Then, after four steps, the resultant +probabilities of the walker position are: +0; with probability 3/8, +−2 or 2; with probability 2/8, +−4 or 4; with probability 1/16. +The probability values (under truncation) in the closed interval [−2, 2] for the values (−2, −1, 0, 1, 2) are, respectively: +(2/7, 0, 3/7, 0, 2/7) +With fixed boundaries, considering reflections at [−2, 2], we can construct the following Markov transition matrix P: +P = +0 +1 +0 +0 +0 +1/2 +0 +1/2 +0 +0 +0 +1/2 +0 +1/2 +0 +0 +0 +1/2 +0 +1/2 +0 +0 +0 +1 +0 +At the fourth step, after some algebra, we obtain the respectively mass point probability for the position of the walker. +This is provided by the stochastic vector +(1/4, 0, 1/2, 0, 1/4) +(given by the third file of P 4, i.e.: with starting point at the origin). At this point, is clearly the difference between +truncation and “rebounds" (considering reflection on the boundary). +We must modify the density of the position H(t) according to the particle rebounds (see Fig. 1). It is important to note +that the rebounds indicated in the figure in gray colour do not correspond to the physical rebounds of the particles in the +cubic box, but to the contributions of the displaced distribution considering an infinite behavior. +4 + +M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv. +Figure 1: In red colour an arbitrary normal distribution, N(0, σ). We observe in gray colour the folding of the normal +distribution at the edge of the box. We could see that A + B = 2L. +Inside the box, the derived density fB according to the variable X(t) ∼ N(x0, σ2t) with density function fX of the +particle position (for each dimension, see Fig. 1) follows the expression: +fB(x) = [fX(x) + fB+(x) + fB−(x)] × I[−L;L](x) +where: +fB+(x) = f(x + 2A) + f(x + 2A + 2B) + f(x + 2A + 2B + 2A) + ... = += f(x + 2(L − x)) + f(x + 2(L − x) + 2(x − (−L)) + ... += f(−x + 2L) + f(x + 4L) + f(−x + 6L) + ... += +∞ +� +k=1 +f((−1)kx + 2kL) += +∞ +� +k=1 +1 +√ +2πσ2t +exp +� +−1 +2 +�((−1)kx + 2kL) − x0 +σ +√ +t +�2� +and +fB−(x) = f(x − 2B) + f(x − 2B − 2A) + f(x − 2B − 2A − 2B) + ... = += f(x − 2(x − (−L))) + f(x − 2(x − (−L)) − 2(L − x)) + ... += f(−x − 2L) + f(x − 4L) + f(−x − 6L) + ... += +∞ +� +k=1 +f((−1)kx − 2kL) += +∞ +� +k=1 +1 +√ +2πσ2t +exp +� +−1 +2 +�((−1)kx − 2kL) − x0 +σ +√ +t +�2� +5 + +B +B' +B +0 +XM.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv. +The proof that fB is a density function is straightforward its definition. Trivially, fB > 0, and by construction: +� ∞ +−∞ +fB(t)dt = +� L +−L +fB(t)dt = +� ∞ +−∞ +fX(t)dt = 1 +For practical purposes, we now try to find an upper bound to this expression. Looking at in the model proposed, the +next constraint is straightforward |(−1)kx − x0| ≤ 2L. +Following these constraints: +fB+(x) = +∞ +� +k=1 +1 +√ +2πσ2t +exp +� +−1 +2 +�((−1)kx + 2kL) − x0 +σ +√ +t +�2� +≤ +∞ +� +k=1 +1 +√ +2πσ2t +exp +� +−1 +2 +�−2L + 2kL +σ +√ +t +�2� += +∞ +� +k=1 +1 +√ +2πσ2t +exp +� +−1 +2 +�2(k − 1)L +σ +√ +t +�2� += +∞ +� +k=0 +1 +√ +2πσ2t +exp +� +−1 +2 +� 2kL +σ +√ +t +�2� += +1 +√ +2πσ2t +∞ +� +k=0 +exp +� +−1 +2 +�4L2 +σ2t +��k2 +It is known that +∞ +� +k=0 +rk2 = 1 +2 + 1 +2ΘE[3, 0, r] where ΘE is the Jacobi theta elliptic function [23]. So: +fB+(x) ≤ +1 +√ +2πσ2t +∞ +� +k=0 +exp +� +−1 +2 +�4L2 +σ2t +��k2 += +1 +√ +2πσ2t +�1 +2 + 1 +2ΘE +� +3, 0, exp +� +−1 +2 +�4L2 +σ2t +���� +and +fB−(x) = +∞ +� +k=1 +1 +√ +2πσ2t +exp +� +−1 +2 +�((−1)kx − 2kL) − x0 +σ +√ +t +�2� +≤ +∞ +� +k=1 +1 +√ +2πσ2t +exp +� +−1 +2 +�−2L − 2kL +σ +√ +t +�2� += +∞ +� +k=1 +1 +√ +2πσ2t +exp +� +−1 +2 +�−2(k + 1)L +σ +√ +t +�2� += +∞ +� +k=2 +1 +√ +2πσ2t +exp +� +−1 +2 +�−2kL +σ +√ +t +�2� += +1 +√ +2πσ2t +� ∞ +� +k=0 +exp +� +−1 +2 +�4L2 +σ2t +��k2 +− 1 − exp +� +−1 +2 +�4L2 +σ2t +��� += +1 +√ +2πσ2t +�1 +2 + 1 +2ΘE +� +3, 0, exp +� +−1 +2 +�4L2 +σ2t +��� +− 1 − exp +� +−1 +2 +�4L2 +σ2t +��� +6 + +M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv. +Then, +fB+(x) + fB−(x) ≤ +1 +√ +2πσ2t +� +ΘE +� +3, 0, exp +� +−1 +2 +�4L2 +σ2t +��� +− exp +� +−1 +2 +�4L2 +σ2t +��� += CB +For each x ∈ [−L, L], fB(x) ≤ fX(x) + CB. Besides CB does not depends on x. Consequently, we have a maximum +for the density fB which is equal to f(x0) + CB. +Calling PB = (f(x0) + CB).2ϵ, we can conclude that: +P(X(t) ∈ (a0 − ϵ, a0 + ϵ)) ≤ PB, for any a0. +Analogous, CB is the same for the variables Y (t) and Z(t), and we know that f(x0) = f(y0) = f(z0). Then, the +same result is available for the variables Y (t) and Z(t). According the bounded CB, it is straightforward the uniform +convergence of the series fB+ and fB− (by the M Weierstrass criteria). An important fact to remark is that PB is not +even a probability, but in the case in we are interested, we know that is a real number bigger than the probability desired, +and then, under certain conditions, we can work with it. +For practical purposes, the error through the CB implementation can be minimized, since the first S terms are available, +and the tail can be compared with +S−1 +� +k=0 +rk2 ≤ +∞ +� +k=0 +rk2 = +S−1 +� +k=0 +rk2 + +∞ +� +k=S +rk2 +And, +∞ +� +k=S +rk2 = +∞ +� +k=0 +r(k+S)2 = +∞ +� +k=0 +rk2+2kS+S2 = rS2 +∞ +� +k=0 +rk2r2kS ≤ rS2 +∞ +� +k=0 +rk2 +Then, +∞ +� +k=S +rk2 ≤ rS2 �1 +2 + 1 +2ΘE[3, 0, r] +� +Controlling the value of S controls the value of the error made by truncating the sum. As we said, CB does not depend +on x, thus, the desired probability can be estimated with any degree of accuracy, according the computational cost +necessary to this development. +Taking into consideration the Brownian Motion Theory, in the time lapse of 1 second, the particle position under +unbounded conditions follows a N(x0, σ2) distribution. To discretize the problem, if we partitioned the time axis of τ +seconds in τ intervals of 1 second each one, then: +P(H(t) ∈ Bϵ(a, b, c)) ≤ P(H(t) ∈ Qϵ(a, b, c)) +where Qϵ(a, b, c) denotes the cube centered in (a, b, c) with side size 2 × ϵ. And, considering the independence +between X(t), Y (t) and Z(t), with X(t) ∈ (a − ϵ, a + ϵ), Y (t) ∈ (b − ϵ, b + ϵ) and Z(t) ∈ (c − ϵ, c + ϵ), +P(H(t) ∈ Qϵ(a, b, c)) = P(H ∈ Qϵ) is +P(H ∈ Qϵ) = P(X(t)) × P(Y (t)) × P(Z(t)) ≤ PB × PB × PB = P 3 +B +For each second τj for τj ∈ {1 : τ}, P(H(t) ∈ Qϵ(a, b, c)) ≤ P 3 +B. Then, under the Wiener process formulation, +H(τj) ⊥ H(τk|τj) if j ̸= k, j ≤ k. +7 + +M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv. +P(H(τj) ∈ Qϵ(a, b, c)) ≤ P 3 +B, ∀τj ∈ {1 : τ}. Calling F :=“# of τj ∈ {1 : T} in which H(τj) ∈ Qϵ(a, b, c)", we are +interesting in the event F = 0. +According its nature, F is a Binomial random variable B(τ, P 3 +B). Consequently, the non-collision probability is +pNC = P(F = 0) ≤ (1 − P 3 +B)τ. At this point, we only can conclude that the probability of the encounter between +a hydrogen molecule and a Men cluster in a time τ is less than p. We proceed to analyze what happens when the +number of hydrogen molecules and metallic clusters increase. We emphasize that the H2 molecules have a random +movement while the clusters are confined in a fixed region of space. Since p is the probability that a random hydrogen +molecule meets in the cube Qϵ in which a Men cluster is, the most unfavorable case with M clusters is when there is no +intersection among the cubes that contain it. In this case: +pA = P(Rτ1 = 0, ..., RτM = 0) += 1 − +M +� +i=1 +P(Rτi = 1) +≥ 1 − +� M +� +i=1 +P(Rτi = 1) +� += 1 − M × p +In view of this analysis, we can conclude that the non-collision probability is higher than pNC. +In regular conditions, when this approach is used, the values of pNC and N outcomes into a several numerical instability. +In this case, the small value of pNC and the large value of N place us in conditions to use the Poisson approach to +the Binomial distribution (with parameter λ = N × p). Then, P(X = 0) ≈ exp(−λ). Even in the cases when the +probability is still unavailable, the expected number of collisions is presented according a time window, and then we can +estimate the probability of collisions in a time window T using the relationship between the Poisson and Exponential +distributions[26]. +Next, we present the results of the analysis whit different box dimensions (in meters) and number of hydrogen molecules +(N), according to M = 1.9 × 10 +20 Cu20-clusters [27], where the Cu20-clusters have been considered as spheres. +3 +Results and analysis +3.1 +Obtaining non-collision probability values +The situation we consider is approximately a “realistic”situation, with M = 1.9 × 10 +20 Cu20-clusters in a cubic box +according to the standard dimensions of reaction chambers (0.1 +3, 0.2 +3 and 0.4 +3 m +3), and a variable N-H2-molecules +“contamination”(10 +1 ≤ N ≤ 10 +10). It worked with three temperatures, T, 293, 373 and 473 K. The choice of T is +arbitrary, conditioned by the possible reaction temperatures [28]. +In Fig. 2 we observe the results obtained for the simulations, considering the maximum sum. That is, take S = 10 +6, +perform the sum, and add the maximum level for the error. Clearly, a greater probability of non-collision, pNC, is +observed depending on the increase in volume. +For a detailed study, we proceed as follows: we model the data obtained through a non-linear graphic fitting considering +a Boltzmann decrease function, g(x) = A2 + +A1−A2 +1+exp +� x−x0 +dx +� (see Fig. 3). In the Appendix A.2 we show the statistical +results for each parameter in each data fitting. +Under these considerations, we can calculate the critical value (criticality)[29, 30] of hydrogen molecules, that is “what +is the value of N for which the non-collision probability is greater than 1 +2”, i.e. the value of the exponent for which +1 +2 < pNC. +It should be clarified that, in the strict physical sense, there is no abrupt phase transition to consider “criticality”. As +we assumed in the introduction, we consider that there is a chemical reaction if there is an encounter between two +molecules, and under this assumption we are considering as critical the level of presence of hydrogen for a chemical +reaction to occur. In any case, it can be demonstrated that there is an “abrupt”transition behavior, for a well defined +interval in the number of molecules. In Fig. 3 we can observe this behavior. +8 + +M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv. +Figure 2: Results for the non-collision probability, pNC, vs. ln(N) for L = 0.05m (V1), L = 0.1m (V2) and L = 0.2m +(V3), at T = 293 K (blue square), 373 K (black star) and 493 K (red triangle). +Figure 3: Data (blue square) modeling using a non-linear Boltzmann decrease function (green line). +9 + +V1 +V2 +1.0 - +4 +GD +△ +0.8 +0 +★ +口 +293 K +0.6 +V +373 K +0 +473 K +文 +0.4 - +★ +0.2 - +0.0- +支立文支安 +123456789101234567891012345678910 +Ln(N)293 K +1.0 - +Boltzmann Fit 293 K +0.8 +0.6 +0.4 +0.2 +0.0 +0 +3 +4 +5 +6 +7 +8 +10 +Ln(N)M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv. +Table 1: Critical values obtained from the decrease model for each box and each temperature, for mathematical +robounds. +L [m] +293 K +373 K +473 K +0.05 +3.25 +3.16 +3.09 +0.10 +4.97 +4.86 +4.72 +0.20 +5.96 +5.96 +5.96 +Figure 4: Results for the non-collision probability, pNC, vs. ln(N) for L = 0.05m (V1), at T = 293 K, 373 K and 493 +K. Comparison between models:“xxx Trunc”correspond to the truncated normal model and “xxx K”to the mathematical +rebound model. +In Table 1 we can see the critical values obtained from the decrease model for each box and each temperature. For the +smallest volumes, V1 and V2, it is observed that the critical value of N depends more strongly on the temperature than +in the case of the larger volume (V3). Although it is remarkable the fact of dependence with the size of the box, it can +be seen directly from Fig. 2. In this way, and under these simplified assumptions, we can obtain control of contaminant +molecules in relation to the volume and temperature parameters. Linear behavior is evident from the values obtained +(Table 1, N vs. temperature). Moreover, as the volume increases the slope increases from negative values to null value. +4 +Conclusion +By way of conclusion, it can be indicated that considering a Wiener stochastic process, for thermodynamic-statistical +movements of a gas confined in a box, and considering mathematical rebounds bounded by the physical-geometric +contour of the problem, the analytical expression could be obtained for the probability density function of encounters +between two differentiated species of molecules (one of the species fixed in the box -solid or liquid- and the other +species is a gas whose molecules move stochastically). In addition, the function obtained can be calculated numerically +or can be bounded. The bounded process allows to reduce the computational cost, and to limit the error from cutting the +Table 2: Critical values obtained from the decrease model for each box and each temperature, for truncated normal +model. +L [m] +293 K +373 K +473 K +0.05 +3.27 +3.18 +3.12 +0.10 +5.01 +4.91 +4.76 +0.20 +6.00 +5.98 +5.96 +10 + +293 Trunc +☆373 Trunc +473 Trunc +口 +293 K +373 K +473 K +1.0 0 +0 +★ +Non-collision probability +0.8 +8 +0.6 - +0.4 +0.2 + +0.0+ +口口OO口 +1 2 3 4 5 6 7 8 9101 2 3 4 5 6 7 8 9101 2 3 4 5 6 7 8 910 +Ln(N)M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv. +Figure 5: Results for the non-collision probability, pNC, vs. ln(N) for L = 0.1m (V1), at T = 293 K, 373 K and 493 K. +Comparison between models:“xxx Trunc”correspond to the truncated normal model and “xxx K”to the mathematical +rebound model. +Figure 6: Results for the non-collision probability, pNC, vs. ln(N) for L = 0.2m (V1), at T = 293 K, 373 K and 493 K. +Comparison between models:“xxx Trunc”correspond to the truncated normal model and “xxx K”to the mathematical +rebound model. +11 + +293 Trunc +☆373 Trunc +→ 473 Trunc +口 +293 +★373 +473 +★★★ +口 +Non-collision probability +0.8. +0.6 - +0.4 +0.2 + +0.0 - +OOOOG +1 2 3 4 5 6 7 8 9101 2 3 4 5 6 7 8 9101 2 3 4 5 6 7 8 910 +Ln(N)293 Trunc +☆373 Trunc +→ 473 Trunc +口 +293 K +373 K +473K +1.0- +Non-collision probability +0.8 +0.6 +0.4 +0.2 +0.0- +123456789101234567891012345678910 +Ln(N)M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv. +sum in a finite number. In particular, there is an error control that can be made, and it is possible to refine the process +according to the precision required. +From the physical-chemical point of view, it is observed that both the number of gas molecules and the dimensions of +the box affect the probability of encounter. For this model, temperature is a parameter that has a lower incidence on the +values of the probability of encounter. At this point some considerations have to be made. The first is that in a strict +sense a chemical reaction is more than the encounter of two chemical entities. The second is the exceptional chemical +nature of metal clusters, which make them highly reactive. Despite the simplicity of the model we are proposing, this +model can account in an experiment design about the collision probability between two chemical entities (and this +collision can lead to a chemical reaction). +From the point of view of computation, it is a system that requires less computational cost (time + memory) than the +algorithmic systems developed for this type of problems, so it contributes as a test method in the design of experiments. +The comparison with an established method (truncated normal model) was optimal. In the method of mathematical +rebounds the number of molecules needed for a reaction is less than the number obtained by the truncated normal +model. This is an advantage when strict contamination control is needed. +On the other hand, in terms of obtaining the density function, mathematical results can be generalized for volumes of +rectangular prisms of uneven sides. In addition, it remains to calculate the first and second order moments of the density +function obtained, work that exceeded the purposes of present communication. +Acknowledgments +This was was supported in part by PICT-2019-0784, PICT-2017-3944, PICT-2017-1220, PICT-2017-3150 (PICT, +Agencia Nacional de Promoción de la Investigación, el Desarrollo Tecnológico y la Innovación) and PPID-I231 (PPID, +Universiad Nacional de La Plata). +References +[1] Bruce C. Gates. Supported metal clusters: synthesis, structure, and catalysis. Chemical reviews, 95(3):511–522, +1995. +[2] M Arturo López-Quintela. Synthesis of nanomaterials in microemulsions: formation mechanisms and growth +control. Current Opinion in Colloid & Interface Science, 8(2):137–144, 2003. +[3] Puru Jena and A. Welford Castleman Jr. Clusters: A bridge across the disciplines of physics and chemistry. +Proceedings of the National Academy of Sciences, 103(28):10560–10569, 2006. +[4] Shahana Huseyinova, Joseé Blanco, Feélix G. 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Exceptional oxidation activity with size-controlled supported gold clusters of low atomicity. +Nature Chemistry, 5(9):775–781, 2013. +[29] Per Bak and Maya Paczuski. Complexity, contingency, and criticality. Proceedings of the National Academy of +Sciences, 92(15):6689–6696, 1995. +[30] Terrie M. Williams. Criticality in stochastic networks. Journal of the Operational Research Society, 43(4):353–357, +1992. +Appendix +A.1 +Errors in the Boltzmann model for the probability calculated according to mathematical rebounds. +Program used: Origin 9.1 +In all cases, number of points is 10, and degrees of freedon is 6. +13 + +M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv. +L=0.05 m, T = 293 K +Parameter +Value +Standard Error +A1 +0.991 +0.009 +A2 +-0.0060 +0.0008 +x0 +4.98 +0.02 +dx +0.30 +0.03 +Reduced Chi-Sqr 2.66387 × 10 +−4 +Residual Sum of Squares: 0.0016 +Adj. R-Square: 0.99888 +L=0.05 m, T = 373 K +Parameter +Value +Standard Error +A1 +0.981 +0.006 +A2 +-0.005 +0.003 +x0 +3.16 +0.01 +dx +0.22 +0.02 +Reduced Chi-Sqr 7.93743 × 10 +−5 +Residual Sum of Squares: 4.76246 × 10 +−4 +Adj. R-Square: 0.99957 +L=0.05 m, T = 473 K +Parameter +Value +Standard Error +A1 +0.976 +0.008 +A2 +-0.0011 +0.0009 +x0 +3.10 +0.02 +dx +0.21 +0.03 +Reduced Chi-Sqr 1.30175 × 10 +−4 +Residual Sum of Squares: 7.8105 × 10 +−4 +Adj. R-Square: 0.99927 +L=0.1 m, T = 293 K +Parameter +Value +Standard Error +A1 +0.991 +0.009 +A2 +-0.0060 +0.0011 +x0 +4.98 +0.02 +dx +0.30 +0.03 +Reduced Chi-Sqr 2.66387 × 10 +−4 +Residual Sum of Squares: 0.0016 +Adj. R-Square: 0.99888 +L=0.1 m, T = 373 K +Parameter +Value +Standard Error +A1 +0.994 +0.008 +A2 +-0.006 +0.002 +x0 +4.88 +0.02 +dx +0.33 +0.02 +Reduced Chi-Sqr 1.88888 × 10 +−4 +Residual Sum of Squares: 0.00113 +Adj. R-Square: 0.9992 +14 + +M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv. +L=0.1 m, T = 473 K +Parameter +Value +Standard Error +A1 +0.996 +0.005 +A2 +-0.0043 +0.0019 +x0 +4.73 +0.01 +dx +0.33 +0.01 +Reduced Chi-Sqr 7.76599 × 10 +−5 +Residual Sum of Squares: 4.65959 × 10 +−4 +Adj. R-Square: 0.99967 +L=0.2 m, T = 293 K +Parameter +Value +Standard Error +A1 +0.993 +0.003 +A2 +-0.0082 +0.0025 +x0 +5.97 +0.02 +dx +0.31 +0.03 +Reduced Chi-Sqr 2.64593 × 10 +−4 +Residual Sum of Squares: 0.00159 +Adj. R-Square: 0.9989 +L=0.2 m, T = 373 K +Parameter +Value +Standard Error +A1 +0.993 +0.008 +A2 +-0.0082 +0.0025 +x0 +5.97 +0.02 +dx +0.31 +0.03 +Reduced Chi-Sqr 2.64587 × 10 +−4 +Residual Sum of Squares: 0.00159 +Adj. R-Square: 0.9989 +L=0.2 m, T = 473 K +Parameter +Value +Standard Error +A1 +0.993 +0.008 +A2 +-0.0082 +0.0025 +x0 +5.97 +0.02 +dx +0.31 +0.03 +Reduced Chi-Sqr 2.64587 × 10 +−4 +Residual Sum of Squares: 0.00159 +Adj. R-Square: 0.9989 +A.2 +Errors in the Boltzmann model for the probability calculated according to the truncated normal model. +L=0.05 m, T = 293 K +Parameter +Value +Standard Error +A1 +0.982 +0.006 +A2 +-0.0003 +0.0001 +x0 +3.29 +0.01 +dx +0.24 +0.01 +Reduced Chi-Sqr 6.6611 × 10 +−5 +Residual Sum of Squares: 0.000399 +15 + +M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv. +Adj. R-Square: 0.99965 +L=0.05 m, T = 373 K +Parameter +Value +Standard Error +A1 +0.982 +0.006 +A2 +-0.004 +0.003 +x0 +3.19 +0.02 +dx +0.22 +0.02 +Reduced Chi-Sqr 6.69947 × 10 +−5 +Residual Sum of Squares: 4.0196 × 10 +−4 +Adj. R-Square: 0.99960 +L=0.05 m, T = 473 K +Parameter +Value +Standard Error +A1 +0.982 +0.005 +A2 +-0.0004 +0.0003 +x0 +3.19 +0.01 +dx +0.22 +0.02 +Reduced Chi-Sqr 6.69508 × 10 +−4 +Residual Sum of Squares: 4.01705 × 10 +−4 +Adj. R-Square: 0.99964 +L=0.1 m, T = 293 K +Parameter +Value +Standard Error +A1 +0.991 +0.003 +A2 +-0.0022 +0.0009 +x0 +5.02 +0.07 +dx +0.21 +0.03 +Reduced Chi-Sqr 5.5392 × 10 +−5 +Residual Sum of Squares: 0.00033 +Adj. R-Square: 0.99977 +L=0.1 m, T = 373 K +Parameter +Value +Standard Error +A1 +0.992 +0.006 +A2 +-0.0047 +0.0025 +x0 +4.92 +0.01 +dx +0.29 +0.02 +Reduced Chi-Sqr 1.20178 × 10 +−4 +Residual Sum of Squares: 0.000721 +Adj. R-Square: 0.9995 +L=0.1 m, T = 473 K +Parameter +Value +Standard Error +A1 +0.995 +0.005 +A2 +-0.0045 +0.0025 +x0 +4.77 +0.05 +dx +0.33 +0.01 +Reduced Chi-Sqr 8.70051 × 10 +−5 +Residual Sum of Squares: 5.2203 × 10 +−4 +16 + +M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv. +Adj. R-Square: 0.99963 +L=0.2 m, T = 293 K +Parameter +Value +Standard Error +A1 +0.992 +0.003 +A2 +-0.0078 +0.0065 +x0 +6.01 +0.02 +dx +0.29 +0.03 +Reduced Chi-Sqr 2.75184 × 10 +−4 +Residual Sum of Squares: 0.00165 +Adj. R-Square: 0.99885 +L=0.2 m, T = 373 K +Parameter +Value +Standard Error +A1 +0.990 +0.007 +A2 +-0.0068 +0.0075 +x0 +5.99 +0.02 +dx +0.28 +0.03 +Reduced Chi-Sqr 2.07471 × 10 +−4 +Residual Sum of Squares: 0.00124 +Adj. R-Square: 0.99913 +L=0.2 m, T = 473 K +Parameter +Value +Standard Error +A1 +0.993 +0.008 +A2 +-0.0082 +0.0025 +x0 +5.97 +0.02 +dx +0.31 +0.03 +Reduced Chi-Sqr 2.64587 × 10 +−4 +Residual Sum of Squares: 0.00159 +Adj. R-Square: 0.9989 +17 + diff --git a/9tAzT4oBgHgl3EQfg_wg/content/tmp_files/2301.01476v1.pdf.txt b/9tAzT4oBgHgl3EQfg_wg/content/tmp_files/2301.01476v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c1f934e400af53a1c4757094c76c8bd889c34594 --- /dev/null +++ b/9tAzT4oBgHgl3EQfg_wg/content/tmp_files/2301.01476v1.pdf.txt @@ -0,0 +1,1696 @@ +Lessons Learned Applying Deep Learning Approaches to +Forecasting Complex Seasonal Behavior + +Andrew T. Karl1, James Wisnowski1, Lambros Petropoulos2 + +1Adsurgo LLC, Pensacola, FL +2USAA, San Antonio, TX + + + +Abstract +Deep learning methods have gained popularity in recent years through the media and the +relative ease of implementation through open source packages such as Keras. We +investigate the applicability of popular recurrent neural networks in forecasting call +center volumes at a large financial services company. These series are highly complex +with seasonal patterns - between hours of the day, day of the week, and time of the year - +in addition to autocorrelation between individual observations. Though we investigate the +financial services industry, the recommendations for modeling cyclical nonlinear +behavior generalize across all sectors. We explore the optimization of parameter settings +and convergence criteria for Elman (simple), Long Short-Term Memory (LTSM), and +Gated Recurrent Unit (GRU) RNNs from a practical point of view. A designed +experiment using actual call center data across many different “skills” (income call +streams) compares performance measured by validation error rates of the best observed +RNN configurations against other modern and classical forecasting techniques. We +summarize the utility of and considerations required for using deep learning methods in +forecasting. +Key Words: ARIMA, Time Series + + +1. Introduction + +Member contact call centers receive fluctuating call volumes depending on the day of the +week, the time of day, holidays, business conditions, and other factors. It is important for +call center managers to have accurate predictions of future call volumes in order to manage +staffing levels efficiently. The call center arrival process has been well documented and +explored in the literature (Gans, Koole, & Mandelbaum, 2003). In the application presented +here, there are several different “skills” (or “splits”) to which an incoming call may be +routed – depending on the capabilities of the call center agents – and an arrival volume +forecast is required for each skill in the short term for day-ahead or week-ahead predictions. + +The weekly seasonality found in call arrivals can be modeled effectively through a variety +of methods to include Winter’s Seasonal Smoothing (Winters, 1960) or Autoregressive +Integrated Moving Average (Box & Jenkins, 1970). Some accessible references for many +of these concepts aimed at the practitioner are well documented in the literature (e.g. +Bisgaard & Kulachi, (2007, 2008)) while recommended texts are Bisgaard & Kulachi +(2011) and Montgomery, Jennings & Kulachi (2015). + +Aiming to improve on these classic methods, “doubly stochastic” linear mixed models +(Aldor-Noiman, Feigin, & Mandelbaum, 2009) have effectively modeled additional + +complexities as outlined in a recent review paper from Ibrahim, Ye, L'Ecuyer, & Shen +(2016). Similarly, Recurrent Neural Networks (RNNs) have been recommended as deep +learning approaches to forecast call volume for a wireless network (Bianchi et al., 2017) in +addition to numerous other applications including ride volumes with Uber (Zhu & Laptev, +2017). While the doubly stochastic and RNN approaches to predicting call volumes offer +greater flexibility in modeling complex arrival behavior by incorporating exogenous +variables, this flexibility comes at the cost of greater computational and programming +complexity (as well as greater prediction variance). This paper explores practical aspects +of managing that complexity for these models, applies the models to actual call volumes +recorded by a large financial services company, and compares the prediction capability to +that of the more traditional Winters smoothing and ARIMA models. + +First, we modify computational aspects of the doubly stochastic approach proposed by +Aldor-Noiman, Feigin, & Mandelbaum (2009) to improve call center forecasting +performance. Doubly stochastic implies a two-level randomization where not only are call +arrivals random variables, but also the call arrival mean parameter. Forecasts are produced +by taking advantage of the unique correlation structure for each split while accounting for +trend, seasonality, cyclical behavior, and serial dependence. The doubly stochastic model +is more complex than ordinary regression as it accounts for both inter- and intra-day +correlation. We suggest modifications to the originally proposed approach that lead to more +stable convergence and more flexible behavior when many splits need to be fit. + +Secondly, we consider how RNNs may be used to model incoming call volume. Whereas +“traditional” densely connected, feedforward neural networks process each data point +independently, RNNs process sequences according to temporal ordering and retain +information from previous points in the sequence. As it processes points within sequences, +the RNN maintains states that contain information about what it has seen previously in the +sequence (Chollet & Allaire, 2018). This intra-sequence memory is useful in time series +applications to autocorrelated data. In the context of call center volumes, these sequences +could be constructed to correspond to individual days of observations over a fixed number +of (e.g. 30 minute) periods. Bianchi et al. (2017) consider three different RNN architectures +to model incoming call volume over a mobile phone network: Elman Recurrent Neural +Networks (ERNN) (Elman, 1990), Gated Recurrent Units (GRU) (Cho, et al., 2014), and +Long Short Term Memory (LSTM) (Hochrieter & Schmidhuber, 1997), listed in order of +increasing complexity. +These three RNNs, along with the dense neural network, are now available via the R Keras +package (Allaire & Chollet, 2018). Once code has been written for one of the RNNs, the +user can switch between the other two by toggling a single option (and, after data +reformatting, switch to a dense network). This offers the potential – via a designed +experiment – to produce a pragmatic answer to the question of which type of (R)NN +provides the best fit to the process at hand. Whereas Bianchi, Maiorino, Kampffmeyer, +Rizzi, & Jenssen (2017) created their experimental design by randomly generating points +within the design space and then selected the design that lead to the minimum error rate, +we create a full factorial design (treating all factors as categorical to allow arbitrary shape +in the otherise (discrete) continuous factor of number of nodes) and then explore the +behavior of the error rates across the design space with a profiler for the resulitng linear +model for the error rate as a function of the NN settings. Unlike ARIMA or regression +(including doubly stochastic) modeling approaches for time series, there is a stochastic +behavior in the predictions made by neural networks due to the use of randomly initialized +weights. Unless the seed for the software’s random number generator is fixed, repeated +fitting of the same neural network will lead to different predictions. The amount of + +variation in the resulting predictions depends on the complexity of the network and on the +steps that have been taken to avoid overfitting, including early stopping of the optimizer. +When selecting a model configuration, we will not only want to minimize the expected +error rate, but also minimize the variability in the error rates. To this end, we seek to +minimize the upper 95% prediction interval on the testing error rate. The NN study +proceeds in two phases where a screening experiment first identifies the most useful +(R)NN, followed by a more comprehensive performance study against common +forecasting approaches across many more skills. + +Section 2 describes the doubly stochastic model for call volumes and how modifications to +the originally proposed computational approach can lead to improved convergence. +Section 3 details how a full factorial design is used to characterize the performance of RNN +options as a function of five factors (and their interactions) on the resulting short-term +forecast error rate. Additionally, Section 3 describes the selection of the model +configuration that leads to the minimum upper bound on the 95% prediction interval for +the testing error rate. Due to the number of different model configurations that must be run +along with the computational complexity of RNNs, the first phase discussed in Section 3 +considers only a limited number of skills and validation days. In Section 4, the best +performing RNNs are run over a larger validation set and over all call center skills to +compare the performance to the doubly stochastic mixed model approach, and to ARIMA +as well as Winters smoothing. + +2. Stable Settings for Fitting the Mixed Model + +There are two distinct influences on call volumes that induce a correlation between the +observed call counts, violating the independence assumption made by ordinary least +squares regression models that might be used to model the volumes (Ibrahim & L'Ecuyer, +2013). Within a given day, some event may lead to more/fewer calls than expected. For +example, unexpected behavior in the stock market in the morning may lead to an increased +number of calls for the rest of the day at a financial services contact center. This is intra- +day correlation. Likewise, there are systemic processes responsible for inter-day +correlation. Heuristically, if we noticed that the residuals are very large and positive +throughout the day today caused by a weather event for example, we might also expect a +larger-than-average call load tomorrow. Ignoring correlation between subsequent +observations leads to inaccurate standard errors and prediction intervals. In addition, +although the estimates from a linear regression may be unbiased in the presence of +correlated residuals, they will not be efficient (Demidenko, 2013). + +It is typical for call center regression models to include a day-of-week by period-of-day +interaction (Ibrahim, Ye, L'Ecuyer, & Shen, 2016). In a call center open five days per +week with 32 half-hour periods per day, this interaction involves 160 fixed-effect +parameters. In addition, a call center may require forecasting for holidays. Aldor- +Noiman, Feigin, & Mandelbaum (2009) exclude holidays when training their model; +however, we cannot ignore these days because some splits operate on holidays and may +exhibit different behavior on those days. In order to capture this behavior, we include a +holiday indicator (holiday_ind) by period-of-day interaction effect in the model. +However, some training data sets may include only a single holiday, leading to high +variance in the parameter estimates for this effect (each period observation from that one +day becomes the new estimate for that period during holidays). To reduce the variability +of these estimates, we combine groups of 3 periods together on holidays. That is, periods +{1, 2, 3} are assigned p_group = 1, periods {4, 5, 6} are assigned p_group = 2, etc. The + +p_group*holiday_ind interaction is included in the fixed effect structure as an additive +effect. + +Following Aldor-Noiman et al. (2009), we fit a linear mixed model with correlated errors +to the transformed call counts +𝑌 = 𝑿𝛽 + 𝒁𝑏 + 𝜀 +where +• +𝑌 is the vector transformed call counts, 𝑌 = √𝑐𝑜𝑢𝑛𝑡 + 0.25 +• +𝑿 is a matrix containing the levels of the fixed effects for each observation +• +𝛽 is the vector of fixed effects parameters containing a day-of-week*period-of-day +interaction and a p_group*holiday-indicator interaction +• +𝒁 is a binary coefficient matrix for the random day-to-day effects in the model. +There is one column for each day in the data. +• +𝑏~𝑁(0, 𝑮 ) is the vector of random day-to-day effects. Each unique day in the data +set is represented by one random effect in b. G follows a first-order autoregressive +structure, AR(1). +• +𝜀~𝑁(0, 𝑹 ) is the vector of error terms (residuals), allowing 𝜀 to potentially follow +an AR(1) process within days. Thus, R is a block-diagonal matrix, with one AR(1) +block for each day in the data set. This accounts for the potential correlation in +residuals from proximal periods within days. +The full model allows for complex correlation structures. However, for some splits (within +particular training data sets), there may be only sporadic and sparse occurrences of call +arrivals. This can lead to slow or failed model convergence in some cases. Aldor-Noiman +et al. (2009) address this by estimating the doubly stochastic model in two steps: first, the +inter-day correlation (G) is estimated using the aggregated total call counts from each day. +These parameters are then held constant in a second call to SAS PROC MIXED while 𝛽 +and 𝑹 are estimated. + +Indeed, PROC MIXED can experience convergence problems when the solutions lie on +the boundary of the parameter space, such as when variance components are zero (Karl, +Yang, & Lohr, 2013). However, after making modifications to the default PROC MIXED +settings, we were reliably able to achieve convergence of the full model with the joint +optimization of (𝛽, 𝑮, 𝑹) in a single call to PROC MIXED. In this regard, our approach +differs from that of Aldor-Noiman et al. (2009): we fit all of the model parameters jointly +(with a single call to PROC MIXED). This will lead to reduced bias in the estimates for +the models that do converge. + +We improved convergence rates by changing the convergence criterion used by SAS +PROC MIXED. By default, SAS ensures that the sum of squared parameter gradients +(weighted by the current Hessian of the parameter estimates) is sufficiently small. +However, in the presence of strong correlations in the doubly stochastic model, the +parameter estimates may lie near the boundary of the parameter space, meaning the +gradients may not approach 0 with convergence (Demidenko, 2013). As an alternative, we +declare convergence when the relative change in the loglikelihood between iterations is +sufficiently small. Additionally, we employ Fisher scoring during the estimation process. +Fisher scoring is more stable for models with complex covariance structures and can lead +to better estimates of the asymptotic covariance (Demidenko, 2013). Finally, since our + +application only uses the call volume point estimates and not the associated standard errors +or tests of significance, we specify ddfm=residual to avoid spending substantial time +calculating appropriate degrees of freedom for the approximate F-tests. If confidence or +prediction intervals are needed, this value should be set to ddfm=kenwardrodger2 in order +to calculate Satterthwaite approximations for the degrees of freedom and to apply the +Kenward-Rodger correction (Kenward & Roger, 2009) to the standard errors. The code for +our modified approach appears in Figure 1. + +Figure 1 Modified SAS code for the Doubly Stochastic Model +The square root transformation is applied to reduce the right skew in the observed call +volumes, and to stabilize the variance of the observations since quantities such as call +volumes tend to follow a Poisson distribution. The approach in Figure 1 employs a normal +approximation of this process. We experimented with fitting a mixed Poisson regression to +the untransformed call volumes (via PROC GLIMMIX), but found that the run times +became unfeasibly long (even when using the default pseudolikelihood approach and +avoiding integral approximation) with no noticeable improvement in error rates. + +3. Choosing Recurrent Neural Network Configurations with a Designed Experiment + +Generally, neural networks consist of layers of weights and nonlinear activation functions +that are used to relate inputs (predictors) to outputs (targets). Outputs from each layer are +passed sequentially to the next layer as an input vector. The complexity of each layer is +determined by the length of the output vector (number of nodes) it produces. A loss +function is used to compare the output of the final layer of the neural network to the +provided targets (e.g. call volumes), and an optimizer function provides updated values of +the weights each node that will decrease the resulting loss. The “depth” of the model is +controlled by the number of layers that are used. This “depth” is the source of the phrase +“deep learning”. For example, in image processing applications with convolutional neural +networks, the different layers can be shown to represent different levels of granularity of +detail in an image (Chollet & Allaire, 2018). Besides the number of layers and the +number of nodes per layer, there are a number of choices that must be made regarding the +properties of the optimizer, the distribution of the random initialization of the parameter +weights, and the shape of the activation function(s). + +In a traditional, densely connected network, the individual observations are assumed to be +independent. A simple example using output from JMP Pro 14.1 helps to illustrate. +Suppose we want to fit a densely connected neural network to predict the standardized +call count using only the previous day’s standardized call count at the same period (the +lag-32 of the call count, since there are 32 periods per day in the example) as a predictor +with one node in one layer, using a hyperbolic tangent activation function. This network +shown in Figure 2 with resulting weights shown in Figure 3. + +proc mixed data=training_data scoring=50 maxiter=150 maxfunc=10000 convf=1E-6; +class day_of_week period day_num split p_group; +by split; +/* The fixed effects */ +model transf_call_count=day_of_week*period p_group*holiday_ind/ +noint ddfm=residual outp=pred_call_count_output notest; +/* The day-level random effects */ +/* Note: day num copy is not included in the clAss statment and is numeric * +random day_num / type=sp(pow)(day_num_copy); +/* The period-level correlated residuals */ +run; +Figure 2 Densely connected neural network with one node in one layer. + +Figure 3 Fitted weights from the network with one node in one layer. +Suppose that the lag-32 standardized call count is equal to 1 at a given period, t. Then the +neural network predicts a standardized call volume of +−0.6354 + 3.3923 ∗ TanH(0.5 ∗ (0.4046 + 0.5323 ∗ 1)) = 0.847 +for the current period, t, where TanH is the hyperbolic tangent function and the 0.5 +parameter is a fixed value. The nonlinear activation function provides the network with +the flexibility to model nonlinear relationships between the inputs and the response, as +well as interactions between the inputs. + +We next consider a slightly more complex network with two layers using 2 nodes in the +first layer and 1 node in the second layer (Figure 4) with parameter estimates shown in +Figure 5. + +Figure 4 A densely connected neural network with 2 layers using two nodes in the first +layer and one node in the second layer. + +Lag[Standardize[cnt_ call], 32] +Standardize[cnt_call]Estimates +Parameter +Estimate +H1_ 1:Lag[Standardize[cnt_ call], 32] +0.5323 +H1_1:Intercept +0.4046 +Standardize[cnt call]_ 1:H1_ 1 +3.3923 +Standardize[cnt call]_2:Intercept +-0.6354Lag[Standardize[cnt_ call], 32] +Standardize[cnt_call +Figure 5 Fitted weights from the network with two layers. +Again assuming that the lag-32 standardized call count is 1, the predicted value is +−0.82 + 4.9517 +∗ 𝑇𝑎𝑛𝐻 (0.5 +∗ (0.2416 − 0.7382 ∗ 𝑇𝑎𝑛𝐻(0.5 ∗ (−0.3633 − 0.8785 ∗ 1)) + 0.3057 +∗ 𝑇𝑎𝑛𝐻(0.5 ∗ (−0.0824 + 0.4124 ∗ 1)))) = 0.843 +In the densely connected network, each observation is processed independently and there +is no “memory” of what happened in the previously processed observation. In time series +applications, however, there is a temporal ordering that the data are recorded in, and there +may be correlation between nearby observations. For example, a spike or drop in call +volume might persist over several periods. To address this potential, recurrent neural +networks record information when fitting each observation that is then provided as a +model input when fitting later observations. + +In a simple (Elman) RNN layer, the output from each node (the output of the TanH +functions, referred to as the “state”) is recorded and stored, and used as an input for the +same node when processing the next observation. Note that there is one state recorded for +each node in the layer. For the single layer network example (Figure 3), the state was +calculated as 𝑠𝑡 = TanH(0.5 ∗ (0.4046 + 0.5323 ∗ 1)) = 0.44 when the lagged +standardized call volume was equal to 1. A simple RNN learns an extra parameter (say, +u) to act as a coefficient for the stored state, and the activation function TanH(0.5 ∗ +(𝑤0 + 𝑤1 ∗ 𝑋𝑡)) that is used by the dense network would be replaced by 𝑠𝑡 = +TanH(0.5 ∗ (𝑤0 + 𝑤1 ∗ 𝑋𝑡 + 𝑢 ∗ 𝑠𝑡−1)) in order to fit a simple RNN. The LSTM and +GRU RNNs also use the recorded state when making predictions for the current time +period, along with products of additional activation functions that are designed to carry +state information further in time. Details of these additional structures are explained in +Section 3 of Bianchi et al. (2018). For our purpose it is sufficient to note that the GRU +network is extremely similar to the LSTM, albeit less complex due to the ommison of a +group of paramters. Chollet & Allaire (2018) remark that Google Translate currently runs +using an LSTM with seven large layers. +It is not clear a priori which of these four neural networks is most appropriate for a +particular call center. Furthermore, it is possible that each of these networks may have a +different optimal depth and structure when applied to the call center data. A designed +experiment is run to identify the optimal model type and structure. + + +Estimates +Parameter +Estimate +H2_1:Lag[Standardize[cnt _ call], 32] +-0.8785 +H2_1:lntercept +-0.3633 +H2_2:Lag[Standardize[cnt_call], 32] +0.4124 +H2_2:Intercept +-0.0824 +H1_1:H2_1 +-0.7382 +H1_1:H2 2 +0.3057 +H1_1:lntercept +0.2416 +Standardize[cnt_ call]_1:H1_1 +4.9517 +Standardize[cnt_ call]_2:Intercept +-0.82003.1 Data for the Experiment +We analyze call volumes aggregated into 30 minute periods from 3 different large-volume +skills in an operational call center for the months of March-June 2018. All of the models +under consideration are used to forecast next-day call volumes using 5 weeks of training +data. Each day contains 32 30-minute periods during which the call center is operating. +This application considers the Monday through Friday behavior of the call skills. Due to +the use of the one-week lagged observations as a predictor in the neural networks, the first +week of training data is not included in the predictor matrix (since the prior week’s call +volumes are unknown), meaning each training data set consists of 4*5*32=640 +observations. The designed experiment in this section will evaluate the methods using a +holdout period of the five one-day ahead predictions during last week in June using the +three largest skills from the call center. Section 4 will then fit a reduced set of models over +all skills for 60 day-ahead predictions. + +3.2 Neural Network Input Factors +Each network makes use of the one-hot encoding (via a binary indicator matrix) of the day +of week and the one-hot encoding of period of day. Furthermore, to capture the day- and +week-long correlations, the networks are also fed the call volumes for the same period in +the previous day when modeling the current day, as well as the call volumes for the same +period on the same day of last week. One-period lagged call volumes are not included, as +this is the purpose of the within-sequence memory of the RNNs. Other inputs include a +binary indicator for whether the current day is a holiday, a binary indicator for whether +yesterday was a holiday, a binary indicator for whether or not last week’s observation was +recorded on a holiday, and day number. The day number is a continuous counter for the +number of the given day in the data set, which would potentially allow the neural network +to detect trends across time. All told, these account for 42 vectors of input for the neural +networks. There is no need to create indicator columns for interactions between day of +week and period (or any other factors) as the neural network will automatically detect them. + +Bianchi et al. (2017) application of hourly call volumes displays strong lag-24 correlation, +representing a period-of-day effect. They remove this seasonality by differencing the data +at lag-24. By contrast, we do not difference the call volumes, but instead include period- +of-day (along with day-of-week) as exogenous variables and allow the neural network to +detect this seasonality. This approach allows the network to detect the expected interactions +between period-of-day and day-of-week, as well as any other input factors. + +While these input vectors are included for all models, there are three final input vectors +whose (joint) inclusion is treated as an experimental factor: the same-period predictions +from the mixed model approach (Aldor-Noiman, Feigin, & Mandelbaum, Workload +forecasting for a call center: Methodology and a case study, 2009), from a Winters +smoothing model, and from a seasonal ARIMA(1, 0, 1)(0, 1, 1)160 model. The inclusion +of the predictions from these models as an input to the neural network is an original +approach that gives the network the opportunity to form predictions that may be thought +of as corrections to those from these traditional models, based on potential interactions +with other included factors. For brevity, we refer to this as the mixed.cheat option, since it +allows the neural networks to “cheat” by looking at the predictions generated by these other +three models when forming its own predictions for the same time periods. If none of the +other 42 input vectors were included with these three, this would represent a supervised +learning approach to forming a dynamically weighted average of these three model +predictions in order to create a single “bagged” prediction. + + +3.3 Network Configuration Aspects Treated as Experimental Factors +The designed experiment considered five different factors of structural settings of the +neural networks: model.type {dense, simple (Elman) RNN, GRU, LSTM}, nlayers {1, +2}, nnodes (per layer) {25, 50, 75, 100}, kernel.L2.reg {0, 0.0001} and mixed.cheat +{FALSE, TRUE}. The L2 regularizer adds kernel.L2.reg times each weight coefficient to +the total loss function for the network. Similar to Lasso regression, this helps bound the +magnitude of the model coefficients and could potentially help prevent overfitting the +training data. + +We fit a full factorial design (requiring 128 runs) for these five factors, which allows us +to test for the presence of up to five-way interactions between the five factors. Figure 6 +shows the first 6 runs of the design. The design is replicated over the 3 largest splits and +across the 5 subsequent one-day ahead forecasts. This produces a total of 1920 runs for +the entire experiment. The replication allows for behavior to be averaged over different +days and for a more detailed exploration of how the variance of the error rates depends on +each factor. + +Figure 6 First 6 runs of the designed experiment +3.4 Static Considerations for Neural Network Configuration +The input for the classical dense neural network is a 640x42 matrix (640x45 if +mixed.cheat=TRUE). The dense network does not consider the temporal ordering of the +640 observations (outside of the explicit inclusion of the day- and week-lagged +observations as inputs): the observations are shuffled after each epoch and then processed +in batches (we used a batch size of 32). + +By contrast, the RNNs consider the ordering of the observations. The input for the RNNs +is a 20x32x42 array. This indicates to the RNN that there are 20 batches (days) of 32 +timesteps (periods) with 42 predictors per time step. By default, the batches are treated +independently and the timesteps within each batch are potentially correlated (via the +persistence of the states in the RNN). If the batches themselves are presented in a +temporal order (as is the case in our application), then this can be indicated to the Keras +model via the STATEFUL=TRUE option and by disabling the shuffling of batches +during training. This retains the model weights from batch-to-batch (day-to-day) to allow +for possible long-term behavior. However, we found that using the STATEFUL option +led to a failure to converge in some skill*day combinations and the resulting validation +error rates were not significantly different from those generated without the STATEFUL +option. This seems to indicate that the dependence on prior days’ behavior is already +captured by the inclusion of the one-day and one-week lagged observations. Due to the +occasional convergence issues, the results in the sequel are generated with +STATEFUL=FALSE (and batch shuffling enabled during training). It would have also +been possible to fit the model with week- or month-long batches by training the RNN on +a 4x160x42 or a 1x640x42 array. We did not consider the week-long batches, but the + +model.type +nlayers mixed.cheat nnodes +kernel.L2.reg +layer_simple_rnn +2 +FALSE +50 +0.0001 +layer_ gru +FALSE +25 +0 +layer_Istm +1 +TRUE +100 +0 +layer_gru +TRUE +75 +0.0001 +layer_gru +TRUE +75 +0 +layer_simple_rnn +FALSE +100 +0month-long batches tended to produce inferior predictions to the day-long batches. This +could possibly be due to the lack of long term correlations in the data and the fact that the +smaller batches allow for the shuffling of the ordering that the days are fed through the +gradient-optimization routine of the neural network, which can improve the model fit by +preventing the model from overweighting the first observations that are provided to the +network in each epoch (Chollet & Allaire, 2018). + +Note that the model weights are reset and the model is retrained for each skill. This is in +contrast to the approach taken by Zhu & Laptev (2017) in which a single network is fit to +accommodate disparate behavior from different cities. As discussed in Section 5, a single +multi-output network could potentially be built to model all of the skills at once. + +While they were not included as experimental factors in this application, we also noticed +a significant relationship between the quality of the predictions and the optimization +routine employed. Extensive pilot experimentation led to our use of the AMSGrad variant +(Reddi, Kale, & Kumar, 2018) of the Adam optimizer (Kingma & Ba, 2014), with a +learning rate decay of 0.0001. We would recommend including both the optimizer and +the optional learning rate decay as factors in the designed experiment for parameter +tuning in future problems. + +We found it was important to tune the number of epochs for each model fit using +validation data (the last week in the training set) by first fitting 500 epochs for each +model, taking a moving average (with a window size of 10 epochs) of the resulting +WAPEs on the validation data, and then refitting the model (on both the training and the +validation data in order to predict an additional day which was held out as a test set) with +the number of epochs that produced the minimum validation WAPE. The moving +average is important due to the volatile and non-monotonic behavior we observed in the +individual recorded WAPEs and helps to find a relatively stable region. Consistent with +previous findings (Bianchi, Maiorino, Kampffmeyer, Rizzi, & Jenssen, 2017), we noticed +that the RNNs take many more epochs to converge than the dense neural network. Other +researchers (such as Bianchi et al.) have used more than 500 epochs when fitting RNNs, +so this upper bound should also be considered as an important factor when building an +RNN. + +While we also experimented with recurrent dropout (Gal & Ghahramani, 2015) to +prevent overfitting, it led to degraded performance in the early iterations of our +experiment and we removed it from consideration. However, this could simply be due to +features of this particular application, such as the relatively short training period of five +weeks: Chollet & Allaire (2018) strongly advocate the use of dropout and recurrent +dropout. + +The models experienced improved errror rates after switching the kernel initializer for the +random weights to the He normal initializer (He, Zhang, Ren, & Sun, 2015). We used the +relu activation function exclusively, although this choice could also impact the quality of +the resulting model fit. And while our application did not detect any two-factor +interactions among the five experimental factors, it is possible that some of these +additional factors could depend on the type of (R)NN being used, meaning there would +be an interaction between these factors and model.type. + +A final contributing factor is the batch size. This determines how many observations are +processed before the gradients are updated: when fitting Keras models on a GPU, amount + +of memory available on the GPU can be a limiting factor on the batch size. This choice is +more constrained in the RNNs, where each batch is a single day/week/month (determined +by the number of timesteps specified in the input array to the RNN). By contrast, the +batch size for a dense network can be set between 1 and the number of observations in +the training data. We noticed significant differences in the error rates from the dense +model depending on what batch size was used. + +3.5 Experimental Response +While mean squared error (MSE) is frequently used to evaluate and compare predictive +models, this is a poor metric for the call center application as it will give undue focus to +the low call volume periods at the beginning and end of each day. Instead, the weighted +absolute percentage error (WAPE) is recommended for call volume modeling (Ibrahim, +Ye, L'Ecuyer, & Shen, 2016). This weights the absolute percentage error in each period +by the number of calls received in that period and is defined by +𝑊𝐴𝑃𝐸 = +∑ +|𝑌𝑖 − 𝑌̂𝑖| +𝑛 +𝑖=1 +∑ +𝑌𝑖 +𝑛 +𝑖=1 + +where 𝑌𝑖 and 𝑌̂𝑖 are the observed and predicted volumes, respectively, for each 30 minute +period 𝑖 = 1, … , 𝑛. Because WAPE is our metric of interest, the models are compiled to +use a mean absolute error loss-function, which minimizes the numerator of the WAPE +(the denominator is static). + +For each row of the experimental table (Figure 6), the specified neural network is fit +(independently) to model five subsequent days of each split. That is, day 1 is predicted +using the previous 5 weeks leading up to day 1, then the model is reset and day 2 is +predicted using the 5 weeks leading up to day 2, etc. The vector of five one-day ahead +predictions is compared with the observed call volumes (that were not visible to the +model during training), and the resulting WAPE is recorded. + +3.6 Analysis of Experimental Results +Figure 7 gives a typical output of the prediction error across the different formulations of +the neural networks. GRU often has the lowest forecast error or at least is consistently +close to the lowest. The other procedures tend to have much more unstable performance +based on the choice of nodes and layers as well as across the days and splits. + +Figure 7 Forecast error by model type and settings for forecast day 5 split 3 +Regression analysis is used to determine the statistically significant factors and +interactions. In order to control a heavy right-skew in the recorded WAPEs, an inverse + +model.type +NN Classic +RNN GRU + RNN LSTM RNN Simple +WAPE +WAPE +WAPE +WAPE +nlayers +nnodes +Mean +Mean +Mean +Mean +1 +25 +6.7% +6.2% +8.5% +6.0% +50 +7.1% +6.2% +6.5% +6.7% +75 +7.1% +5.8% +6.8% +5.9% +100 +7.0% +6.4% +6.9% +6.8% +2 +25 +7.5% +6.6% +8.1% +6.9% +50 +7.3% +5.8% +6.9% +7.0% +75 +7.1% +6.0% +6.6% +6.5% +100 +8.0% +5.9% +7.5% +6.0%transformation is applied to use as the response in the regression models. Figure 12 +displays the original skewness and the transformation to normality after taking the +inverse of the WAPEs. + +3.7 Loglinear Variance Regression Model +The mean structure of 1/WAPE is modeled by including the main effects of the five +experimental factors (model.type, nlayers, mixed.cheat, nnodes, kernel.L2.reg) and all of +the interactions. In addition, split, file (which represents the validation day), and split*file +are included as blocking factors. + +While an ordinary regression model for the full factorial design would assume the same +error variance for all responses across the design space, the loglinear variance model +allows for the error variance itself to be modelled as a function of the input factors. This +is appealing for this application, where parsimonious networks may be expected to +demonstrate less variability during repeated fittings. + +For each factor setting, the loglinear variance model produces a predicted mean of +1/WAPE and a predicted standard deviation of 1/WAPE, representing the variability due +to the random initial weighting of the NN. +We include model.type, nlayers, mixed.cheat, nnodes, kernel.L2.reg, file, and split as +factors in the loglinear variance model (using JMP Pro 14.1), which models the log of the +error variance as a linear combination of these factors (which are all treated as +categorical). With the exception of kernel.L2.reg, all of these effects are found to be +significantly associated with the error variance (Figure 8). + +Figure 8 Factors with a significant impact on WAPE variability +For the mean model, model.type, nlayers, mixed.cheat, nnodes, day, split, and file*split +were found to be significantly associated with 1/WAPE (Figure 9). Neither kernel.L2.reg +nor any of the interaction terms involving the NN architecture were found to be +significant. + + + +Variance Effect Likelihood Ratio Tests +Source +Test Type +DF + ChiSquare +Prob>ChiSq +model.type +Likelihood +3 +114.126 +<.0001* +nlayers +Likelihood +1 +4.6333 +0.0314* +mixed.cheat +Likelihood +1 +53.6884 +<.0001* +nnodes +Likelihood +3 +16.5624 +0.0009* +kernel.L2.reg +Likelihood +1 +0.0042 +0.9482 +file +Likelihood +4 +36.8213 +<.0001* +split +Likelihood +2 +56.3123 +<.0001* +Figure 9 Factors that are significantly associated with the mean WAPE +Removing the insignificant interaction terms (but allowing kernel.L2.reg to remain in +both the mean and variance models) produces the profiler shown in Figure 10. Notice the +large standard deviation associated with the LSTM model. The dependence on file and +split is not shown, since there are no interactions modeled between these and the other +experimental factors. While we want to maximize 1/WAPE, we also want to minimize +the standard deviation of 1/WAPE. That is, we would like to maximize the lower 95% +prediction interval of 1/WAPE (or minimize the upper 95% PI of WAPE) in order to find +the model configuration that will give a combination of relatively good results on average +while being protected against large errors. + +Fixed Effect Tests +Source +Nparm +DF +DFDen +F Ratio +Prob > F +model.type +m +m +796.5 +44.6120 +<.0001* +nlayers +1 +1 +1258 +5.5213 +0.0189* +model.type*nlayers +3 +m +796.5 +1.0691 +0.3614 +mixed.cheat +1 +1 +1258 +17.0652 +<.0001* +model.type*mixed.cheat +3 +3 +796.5 +1.3181 +0.2672 +nlayers*mixed.cheat +1 +1 +1258 +2.3413 +0.1262 +model.type*nlayers*mixed.cheat +3 +3 +796.5 +1.9528 +0.1196 +nnodes +m +786.2 +4.5600 +0.0036* +model.type*nnodes +9 +5 +839.4 +0.7864 +0.6290 +nlayers*nnodes +3 +3 +786.2 +0.7371 +0.5301 +model.type*nlayers*nnodes +9 +6 +8394 +0.6668 +0.7395 +mixed.cheat*nnodes +3 +3 +786.2 +0.6624 +0.5753 +model.type*mixed.cheat*nnodes +9 +5 +839.4 +0.2224 +0.9913 +nlayers*mixed.cheat*nnodes +m +786.2 +0.9291 +0.4261 +model.type*nlayers*mixed.cheat*nnodes +9 +9 +839.4 +0.4312 +0.9186 +kernel.L2.reg +1 +1 +1258 +0.0026 +0.9590 +model.type*kernel.L2.reg +m +796.5 +0.1263 +0.9445 +nlayers'kernel.L2.reg +1 +1 +1258 +0.2313 +0.6306 +model.type*nlayers*kernel.L2.reg +3 +m +796.5 +0.7812 +0.5046 +mixed.cheat*kernel.L2.reg +1 +1 +1258 +0.0295 +0.8636 +model.type*mixed.cheat*kernel.L2.reg +3 +m +796.5 +0.1395 +0.9364 +nlayers*mixed.cheat*kernel.L2.reg +1 +1 +1258 +0.1498 +0.6988 +model.type*nlayers*mixed.cheat*kernel.L2.reg +3 +3 +796.5 +0.3776 +0.7692 +nnodes*kernel.L2.reg +m +m +786.2 +0.9118 +0.4347 +model.type*nnodes*kemel.L2.reg +9 +6 +839.4 +0.8905 +0.5331 +nlayers*nnodes*kernel.L2.reg +3 +m +786.2 +0.7442 +0.5259 +model.type*nlayers*nnodes*kernel.L2.reg +9 +839.4 +0.6535 +0.7513 +mixed.cheat*nnodes*kernel.L2.reg +3 +3 +786.2 +0.0812 +0.9703 +model.type*mixed.cheat*nnodes*kernel.L2.reg +9 +8394 +0.2691 +0.9827 +nlayers*mixed.cheat*nnodes*kernel.L2.reg +3 +m +786.2 +0.8510 +0.4662 +model.type*nlayers*mixed.cheat*nnodes*kernel.L2.reg +9 +6 +8394 +0.6614 +0.7442 +file +4 +4 +668.1 +1258.316 +<.0001* +ds +2 +2 +857.3 +1867.911 +<.0001* +file*split +8 +8 +746.8 +244.5502 +<.0001* +Figure 10 Mean and Standard Deviation of 1/WAPE as a function of the experimental +factors. Goal is to maximize the top (1/mean forecast error) and minimize the bottom +(standard deviation) +Figure 11 plots the upper 95% prediction interval for WAPE against the experimental +factors. The model configuration that minimizes the upper 95% PI of WAPE is a GRU +model that is allowed to use the MIXED, ARIMA, and Winters predictions as inputs, +with 2 layers and 50 nodes per layer and an L2 penalty of 0.0001 on the kernel weights. +However, the L2 penalty was not found to be significant in either the mean or the +variance models and the contribution of nlayers is relatively flat. + +Figure 11 Upper 95% Prediction Interval for WAPE +To summarize, the designed experiment provided insight to effectively select the +appropriate levels across several neural network models and parameters. The goal is to +balance model performance in minimizing forecast error (that is, maximizing 1/forecast +error) and minimizing the variance of this forecast error. Figure 10 and Figure 11 are +interpretable across all factors and settings as displayed due to the absence of significant +interactions between the factors. The top half of Figure 10 displays the reciprocal forecast +error (the larger the value the better) which indicates preferred settings of GRU or Elman +(simple) for the model, 1 layer, using the mixed model forecasts, and 25 nodes while the +selection of L2.kernel makes little difference. The lower half of Figure 10 displays the +variance (lower is better) where the traditional neural net would be preferred along with 2 + +Prediction Profiler +30 - +28.87292 +29 +[27.9033, +28 +29.8426] +27 +26 +Dev +2.5 + [1.66636, +1.5 +dense +layer_gru +layer_Istm +2 +FALSE- +TRUE- +5 +0.0001 +layer_gru +2 +TRUE +50 +0.0001 +model.type +nlayers +mixed.cheat +nnodes +kemel.L2.regPrediction Profiler +0.044 +0.0435 +0.043 +0.040571 +0.0425 +0.042 +0.0415 +0.041 +0.0405 +dense +layer_gru +layer_Istm +layer_simple_rnn +2 +FALSE- +TRUE- +5 +9 +5 +100- +0.0001 +layer_gru +TRUE +50 +0.0001 +model.type +nlayers +mixed.cheat +nnodes +kenel.L2.reglayers, using mixed model forecasts, with 50 nodes while robust to the regularization +parameter value. Note that the traditional dense neural network does have consistently +worse forecast error across all scenarios and could not be recommended despite having +the lowest variance. The prediction interval on forecast error is an alternative view that is +preferred by many practitioners where the goal is to minimize the width. Figure 11 (lower +is better) clearly shows GRU is the preferred solution using the mixed model forecasts +with 50 nodes. + + +Figure 12 Representative distribution of WAPE and 1/WAPE resulting from the designed +experiment + +4. Comprehensive Performance Study + +Based on the results of the designed experiment for NN error rates, we perform a further +study with actual call center data across 36 skills rather than only 3. We consider a +consistent 5-week training period advancing across multiple months of data to produce 60 +one-day ahead predictions. These predictions do not use the actual data for the forecasted +day during model training and can be viewed as a validation set for the trained models. +This allows us to compare the relative performance of the mixed model and the neural +networks. We also include the error rates from the ARIMA and Winters seasonal +smoothing models for comparison. + +Though we could have simply chosen the GRU RNN with 50 nodes on each of 2 layers +with kernel.L2.reg complexity parameter set to 0.0001 as the only representative RNN +based on the designed experiment, we decided to include all 4 neural network methods + +Distributions +WAPE +0 0.1 +0.3 +0.5 +0.7 +0.9 1 1.1 +1.3 +1.5 +1/WAPE +0 +5 +10 +15 +20 +25 +30screened by using these same parameter settings. It is possible with the added skills – +with many having much lower call volumes – that another RNN model could work better +than GRU. + +4.1 Results: One-day ahead Predictions Over 60 Separate Validation Days +Figure 13 provides the forecast errors averaged across all 60 days for the 12 splits with +the highest call volume sorted in descending call volume order. Note that results in this +section are generated by the (R)NNs with the mixed.cheat option disabled. The Doubly +Stochastic Mixed Model has the lowest average forecast error for all but Split 5540 and is +often significantly lower than the competitors. Winters Seasonal Exponential Smoothing +also performs quite well given its relative simplicity. The GRU performance confirms the +results from the designed experiment as usually the best RNN and always competitive +with the best. The highly complex LSTM recurrent neural network has very large forecast +errors for many of these splits and cannot be recommended. Note also that forecast error +generally increases for all methods as call volume decreases. + +Figure 13 Average WAPE forecast errors across 60 separate validation days for high +call-volume splits +Figure 14 shows the similar trend of increasing error rates with decreasing call volumes +for the medium call volume splits. The GRU recurrent neural network is usually +outperforming the other neural network methods and is closer to the error rates of the +Doubly Stochastic Mixed Model. + +Figure 14 Average WAPE forecast errors across 60 separate validation days for medium +call-volume splits +For the low call volume splits displayed in Figure 15, the GRU and Simple recurrent +neural networks perform similarly and slightly better than Doubly Stochastic and Winters +for most of the splits. The very low call volumes (last 3 splits) do seem to benefit from +the recurrent neural network formulation. + +Split +Sum Call Vol +ARIMA +Doubly Stoch +NN_Classic +RNN_GRU +RNN_LSTM +RNN_Simple +Winters +5000&5240 +929754 +9.0% +8.4% +10.2% +11.2% +38.0% +13.9% +8.3% +5400&5570 +461256 +7.4% +6.6% +8.8% +8.4% +26.1% +14.3% +7.1% +5620 +162996 +8.6% +7.7% +9.6% +9.1% +28.7% +9.6% +8.4% +5660 +137759 +10.2% +10.1% +13.2% +12.0% +17.1% +14.0% +10.5% +5020 +71930 +11.5% +11.2% +14.5% +12.3% +27.3% +14.2% +11.7% +5630 +65079 +15.4% +14.9% +16.7% +15.5% +26.4% +15.5% +14.1% +5840 +63984 +13.1% +12.1% +14.6% +13.4% +25.3% +12.9% +13.0% +5200 +48728 +13.2% +13.4% +15.6% +13.5% +52.4% +13.5% +13.6% +5540 +38236 +15.9% +15.8% +16.8% +16.3% +28.5% +16.6% +16.0% +5670 +34793 +14.0% +13.2% +15.6% +14.2% +27.3% +13.9% +14.0% +6500 +30534 +16.8% +16.4% +18.5% +17.3% +23.0% +18.0% +17.1% +5260 +23849 +21.9% +20.9% +22.7% +20.3% +32.5% +20.2% +21.2%Split +Sum Call Vol +ARIMA +Doubly Stoch +NNClassic +RNN_GRU +RNN_LSTM +RNN_Simple +Winters +5460 +20922 +18.2% +18.1% +19.5% +18.1% +26.0% +17.7% +18.2% +5410 +19461 +19.8% +19.2% +21.4% +20.0% +59.1% +20.8% +19.7% +6350 +16874 +30.8% +26.8% +29.3% +28.7% +31.1% +29.0% +29.4% +5060 +16765 +19.3% +19.1% +20.7% +19.6% +24.4% +19.6% +19.6% +5650 +9911 +23.8% +23.4% +24.6% +24.0% +27.3% +24.3% +24.0% +5030 +8102 +40.0% +26.0% +28.5% +27.8% +33.2% +27.7% +26.9% +5680 +7525 +27.3% +27.1% +28.2% +26.7% +31.9% +27.8% +27.7% +5440 +7402 +28.5% +28.1% +30.3% +28.3% +33.8% +29.7% +29.2% +5070 +6446 +34.6% +34.5% +34.6% +33.5% +43.6% +35.0% +34.7% +5420 +5247 +35.0% +34.6% +36.2% +33.9% +38.1% +34.7% +35.0% +5899 +4844 +36.4% +35.8% +37.0% +35.2% +64.8% +36.6% +36.7% +5100 +4019 +34.7% +34.0% +36.6% +34.3% +40.1% +35.5% +34.8% +Figure 15 Average WAPE forecast errors across 60 separate validation days for low +call-volume splits +Overall, the best performing procedures were the Doubly Stochastic Mixed Model and +the GRU Recurrent Neural Network. Figure 16 shows forecast error by split sorted by +call volume. Generally, the mixed model does better for large and medium volume splits +(lower is better on the graph) while the RNN +model is more effective for the small volume—particularly the very small volume splits. + + +Figure 16 Forecast error rates ordered by call volume by split for Doubly Stochastic +(blue) and GRU (red) +Figure 17 presents a different view of this same pattern. For each of the 60 one-day ahead +forecasts within each split, the GRU and Doubly Stochastic WAPEs are recorded, along +with the number of calls recorded for that split over the training and validation data +(sum_all_calls). For each split, Figure 17 plots the percent of the 60 day-ahead forecasts +for which GRU “won” (GRU WAPE < Doubly Stochastic WAPE) against the log of the +median call volume recorded by that split over the 60 different pairs of training and +validation data. There appears to be a linear decrease of the relative performance of GRU +against Doubly Stochastic in the log of the call volume. + +Split +Sum Call Vol +ARIMA +Doubly Stoch +NN_Classic +RNN_GRU +RNN_LSTM +RNN_Simple +Winters +5710 +3742 +39.6% +38.8% +40.7% +39.1% +45.7% +40.2% +38.9% +5820 +2949 +44.1% +43.1% +46.1% +43.2% +50.4% +44.0% +43.4% +5690 +2238 +56.1% +51.1% +51.6% +51.2% +58.9% +51.8% +52.5% +5220 +2089 +49.7% +49.5% +50.5% +49.3% +54.0% +50.1% +49.7% +5470 +1975 +49.2% +48.8% +52.0% +50.5% +57.0% +50.7% +50.2% +6330 +1556 +60.5% +60.1% +61.0% +60.2% +65.1% +61.6% +60.7% +5040 +1398 +79.0% +69.0% +69.8% +73.4% +72.2% +71.1% +80.7% +6310 +922 +98.5% +92.0% +92.2% +88.1% +91.0% +89.5% +92.8% +5720 +918 +80.2% +77.8% +77.5% +77.6% +85.0% +78.4% +80.1% +6370 +485 +95.3% +93.6% +96.5% +91.3% +111.3% +94.0% +95.2% +6360 +171 +150.4% +136.8% +126.5% +107.7% +118.7% +111.7% +142.5% +6340 +14 +189.7% +161.4% +188.2% +102.7% +108.9% +107.8% +187.7%Forecast ErrorbySplit AverageOver 6oDays +1.6 +IDoubly Stoch +1.5 +RNN GRU +1.4 +1.3 +1.2 +1.1 - +Errot +aber +1.0 +0.9 +0.8 +0.7 +pa +ubiay +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +HighCallVolum +Medium CallVolume +Low Call Volume +Figure 17 Percent of the 60 day-ahead forecasts for which the GRU WAPE was less than +the Doubly Stochastic WAPE for each of the 36 splits plotted against the log of the +median sum of all calls recorded for each split over each pair of training and validation +data. +For the large call-volume splits, the extra flexibility of the GRU model does not lead to +improvements over the predictions generated by the mixed model approach. Because the +mixed model computations are faster and the implementation is less complex, there does +not appear to be any benefit to running the neural networks for the high-volume splits for +this short-term application. This is consistent with the findings of the Uber traffic volume +study when short-term predictions were considered (Zhu & Laptev, 2017), and would +possibly change if a longer training period (several months or multiple years) were used +for the call center data. + +4.2 Improving GRU RNN by Using Doubly Stochastic Forecasts as a Covariate +Based on pilot studies during the designed experiment, the forecasting performance of the +GRU recurrent neural network often improved by integrating the forecasted value for the +validation days from the doubly stochastic, ARIMA, and Winters models. This +“cheating” by using other models’ forecasts (shown as mixed.cheat) proved to be a +significant benefit for the GRU model over these 60 predictions for each skill. The +WAPE for the held-out validation data was again the primary measure of performance. + +Figure 18 displays the forecast errors for the high call volume skills averaged over the 60 +validation days for the each of the neural networks when they use the mixed forecasts. +Note the side-by-side comparison of the RNN_GRU (no cheat) and GRU_cheat columns +where in most cases the “cheating” does result in improved forecasts and in those cases +where it is not better, it has only marginally declined. Additionally, the Simple_cheat +error rates compare favorably with the GRU_cheat while the LSTM_cheat suffers from +significantly poorer performance and instability issues. The doubly stochastic forecast is +still quite good and often the best choice. These results are also consistent when looking +at the medium and low call volume splits. Therefore, we recommend using the forecasts +from a doubly stochastic or Winters model as inputs to recurrent neural networks. + +Percent won by GRU vs. Log[Median(sum all calls)] +100% + Percent won by GRU +%06 +%08 +70% +Percent won by GRU +%09 +50% +40% +30% +20% +10% +0% +4 +6 +8 +10 +12 +14 +Log[Median(sum all_ calls)] +Figure 18 Forecast errors for high call volume splits averaged over 60 validation +forecast days allowing NN to “cheat” to improve predictions +The median of the GRU WAPE (mixed.cheat=FALSE) minus the GRU WAPE +(mixed.cheat=TRUE) (within each split/day combination, for a sample size of +36*60=2160) is 0.002 with a p-value of 0.0005 from the Wilcoxon signed rank test with a +null hypothesis that the differences were drawn from a population with median equal to +0, indicating that mixed.cheat does tend to improve the performance of the GRU model. + +A similar comparison of paired differences of error rates (all with mixed.cheat=FALSE) +confirms that GRU outperforms the other (R)NNs: LSTM - GRU produces a median of +0.0247 with a p-value of 1e-129, NN Classic - GRU produces a median of 0.0690 with a +p-value of <1e-185, and Simple RNN - GRU produces a median of 0.0021 with a p-value +of 1e-04. + + +References +Aldor-Noiman, S., Feigin, P. 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An Improved Approximation to the Precision of +Fixed Effects from Restricted Maximum Likelihood. Computational Statistics +and Data Analysis(53), 2583–2595. +Kingma, D., & Ba, J. (2014). Adam: A Method for Stochastic Optimization. arXiv. +Retrieved from https://arxiv.org/abs/1412.6980v8 + +Laptev, N., Yosinski, J., Li, L. E., & Smyl, S. (2017). Time-series Extreme Event +Forecasting with Neural Networks at Uber. ICML 2017 Time Series Workshop. +Sydney. Retrieved from http://roseyu.com/time-series- +workshop/submissions/TSW2017_paper_3.pdf +Reddi, S., Kale, S., & Kumar, S. (2018). On the Convergence of Adam and Beyond. +International Conference on Learning Representations. Retrieved from +https://openreview.net/forum?id=ryQu7f-RZ +Rushing, H., Karl, A., & Wisnowski, J. (2014). Design and Analysis of Experiments by +Douglas Montgomery: A Supplement for Using JMP. Cary: SAS Institute. +Winters, P. (1960). Forecasting sales by exponentially weighted moving averages. +Management Science, 6(1): 127-137. +Zhu, L., & Laptev, N. (2017). Deep and Confident Prediction for Time Series at Uber. +arXiv. Retrieved from https://arxiv.org/abs/1709.01907 + + + diff --git a/9tAzT4oBgHgl3EQfg_wg/content/tmp_files/load_file.txt b/9tAzT4oBgHgl3EQfg_wg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8d0ef528432e7f24b7ed8f38b922cab71ee320e9 --- /dev/null +++ b/9tAzT4oBgHgl3EQfg_wg/content/tmp_files/load_file.txt @@ -0,0 +1,1156 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf,len=1155 +page_content='Lessons Learned Applying Deep Learning Approaches to Forecasting Complex Seasonal Behavior Andrew T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Karl1, James Wisnowski1, Lambros Petropoulos2 1Adsurgo LLC, Pensacola, FL 2USAA, San Antonio, TX Abstract Deep learning methods have gained popularity in recent years through the media and the relative ease of implementation through open source packages such as Keras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' We investigate the applicability of popular recurrent neural networks in forecasting call center volumes at a large financial services company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' These series are highly complex with seasonal patterns - between hours of the day, day of the week, and time of the year - in addition to autocorrelation between individual observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Though we investigate the financial services industry, the recommendations for modeling cyclical nonlinear behavior generalize across all sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' We explore the optimization of parameter settings and convergence criteria for Elman (simple), Long Short-Term Memory (LTSM), and Gated Recurrent Unit (GRU) RNNs from a practical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' A designed experiment using actual call center data across many different “skills” (income call streams) compares performance measured by validation error rates of the best observed RNN configurations against other modern and classical forecasting techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' We summarize the utility of and considerations required for using deep learning methods in forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Key Words: ARIMA, Time Series 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Introduction Member contact call centers receive fluctuating call volumes depending on the day of the week, the time of day, holidays, business conditions, and other factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' It is important for call center managers to have accurate predictions of future call volumes in order to manage staffing levels efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The call center arrival process has been well documented and explored in the literature (Gans, Koole, & Mandelbaum, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' In the application presented here, there are several different “skills” (or “splits”) to which an incoming call may be routed – depending on the capabilities of the call center agents – and an arrival volume forecast is required for each skill in the short term for day-ahead or week-ahead predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The weekly seasonality found in call arrivals can be modeled effectively through a variety of methods to include Winter’s Seasonal Smoothing (Winters, 1960) or Autoregressive Integrated Moving Average (Box & Jenkins, 1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Some accessible references for many of these concepts aimed at the practitioner are well documented in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Bisgaard & Kulachi, (2007, 2008)) while recommended texts are Bisgaard & Kulachi (2011) and Montgomery, Jennings & Kulachi (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=" Aiming to improve on these classic methods, “doubly stochastic” linear mixed models (Aldor-Noiman, Feigin, & Mandelbaum, 2009) have effectively modeled additional complexities as outlined in a recent review paper from Ibrahim, Ye, L'Ecuyer, & Shen (2016)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Similarly, Recurrent Neural Networks (RNNs) have been recommended as deep learning approaches to forecast call volume for a wireless network (Bianchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=', 2017) in addition to numerous other applications including ride volumes with Uber (Zhu & Laptev, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' While the doubly stochastic and RNN approaches to predicting call volumes offer greater flexibility in modeling complex arrival behavior by incorporating exogenous variables, this flexibility comes at the cost of greater computational and programming complexity (as well as greater prediction variance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' This paper explores practical aspects of managing that complexity for these models, applies the models to actual call volumes recorded by a large financial services company, and compares the prediction capability to that of the more traditional Winters smoothing and ARIMA models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' First, we modify computational aspects of the doubly stochastic approach proposed by Aldor-Noiman, Feigin, & Mandelbaum (2009) to improve call center forecasting performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Doubly stochastic implies a two-level randomization where not only are call arrivals random variables, but also the call arrival mean parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Forecasts are produced by taking advantage of the unique correlation structure for each split while accounting for trend, seasonality, cyclical behavior, and serial dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The doubly stochastic model is more complex than ordinary regression as it accounts for both inter- and intra-day correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' We suggest modifications to the originally proposed approach that lead to more stable convergence and more flexible behavior when many splits need to be fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Secondly, we consider how RNNs may be used to model incoming call volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Whereas “traditional” densely connected, feedforward neural networks process each data point independently, RNNs process sequences according to temporal ordering and retain information from previous points in the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' As it processes points within sequences, the RNN maintains states that contain information about what it has seen previously in the sequence (Chollet & Allaire, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' This intra-sequence memory is useful in time series applications to autocorrelated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' In the context of call center volumes, these sequences could be constructed to correspond to individual days of observations over a fixed number of (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' 30 minute) periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Bianchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' (2017) consider three different RNN architectures to model incoming call volume over a mobile phone network: Elman Recurrent Neural Networks (ERNN) (Elman, 1990), Gated Recurrent Units (GRU) (Cho, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=', 2014), and Long Short Term Memory (LSTM) (Hochrieter & Schmidhuber, 1997), listed in order of increasing complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' These three RNNs, along with the dense neural network, are now available via the R Keras package (Allaire & Chollet, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Once code has been written for one of the RNNs, the user can switch between the other two by toggling a single option (and, after data reformatting, switch to a dense network).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' This offers the potential – via a designed experiment – to produce a pragmatic answer to the question of which type of (R)NN provides the best fit to the process at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Whereas Bianchi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Maiorino,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Kampffmeyer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Rizzi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' & Jenssen (2017) created their experimental design by randomly generating points within the design space and then selected the design that lead to the minimum error rate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' we create a full factorial design (treating all factors as categorical to allow arbitrary shape in the otherise (discrete) continuous factor of number of nodes) and then explore the behavior of the error rates across the design space with a profiler for the resulitng linear model for the error rate as a function of the NN settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Unlike ARIMA or regression (including doubly stochastic) modeling approaches for time series, there is a stochastic behavior in the predictions made by neural networks due to the use of randomly initialized weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Unless the seed for the software’s random number generator is fixed, repeated fitting of the same neural network will lead to different predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The amount of variation in the resulting predictions depends on the complexity of the network and on the steps that have been taken to avoid overfitting, including early stopping of the optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' When selecting a model configuration, we will not only want to minimize the expected error rate, but also minimize the variability in the error rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' To this end, we seek to minimize the upper 95% prediction interval on the testing error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The NN study proceeds in two phases where a screening experiment first identifies the most useful (R)NN, followed by a more comprehensive performance study against common forecasting approaches across many more skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Section 2 describes the doubly stochastic model for call volumes and how modifications to the originally proposed computational approach can lead to improved convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Section 3 details how a full factorial design is used to characterize the performance of RNN options as a function of five factors (and their interactions) on the resulting short-term forecast error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Additionally, Section 3 describes the selection of the model configuration that leads to the minimum upper bound on the 95% prediction interval for the testing error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Due to the number of different model configurations that must be run along with the computational complexity of RNNs, the first phase discussed in Section 3 considers only a limited number of skills and validation days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' In Section 4, the best performing RNNs are run over a larger validation set and over all call center skills to compare the performance to the doubly stochastic mixed model approach, and to ARIMA as well as Winters smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=" Stable Settings for Fitting the Mixed Model There are two distinct influences on call volumes that induce a correlation between the observed call counts, violating the independence assumption made by ordinary least squares regression models that might be used to model the volumes (Ibrahim & L'Ecuyer, 2013)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Within a given day, some event may lead to more/fewer calls than expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' For example, unexpected behavior in the stock market in the morning may lead to an increased number of calls for the rest of the day at a financial services contact center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' This is intra- day correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Likewise, there are systemic processes responsible for inter-day correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Heuristically, if we noticed that the residuals are very large and positive throughout the day today caused by a weather event for example, we might also expect a larger-than-average call load tomorrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Ignoring correlation between subsequent observations leads to inaccurate standard errors and prediction intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' In addition, although the estimates from a linear regression may be unbiased in the presence of correlated residuals, they will not be efficient (Demidenko, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=" It is typical for call center regression models to include a day-of-week by period-of-day interaction (Ibrahim, Ye, L'Ecuyer, & Shen, 2016)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' In a call center open five days per week with 32 half-hour periods per day, this interaction involves 160 fixed-effect parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' In addition, a call center may require forecasting for holidays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Aldor- Noiman, Feigin, & Mandelbaum (2009) exclude holidays when training their model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' however, we cannot ignore these days because some splits operate on holidays and may exhibit different behavior on those days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' In order to capture this behavior, we include a holiday indicator (holiday_ind) by period-of-day interaction effect in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' However, some training data sets may include only a single holiday, leading to high variance in the parameter estimates for this effect (each period observation from that one day becomes the new estimate for that period during holidays).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' To reduce the variability of these estimates, we combine groups of 3 periods together on holidays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' That is, periods {1, 2, 3} are assigned p_group = 1, periods {4, 5, 6} are assigned p_group = 2, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The p_group*holiday_ind interaction is included in the fixed effect structure as an additive effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Following Aldor-Noiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' (2009), we fit a linear mixed model with correlated errors to the transformed call counts 𝑌 = 𝑿𝛽 + 𝒁𝑏 + 𝜀 where 𝑌 is the vector transformed call counts, 𝑌 = √𝑐𝑜𝑢𝑛𝑡 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='25 𝑿 is a matrix containing the levels of the fixed effects for each observation 𝛽 is the vector of fixed effects parameters containing a day-of-week*period-of-day interaction and a p_group*holiday-indicator interaction 𝒁 is a binary coefficient matrix for the random day-to-day effects in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' There is one column for each day in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' 𝑏~𝑁(0, 𝑮 ) is the vector of random day-to-day effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Each unique day in the data set is represented by one random effect in b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' G follows a first-order autoregressive structure, AR(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' 𝜀~𝑁(0, 𝑹 ) is the vector of error terms (residuals), allowing 𝜀 to potentially follow an AR(1) process within days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Thus, R is a block-diagonal matrix, with one AR(1) block for each day in the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' This accounts for the potential correlation in residuals from proximal periods within days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The full model allows for complex correlation structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' However, for some splits (within particular training data sets), there may be only sporadic and sparse occurrences of call arrivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' This can lead to slow or failed model convergence in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Aldor-Noiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' (2009) address this by estimating the doubly stochastic model in two steps: first, the inter-day correlation (G) is estimated using the aggregated total call counts from each day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' These parameters are then held constant in a second call to SAS PROC MIXED while 𝛽 and 𝑹 are estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Indeed, PROC MIXED can experience convergence problems when the solutions lie on the boundary of the parameter space, such as when variance components are zero (Karl, Yang, & Lohr, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' However, after making modifications to the default PROC MIXED settings, we were reliably able to achieve convergence of the full model with the joint optimization of (𝛽, 𝑮, 𝑹) in a single call to PROC MIXED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' In this regard, our approach differs from that of Aldor-Noiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' (2009): we fit all of the model parameters jointly (with a single call to PROC MIXED).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' This will lead to reduced bias in the estimates for the models that do converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' We improved convergence rates by changing the convergence criterion used by SAS PROC MIXED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' By default, SAS ensures that the sum of squared parameter gradients (weighted by the current Hessian of the parameter estimates) is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' However, in the presence of strong correlations in the doubly stochastic model, the parameter estimates may lie near the boundary of the parameter space, meaning the gradients may not approach 0 with convergence (Demidenko, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' As an alternative, we declare convergence when the relative change in the loglikelihood between iterations is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Additionally, we employ Fisher scoring during the estimation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Fisher scoring is more stable for models with complex covariance structures and can lead to better estimates of the asymptotic covariance (Demidenko, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Finally, since our application only uses the call volume point estimates and not the associated standard errors or tests of significance, we specify ddfm=residual to avoid spending substantial time calculating appropriate degrees of freedom for the approximate F-tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' If confidence or prediction intervals are needed, this value should be set to ddfm=kenwardrodger2 in order to calculate Satterthwaite approximations for the degrees of freedom and to apply the Kenward-Rodger correction (Kenward & Roger, 2009) to the standard errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The code for our modified approach appears in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Figure 1 Modified SAS code for the Doubly Stochastic Model The square root transformation is applied to reduce the right skew in the observed call volumes, and to stabilize the variance of the observations since quantities such as call volumes tend to follow a Poisson distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The approach in Figure 1 employs a normal approximation of this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' We experimented with fitting a mixed Poisson regression to the untransformed call volumes (via PROC GLIMMIX), but found that the run times became unfeasibly long (even when using the default pseudolikelihood approach and avoiding integral approximation) with no noticeable improvement in error rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Choosing Recurrent Neural Network Configurations with a Designed Experiment Generally, neural networks consist of layers of weights and nonlinear activation functions that are used to relate inputs (predictors) to outputs (targets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Outputs from each layer are passed sequentially to the next layer as an input vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The complexity of each layer is determined by the length of the output vector (number of nodes) it produces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' A loss function is used to compare the output of the final layer of the neural network to the provided targets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' call volumes), and an optimizer function provides updated values of the weights each node that will decrease the resulting loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The “depth” of the model is controlled by the number of layers that are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' This “depth” is the source of the phrase “deep learning”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' For example, in image processing applications with convolutional neural networks, the different layers can be shown to represent different levels of granularity of detail in an image (Chollet & Allaire, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Besides the number of layers and the number of nodes per layer, there are a number of choices that must be made regarding the properties of the optimizer, the distribution of the random initialization of the parameter weights, and the shape of the activation function(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' In a traditional, densely connected network, the individual observations are assumed to be independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' A simple example using output from JMP Pro 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='1 helps to illustrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Suppose we want to fit a densely connected neural network to predict the standardized call count using only the previous day’s standardized call count at the same period (the lag-32 of the call count, since there are 32 periods per day in the example) as a predictor with one node in one layer, using a hyperbolic tangent activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' This network shown in Figure 2 with resulting weights shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' proc mixed data=training_data scoring=50 maxiter=150 maxfunc=10000 convf=1E-6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' class day_of_week period day_num split p_group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' by split;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' /* The fixed effects */ model transf_call_count=day_of_week*period p_group*holiday_ind/ noint ddfm=residual outp=pred_call_count_output notest;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' /* The day-level random effects */ /* Note: day num copy is not included in the clAss statment and is numeric * random day_num / type=sp(pow)(day_num_copy);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' /* The period-level correlated residuals */ run;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Figure 2 Densely connected neural network with one node in one layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Figure 3 Fitted weights from the network with one node in one layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Suppose that the lag-32 standardized call count is equal to 1 at a given period, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Then the neural network predicts a standardized call volume of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='6354 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='3923 ∗ TanH(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='5 ∗ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='4046 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='5323 ∗ 1)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='847 for the current period, t, where TanH is the hyperbolic tangent function and the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='5 parameter is a fixed value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The nonlinear activation function provides the network with the flexibility to model nonlinear relationships between the inputs and the response, as well as interactions between the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' We next consider a slightly more complex network with two layers using 2 nodes in the first layer and 1 node in the second layer (Figure 4) with parameter estimates shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Figure 4 A densely connected neural network with 2 layers using two nodes in the first layer and one node in the second layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Lag[Standardize[cnt_ call], 32] Standardize[cnt_call]Estimates Parameter Estimate H1_ 1:Lag[Standardize[cnt_ call], 32] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='5323 H1_1:Intercept 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='4046 Standardize[cnt call]_ 1:H1_ 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='3923 Standardize[cnt call]_2:Intercept 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='6354Lag[Standardize[cnt_ call], 32] Standardize[cnt_call Figure 5 Fitted weights from the network with two layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Again assuming that the lag-32 standardized call count is 1, the predicted value is −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='82 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='9517 ∗ 𝑇𝑎𝑛𝐻 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='5 ∗ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='2416 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='7382 ∗ 𝑇𝑎𝑛𝐻(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='5 ∗ (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='3633 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='8785 ∗ 1)) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='3057 ∗ 𝑇𝑎𝑛𝐻(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='5 ∗ (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='0824 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='4124 ∗ 1)))) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='843 In the densely connected network, each observation is processed independently and there is no “memory” of what happened in the previously processed observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' In time series applications, however, there is a temporal ordering that the data are recorded in, and there may be correlation between nearby observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' For example, a spike or drop in call volume might persist over several periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' To address this potential, recurrent neural networks record information when fitting each observation that is then provided as a model input when fitting later observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' In a simple (Elman) RNN layer, the output from each node (the output of the TanH functions, referred to as the “state”) is recorded and stored, and used as an input for the same node when processing the next observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Note that there is one state recorded for each node in the layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' For the single layer network example (Figure 3), the state was calculated as 𝑠𝑡 = TanH(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='5 ∗ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='4046 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='5323 ∗ 1)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='44 when the lagged standardized call volume was equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' A simple RNN learns an extra parameter (say, u) to act as a coefficient for the stored state, and the activation function TanH(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='5 ∗ (𝑤0 + 𝑤1 ∗ 𝑋𝑡)) that is used by the dense network would be replaced by 𝑠𝑡 = TanH(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='5 ∗ (𝑤0 + 𝑤1 ∗ 𝑋𝑡 + 𝑢 ∗ 𝑠𝑡−1)) in order to fit a simple RNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The LSTM and GRU RNNs also use the recorded state when making predictions for the current time period, along with products of additional activation functions that are designed to carry state information further in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Details of these additional structures are explained in Section 3 of Bianchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' For our purpose it is sufficient to note that the GRU network is extremely similar to the LSTM, albeit less complex due to the ommison of a group of paramters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Chollet & Allaire (2018) remark that Google Translate currently runs using an LSTM with seven large layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' It is not clear a priori which of these four neural networks is most appropriate for a particular call center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Furthermore, it is possible that each of these networks may have a different optimal depth and structure when applied to the call center data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' A designed experiment is run to identify the optimal model type and structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Estimates Parameter Estimate H2_1:Lag[Standardize[cnt _ call], 32] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='8785 H2_1:lntercept 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='3633 H2_2:Lag[Standardize[cnt_call], 32] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='4124 H2_2:Intercept 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='0824 H1_1:H2_1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='7382 H1_1:H2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='3057 H1_1:lntercept 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='2416 Standardize[cnt_ call]_1:H1_1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='9517 Standardize[cnt_ call]_2:Intercept 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='82003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='1 Data for the Experiment We analyze call volumes aggregated into 30 minute periods from 3 different large-volume skills in an operational call center for the months of March-June 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' All of the models under consideration are used to forecast next-day call volumes using 5 weeks of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Each day contains 32 30-minute periods during which the call center is operating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' This application considers the Monday through Friday behavior of the call skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Due to the use of the one-week lagged observations as a predictor in the neural networks, the first week of training data is not included in the predictor matrix (since the prior week’s call volumes are unknown), meaning each training data set consists of 4*5*32=640 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The designed experiment in this section will evaluate the methods using a holdout period of the five one-day ahead predictions during last week in June using the three largest skills from the call center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Section 4 will then fit a reduced set of models over all skills for 60 day-ahead predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='2 Neural Network Input Factors Each network makes use of the one-hot encoding (via a binary indicator matrix) of the day of week and the one-hot encoding of period of day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Furthermore, to capture the day- and week-long correlations, the networks are also fed the call volumes for the same period in the previous day when modeling the current day, as well as the call volumes for the same period on the same day of last week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' One-period lagged call volumes are not included, as this is the purpose of the within-sequence memory of the RNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Other inputs include a binary indicator for whether the current day is a holiday, a binary indicator for whether yesterday was a holiday, a binary indicator for whether or not last week’s observation was recorded on a holiday, and day number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The day number is a continuous counter for the number of the given day in the data set, which would potentially allow the neural network to detect trends across time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' All told, these account for 42 vectors of input for the neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' There is no need to create indicator columns for interactions between day of week and period (or any other factors) as the neural network will automatically detect them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Bianchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' (2017) application of hourly call volumes displays strong lag-24 correlation, representing a period-of-day effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' They remove this seasonality by differencing the data at lag-24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' By contrast, we do not difference the call volumes, but instead include period- of-day (along with day-of-week) as exogenous variables and allow the neural network to detect this seasonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' This approach allows the network to detect the expected interactions between period-of-day and day-of-week, as well as any other input factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' While these input vectors are included for all models, there are three final input vectors whose (joint) inclusion is treated as an experimental factor: the same-period predictions from the mixed model approach (Aldor-Noiman, Feigin, & Mandelbaum, Workload forecasting for a call center: Methodology and a case study, 2009), from a Winters smoothing model, and from a seasonal ARIMA(1, 0, 1)(0, 1, 1)160 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The inclusion of the predictions from these models as an input to the neural network is an original approach that gives the network the opportunity to form predictions that may be thought of as corrections to those from these traditional models, based on potential interactions with other included factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' For brevity, we refer to this as the mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='cheat option, since it allows the neural networks to “cheat” by looking at the predictions generated by these other three models when forming its own predictions for the same time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' If none of the other 42 input vectors were included with these three, this would represent a supervised learning approach to forming a dynamically weighted average of these three model predictions in order to create a single “bagged” prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='3 Network Configuration Aspects Treated as Experimental Factors The designed experiment considered five different factors of structural settings of the neural networks: model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='type {dense, simple (Elman) RNN, GRU, LSTM}, nlayers {1, 2}, nnodes (per layer) {25, 50, 75, 100}, kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='reg {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='0001} and mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='cheat {FALSE, TRUE}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The L2 regularizer adds kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='reg times each weight coefficient to the total loss function for the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Similar to Lasso regression, this helps bound the magnitude of the model coefficients and could potentially help prevent overfitting the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' We fit a full factorial design (requiring 128 runs) for these five factors, which allows us to test for the presence of up to five-way interactions between the five factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Figure 6 shows the first 6 runs of the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The design is replicated over the 3 largest splits and across the 5 subsequent one-day ahead forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' This produces a total of 1920 runs for the entire experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The replication allows for behavior to be averaged over different days and for a more detailed exploration of how the variance of the error rates depends on each factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Figure 6 First 6 runs of the designed experiment 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='4 Static Considerations for Neural Network Configuration The input for the classical dense neural network is a 640x42 matrix (640x45 if mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='cheat=TRUE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The dense network does not consider the temporal ordering of the 640 observations (outside of the explicit inclusion of the day- and week-lagged observations as inputs): the observations are shuffled after each epoch and then processed in batches (we used a batch size of 32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' By contrast, the RNNs consider the ordering of the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The input for the RNNs is a 20x32x42 array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' This indicates to the RNN that there are 20 batches (days) of 32 timesteps (periods) with 42 predictors per time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' By default, the batches are treated independently and the timesteps within each batch are potentially correlated (via the persistence of the states in the RNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' If the batches themselves are presented in a temporal order (as is the case in our application), then this can be indicated to the Keras model via the STATEFUL=TRUE option and by disabling the shuffling of batches during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' This retains the model weights from batch-to-batch (day-to-day) to allow for possible long-term behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' However, we found that using the STATEFUL option led to a failure to converge in some skill*day combinations and the resulting validation error rates were not significantly different from those generated without the STATEFUL option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' This seems to indicate that the dependence on prior days’ behavior is already captured by the inclusion of the one-day and one-week lagged observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Due to the occasional convergence issues, the results in the sequel are generated with STATEFUL=FALSE (and batch shuffling enabled during training).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' It would have also been possible to fit the model with week- or month-long batches by training the RNN on a 4x160x42 or a 1x640x42 array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' We did not consider the week-long batches, but the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='type nlayers mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='cheat nnodes kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='reg layer_simple_rnn 2 FALSE 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='0001 layer_ gru FALSE 25 0 layer_Istm 1 TRUE 100 0 layer_gru TRUE 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='0001 layer_gru TRUE 75 0 layer_simple_rnn FALSE 100 0month-long batches tended to produce inferior predictions to the day-long batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' This could possibly be due to the lack of long term correlations in the data and the fact that the smaller batches allow for the shuffling of the ordering that the days are fed through the gradient-optimization routine of the neural network, which can improve the model fit by preventing the model from overweighting the first observations that are provided to the network in each epoch (Chollet & Allaire, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Note that the model weights are reset and the model is retrained for each skill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' This is in contrast to the approach taken by Zhu & Laptev (2017) in which a single network is fit to accommodate disparate behavior from different cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' As discussed in Section 5, a single multi-output network could potentially be built to model all of the skills at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' While they were not included as experimental factors in this application, we also noticed a significant relationship between the quality of the predictions and the optimization routine employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Extensive pilot experimentation led to our use of the AMSGrad variant (Reddi, Kale, & Kumar, 2018) of the Adam optimizer (Kingma & Ba, 2014), with a learning rate decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' We would recommend including both the optimizer and the optional learning rate decay as factors in the designed experiment for parameter tuning in future problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' We found it was important to tune the number of epochs for each model fit using validation data (the last week in the training set) by first fitting 500 epochs for each model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' taking a moving average (with a window size of 10 epochs) of the resulting WAPEs on the validation data,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' and then refitting the model (on both the training and the validation data in order to predict an additional day which was held out as a test set) with the number of epochs that produced the minimum validation WAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The moving average is important due to the volatile and non-monotonic behavior we observed in the individual recorded WAPEs and helps to find a relatively stable region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Consistent with previous findings (Bianchi, Maiorino, Kampffmeyer, Rizzi, & Jenssen, 2017), we noticed that the RNNs take many more epochs to converge than the dense neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Other researchers (such as Bianchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=') have used more than 500 epochs when fitting RNNs, so this upper bound should also be considered as an important factor when building an RNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' While we also experimented with recurrent dropout (Gal & Ghahramani, 2015) to prevent overfitting, it led to degraded performance in the early iterations of our experiment and we removed it from consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' However, this could simply be due to features of this particular application, such as the relatively short training period of five weeks: Chollet & Allaire (2018) strongly advocate the use of dropout and recurrent dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The models experienced improved errror rates after switching the kernel initializer for the random weights to the He normal initializer (He, Zhang, Ren, & Sun, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' We used the relu activation function exclusively, although this choice could also impact the quality of the resulting model fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' And while our application did not detect any two-factor interactions among the five experimental factors, it is possible that some of these additional factors could depend on the type of (R)NN being used, meaning there would be an interaction between these factors and model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' A final contributing factor is the batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' This determines how many observations are processed before the gradients are updated: when fitting Keras models on a GPU, amount of memory available on the GPU can be a limiting factor on the batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' This choice is more constrained in the RNNs, where each batch is a single day/week/month (determined by the number of timesteps specified in the input array to the RNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' By contrast, the batch size for a dense network can be set between 1 and the number of observations in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' We noticed significant differences in the error rates from the dense model depending on what batch size was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='5 Experimental Response While mean squared error (MSE) is frequently used to evaluate and compare predictive models, this is a poor metric for the call center application as it will give undue focus to the low call volume periods at the beginning and end of each day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=" Instead, the weighted absolute percentage error (WAPE) is recommended for call volume modeling (Ibrahim, Ye, L'Ecuyer, & Shen, 2016)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' This weights the absolute percentage error in each period by the number of calls received in that period and is defined by 𝑊𝐴𝑃𝐸 = ∑ |𝑌𝑖 − 𝑌̂𝑖| 𝑛 𝑖=1 ∑ 𝑌𝑖 𝑛 𝑖=1 where 𝑌𝑖 and 𝑌̂𝑖 are the observed and predicted volumes, respectively, for each 30 minute period 𝑖 = 1, … , 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Because WAPE is our metric of interest, the models are compiled to use a mean absolute error loss-function, which minimizes the numerator of the WAPE (the denominator is static).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' For each row of the experimental table (Figure 6), the specified neural network is fit (independently) to model five subsequent days of each split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' That is, day 1 is predicted using the previous 5 weeks leading up to day 1, then the model is reset and day 2 is predicted using the 5 weeks leading up to day 2, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The vector of five one-day ahead predictions is compared with the observed call volumes (that were not visible to the model during training), and the resulting WAPE is recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='6 Analysis of Experimental Results Figure 7 gives a typical output of the prediction error across the different formulations of the neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' GRU often has the lowest forecast error or at least is consistently close to the lowest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The other procedures tend to have much more unstable performance based on the choice of nodes and layers as well as across the days and splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Figure 7 Forecast error by model type and settings for forecast day 5 split 3 Regression analysis is used to determine the statistically significant factors and interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' In order to control a heavy right-skew in the recorded WAPEs, an inverse model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='type NN Classic RNN GRU RNN LSTM RNN Simple WAPE WAPE WAPE WAPE nlayers nnodes Mean Mean Mean Mean 1 25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='7% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='2% 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='5% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='0% 50 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='1% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='2% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='5% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='7% 75 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='1% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='8% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='8% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='9% 100 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='0% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='4% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='9% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='8% 2 25 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='5% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='6% 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='1% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='9% 50 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='3% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='8% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='9% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='0% 75 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='1% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='0% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='6% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='5% 100 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='0% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='9% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='5% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='0%transformation is applied to use as the response in the regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Figure 12 displays the original skewness and the transformation to normality after taking the inverse of the WAPEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='7 Loglinear Variance Regression Model The mean structure of 1/WAPE is modeled by including the main effects of the five experimental factors (model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='type, nlayers, mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='cheat, nnodes, kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='reg) and all of the interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' In addition, split, file (which represents the validation day), and split*file are included as blocking factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' While an ordinary regression model for the full factorial design would assume the same error variance for all responses across the design space, the loglinear variance model allows for the error variance itself to be modelled as a function of the input factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' This is appealing for this application, where parsimonious networks may be expected to demonstrate less variability during repeated fittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' For each factor setting, the loglinear variance model produces a predicted mean of 1/WAPE and a predicted standard deviation of 1/WAPE, representing the variability due to the random initial weighting of the NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' We include model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='type, nlayers, mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='cheat, nnodes, kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='reg, file, and split as factors in the loglinear variance model (using JMP Pro 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='1), which models the log of the error variance as a linear combination of these factors (which are all treated as categorical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' With the exception of kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='reg, all of these effects are found to be significantly associated with the error variance (Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Figure 8 Factors with a significant impact on WAPE variability For the mean model, model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='type, nlayers, mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='cheat, nnodes, day, split, and file*split were found to be significantly associated with 1/WAPE (Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Neither kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='reg nor any of the interaction terms involving the NN architecture were found to be significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Variance Effect Likelihood Ratio Tests Source Test Type DF ChiSquare Prob>ChiSq model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='type Likelihood 3 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='0001* split Likelihood 2 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='3123 <.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='0001* Figure 9 Factors that are significantly associated with the mean WAPE Removing the insignificant interaction terms (but allowing kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='reg to remain in both the mean and variance models) produces the profiler shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Notice the large standard deviation associated with the LSTM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The dependence on file and split is not shown, since there are no interactions modeled between these and the other experimental factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' While we want to maximize 1/WAPE, we also want to minimize the standard deviation of 1/WAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' That is, we would like to maximize the lower 95% prediction interval of 1/WAPE (or minimize the upper 95% PI of WAPE) in order to find the model configuration that will give a combination 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='0001 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='type nlayers mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='cheat nnodes kenel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='reglayers, using mixed model forecasts, with 50 nodes while robust to the regularization parameter value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Note that the traditional dense neural network does have consistently worse forecast error across all scenarios and could not be recommended despite having the lowest variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The prediction interval on forecast error is an alternative view that is preferred by many practitioners where the goal is to minimize the width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Figure 11 (lower is better) clearly shows GRU is the preferred solution using the mixed model forecasts with 50 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Figure 12 Representative distribution of WAPE and 1/WAPE resulting from the designed experiment 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Comprehensive Performance Study Based on the results of the designed experiment for NN error rates, we perform a further study with actual call center data across 36 skills rather than only 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' We consider a consistent 5-week training period advancing across multiple months of data to produce 60 one-day ahead predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' These predictions do not use the actual data for the forecasted day during model training and can be viewed as a validation set for the trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' This allows us to compare the relative performance of the mixed model and the neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' We also include the error rates from the ARIMA and Winters seasonal smoothing models for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Though we could have simply chosen the GRU RNN with 50 nodes on each of 2 layers with kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='reg complexity parameter set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='0001 as the only representative RNN based on the designed experiment, we decided to include all 4 neural network methods Distributions WAPE 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='5 1/WAPE 0 5 10 15 20 25 30screened by using these same parameter settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' It is possible with the added skills – with many having much lower call volumes – that another RNN model could work better than GRU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='1 Results: One-day ahead Predictions Over 60 Separate Validation Days Figure 13 provides the forecast errors averaged across all 60 days for the 12 splits with the highest call volume sorted in descending call volume order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Note that results in this section are generated by the (R)NNs with the mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='cheat option disabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The Doubly Stochastic Mixed Model has the lowest average forecast error for all but Split 5540 and is often significantly lower than the competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Winters Seasonal Exponential Smoothing also performs quite well given its relative simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The GRU performance confirms the results from the designed experiment as usually the best RNN and always competitive with the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The highly complex LSTM recurrent neural network has very large forecast errors for many of these splits and cannot be recommended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Note also that forecast error generally increases for all methods as call volume decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Figure 13 Average WAPE forecast errors across 60 separate validation days for high call-volume splits Figure 14 shows the similar trend of increasing error rates with decreasing call volumes for the medium call volume splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The GRU recurrent neural network is usually outperforming the other neural network methods and is closer to the error rates of the Doubly Stochastic Mixed Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Figure 14 Average WAPE forecast errors across 60 separate validation days for medium call-volume splits For the low call volume splits displayed in Figure 15, the GRU and Simple recurrent neural networks perform similarly and slightly better than Doubly Stochastic and Winters for most of the splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The very low call volumes (last 3 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='1% 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='5% 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='8% Figure 15 Average WAPE forecast errors across 60 separate validation days for low call-volume splits Overall, the best performing procedures were the Doubly Stochastic Mixed Model and the GRU Recurrent Neural Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Figure 16 shows forecast error by split sorted by call volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Generally, the mixed model does better for large and medium volume splits (lower is better on the graph) while the RNN model is more effective for the small volume—particularly the very small volume splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Figure 16 Forecast error rates ordered by call volume by split for Doubly Stochastic (blue) and GRU (red) Figure 17 presents a different view of this same pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' For each of the 60 one-day ahead forecasts within each split, the GRU and Doubly Stochastic WAPEs are recorded, along with the number of calls recorded for that split over the training and validation data (sum_all_calls).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' For each split, Figure 17 plots the percent of the 60 day-ahead forecasts for which GRU “won” (GRU WAPE < Doubly Stochastic WAPE) against the log of the median call volume recorded by that split over the 60 different pairs of training and validation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' There appears to be a linear decrease of the relative performance of GRU against Doubly Stochastic in the log of the call volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Split Sum Call Vol ARIMA Doubly Stoch NN_Classic RNN_GRU RNN_LSTM RNN_Simple Winters 5710 3742 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='6% 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='8% 40.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='2% 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='7% 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='9% 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='8% 187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='7%Forecast ErrorbySplit AverageOver 6oDays 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='6 IDoubly Stoch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='5 RNN GRU 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='1 - Errot aber 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='7 pa ubiay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='1 HighCallVolum Medium CallVolume Low Call Volume Figure 17 Percent of the 60 day-ahead forecasts for which the GRU WAPE was less than the Doubly Stochastic WAPE for each of the 36 splits plotted against the log of the median sum of all calls recorded for each split over each pair of training and validation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' For the large call-volume splits, the extra flexibility of the GRU model does not lead to improvements over the predictions generated by the mixed model approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Because the mixed model computations are faster and the implementation is less complex, there does not appear to be any benefit to running the neural networks for the high-volume splits for this short-term application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' This is consistent with the findings of the Uber traffic volume study when short-term predictions were considered (Zhu & Laptev, 2017), and would possibly change if a longer training period (several months or multiple years) were used for the call center data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='2 Improving GRU RNN by Using Doubly Stochastic Forecasts as a Covariate Based on pilot studies during the designed experiment, the forecasting performance of the GRU recurrent neural network often improved by integrating the forecasted value for the validation days from the doubly stochastic, ARIMA, and Winters models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' This “cheating” by using other models’ forecasts (shown as mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='cheat) proved to be a significant benefit for the GRU model over these 60 predictions for each skill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The WAPE for the held-out validation data was again the primary measure of performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Figure 18 displays the forecast errors for the high call volume skills averaged over the 60 validation days for the each of the neural networks when they use the mixed forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Note the side-by-side comparison of the RNN_GRU (no cheat) and GRU_cheat columns where in most cases the “cheating” does result in improved forecasts and in those cases where it is not better, it has only marginally declined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Additionally, the Simple_cheat error rates compare favorably with the GRU_cheat while the LSTM_cheat suffers from significantly poorer performance and instability issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' The doubly stochastic forecast is still quite good and often the best choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' These results are also consistent when looking at the medium and low call volume splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Therefore, we recommend using the forecasts from a doubly stochastic or Winters model as inputs to recurrent neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Percent won by GRU vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Log[Median(sum all calls)] 100% Percent won by GRU %06 %08 70% Percent won by GRU %09 50% 40% 30% 20% 10% 0% 4 6 8 10 12 14 Log[Median(sum all_ calls)] Figure 18 Forecast errors for high call volume splits averaged over 60 validation forecast days allowing NN to “cheat” to improve predictions The median of the GRU WAPE (mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='cheat=FALSE) minus the GRU WAPE (mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='cheat=TRUE) (within each split/day combination, for a sample size of 36*60=2160) is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='002 with a p-value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='0005 from the Wilcoxon signed rank test with a null hypothesis that the differences were drawn from a population with median equal to 0, indicating that mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='cheat does tend to improve the performance of the GRU model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' A similar comparison of paired differences of error rates (all with mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='cheat=FALSE) confirms that GRU outperforms the other (R)NNs: LSTM - GRU produces a median of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='0247 with a p-value of 1e-129, NN Classic - GRU produces a median of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='0690 with a p-value of <1e-185, and Simple RNN - GRU produces a median of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='0021 with a p-value of 1e-04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' References Aldor-Noiman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=', Feigin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=', & Mandelbaum, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Workload forecasting for a call center: Methodology and a case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Annals of Applied Statistics, 3(4), 1403-1447.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Aldor-Noiman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=', Feigin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=', & Mandelbaum, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Workload forecasting for a call center: Methodology and a case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Annals of Applied Statistics(4), 1403– 1447.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Allaire, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=', & Chollet, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=" keras: R Interface to 'Keras'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Retrieved from CRAN: https://CRAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='R-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='org/package=keras Bianchi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=', Maiorino, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=', Kampffmeyer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=', Rizzi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=', & Jenssen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Recurrent Neural Networks for Short-Term Load Forecasting: An Overview and Comparative Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Cham: Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Bianchi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=', Maiorino, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=', Kampffmeyer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=', Rizzi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=', & Jenssen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' An overview and comparative analysis of Recurrent Neural Networks for Short Term Load Forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Retrieved from https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='org/abs/1705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='04378 Box, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=', & Jenkins, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' (1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Time series analysis: Forecasting and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' San Francisco, CA: Holden-Day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content=' Split Sum Call Vol Doubly Stoch RNN_GRU GRU_cheat LSTM_cheat Simple_cheat 5000&5240 926493 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='7% 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='2% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='8% 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='4% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'} +page_content='8% 5400&5570 459961 6.' metadata={'source': 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Applied Sciences, +Germany +wolfgang.konen@th-koeln.de +Sep 2022, +last update Jan 2023 +Abstract +This work describes in detail how to learn and solve the Rubik’s cube game (or +puzzle) in the General Board Game (GBG) learning and playing framework. We cover +the cube sizes 2x2x2 and 3x3x3. We describe in detail the cube’s state representation, +how to transform it with twists, whole-cube rotations and color transformations and +explain the use of symmetries in Rubik’s cube. +Next, we discuss different n-tuple +representations for the cube, how we train the agents by reinforcement learning and +how we improve the trained agents during evaluation by MCTS wrapping. +We present results for agents that learn Rubik’s cube from scratch, with and without +MCTS wrapping, with and without symmetries and show that both, MCTS wrapping +and symmetries, increase computational costs, but lead at the same time to much +better results. We can solve the 2x2x2 cube completely, and the 3x3x3 cube in the +majority of the cases for scrambled cubes up to p = 15 (QTM). We cannot yet reliably +solve 3x3x3 cubes with more than 15 scrambling twists. +Although our computational costs are higher with MCTS wrapping and with sym- +metries than without, they are still considerably lower than in the approaches of McAleer +et al. (2018, 2019) and Agostinelli et al. (2019) who provide the best Rubik’s cube +learning agents so far. +1 +arXiv:2301.12167v1 [cs.LG] 28 Jan 2023 + +Contents +1 +Introduction +4 +1.1 +Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +1.2 +Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +2 +Foundations +6 +2.1 +Conventions and Symbols +. . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +2.1.1 +Color arrangement +. . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +2.1.2 +Twist and Rotation Symbols . . . . . . . . . . . . . . . . . . . . . . . +6 +2.1.3 +Twist Types +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +2.2 +Facts about Cubes +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +2.2.1 +2x2x2 Cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +2.2.2 +3x3x3 Cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +2.3 +The Cube State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +2.4 +Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +9 +2.4.1 +Twist Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . +9 +2.4.2 +Whole-Cube Rotations (WCR) . . . . . . . . . . . . . . . . . . . . . +11 +2.4.3 +Color Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +2.5 +Symmetries +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +3 +N-Tuple Systems +16 +4 +N-Tuple Representions for the Cube +18 +4.1 +CUBESTATE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +18 +4.2 +STICKER +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +18 +4.3 +STICKER2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +20 +4.4 +Adjacency Sets +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +21 +5 +Learning the Cube +21 +5.1 +McAleer and Agostinelli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +21 +5.2 +N-Tuple-based TD Learning +. . . . . . . . . . . . . . . . . . . . . . . . . . +24 +5.2.1 +Temporal Coherence Learning (TCL) . . . . . . . . . . . . . . . . . . +25 +5.2.2 +MCTS +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +25 +5.2.3 +Method Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . +27 +6 +Results +27 +6.1 +Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +27 +6.2 +Cube Solving with MCTS Wrapper, without Symmetries +. . . . . . . . . . . +28 +6.3 +Number of Symmetric States . . . . . . . . . . . . . . . . . . . . . . . . . . +28 +6.4 +The Benefit of Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . +29 +6.5 +Computational Costs +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +30 +7 +Related Work +32 +2 + +8 +Summary and Outlook +33 +A Calculating sloc from fcol +37 +A.1 2x2x2 cube +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +37 +A.2 3x3x3 cube +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +38 +B N-Tuple Representations for the 3x3x3 Cube +38 +B.1 +CUBESTATE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +38 +B.2 +STICKER +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +39 +B.3 +STICKER2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +39 +B.4 +Adjacency Sets +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +40 +C Hyperparameters +40 +3 + +1 +Introduction +1.1 +Motivation +Game learning and game playing is an interesting test bed for strategic decision making. +Games usually have large state spaces, and they often require complex pattern recognition +and strategic planning capabilities to decide which move is the best in a certain situation. +If algorithms learn a game (or, even better, a variety of different games) just by self-play, +given no other knowledge than the game rules, it is likely that they perform also well on +other problems of strategic decision making. +In recent years, reinforcement learning (RL) and deep neural networks (DNN) achieved +superhuman capabilities in a number of competitive games (Mnih et al., 2015; Silver et al., +2016). This success has been a product of the combination of reinforcement learning, +deep learning and Monte Carlo Tree Search (MCTS). However, current deep reinforcement +learning (DRL) methods struggle in environments with a high number of states and a small +number of reward states. +(a) +(b) +Figure 1: (a) Scrambled 3x3x3 Rubik’s Cube. (b) 2x2x2 cube in the middle of a twist. +The Rubik’s cube puzzle is an example of such an environment since the classical +3x3x3 cube has 4.3 · 1019 states and only one state (the solved cube) has a reward. A +somewhat simpler puzzle is the 2x2x2 cube with 3.6 · 106 state and again only one reward +state. Both cubes are shown in Fig. 1. +The difficult task to learn from scratch how to solve arbitrary scrambled cubes (i.e. +without being taught by expert knowledge, whether from humans or from computerized +solvers) was not achievable with DRL methods for a long time. Recently, the works of +McAleer et al. (2018, 2019) and Agostinelli et al. (2019) provided a breakthrough in that +direction (see Sec. 5.1 and 7 for details): Their approach DAVI (Deep Approximate Value +Iteration) learned from scratch to solve arbitrary scrambled 3x3x3 cubes. +This work investigates whether TD-n-tuple learning with much lower computational de- +mands can solve (or partially solve) Rubik’s cube as well. +4 + +1.2 +Overview +The General Board Game (GBG) learning and playing framework (Konen, 2019; Konen and +Bagheri, 2020; Konen, 2022) was developed for education and research in AI. GBG allows +applying the new algorithm easily to a variety of games. GBG is open source and available +on GitHub1. The main contribution of this paper is to take the TD-n-tuple approach from +GBG (Scheiermann and Konen, 2022) that was also successful on other games (Othello, +ConnectFour) and to investigate this algorithm on various cube puzzles. We will show that +it can solve the 2x2x2 cube perfectly and the 3x3x3 cube partly. At the same time it has +drastically reduced computational requirements compared to McAleer et al. (2019). We +will show that wrapping the base agent with an MCTS wrapper, as it was done by McAleer +et al. (2019) and Scheiermann and Konen (2022), is essential to reach this success. +This work is at the same time an in-depth tutorial how to represent a cube and its +transformations within a computer program such that all types of cube operations can be +computed efficiently. As another important contribution we will show how symmetries +(Sec. 2.5, 6.3 and 6.4) applied to cube puzzles can greatly increase sample efficiency and +performance. +The rest of this paper is organized as follows: Sec. 2 lays the foundation for Rubik’s +cube, its state representation, its transformations and its symmetries. In Sec. 3 we in- +troduce n-tuple systems and how they can be used to derive policies for game-playing +agents. Sec. 4 defines and discusses several n-tuple representations for the cube. Sec. 5 +presents algorithms for learning the cube: first the DAVI algorithm of McAleer et al. (2019); +Agostinelli et al. (2019) and then our n-tuple-based TD learning (with extensions TCL and +MCTS). In Sec. 6 we present the results when applying our n-tuple-based TD learning +method to the 2x2x2 and the 3x3x3 cube. Sec. 7 discusses related work and Sec. 8 con- +cludes. +1https://github.com/WolfgangKonen/GBG +5 + +2 +Foundations +2.1 +Conventions and Symbols +We consider in this paper two well-known cube types, namely the 2x2x2 cube (pocket +cube) and the 3x3x3 cube (Rubik’s cube). +2.1.1 +Color arrangement +Each cube consists of smaller cubies: 8 corner cubies for the 2x2x2 cube and 8 corner, +12 edge and 6 center cubies for the 3x3x3 cube. A corner cubie has 3 stickers of different +color on its 3 faces. An edge cubie has two, a center cubie has one sticker. +We enumerate the 6 cube faces with +(ULF) = (Up, Left, Front) and +(DRB) = (Down, Right, Back). +We number the 6 colors with 0,1,2,3,4,5. My cube has these six colors +012 = wbo = (white,blue,orange) in the (ULF)-cubie2 and +345 = ygr = (yellow,green,red) in the opposing (DRB)-cubie. +The solved cube in default position has colors (012345) for the faces (ULFDRB), i.e. the +white color is at the Up face, blue at Left, orange as Front and so on. We can cut the cube +such that up- and bottom-face can be folded away and have a flattened representation as +shown in Figure 2. +w +b +o +g +r +y +Figure 2: The face colors of the default cube in flattened representation +2.1.2 +Twist and Rotation Symbols +Twists of cube faces are denoted by uppercase letters U, L, F, D, R, B. Each of these twists +means a 90◦ counterclockwise rotation.3 If U = U1 is a 90◦ rotation, then U2 is a 180◦ +rotation and U3=U−1 is a 270◦ rotation. +Whole-cube rotations are denoted by lowercase letters u, ℓ, f. (We do not need d, r, b +here, because d = u−1, r = ℓ−1 and so on.) +Further symbols like fc[i], sℓ[i] that characterize a cube state will be explained in Sec. 2.3. +2.1.3 +Twist Types +Cube puzzles can have different twist types or twist metrics: +2We run through the faces of a cubie in counter-clockwise orientation. +3The rotation is counterclockwise when looking at the respective face +6 + +• QTM (quarter turn metric): only quarter twists are allowed: e.g. U1 and U−1. +• HTM (half turn metric): quarter and half turns (twists) are allowed: e.g. U1, U2, U3. +By allowed we mean what counts as one move. In QTM we can realize U2 via U U as +well, but it costs us 2 moves. In HTM, U2 counts as one move. +The twist type influences God’s number and the branching factor of the game, see +Sec. 2.2. +2.2 +Facts about Cubes +2.2.1 +2x2x2 Cube +The number of distinct states for the 2x2x2 pocket cube is (Wikipedia, 2022a) +8! · 37 +24 += 7! · 36 = 3, 674, 160 ≈ 3.6 · 106 +(1) +Why this formula? — We have 8 cubies which we can place in 8! ways on the 8 cube +positions. Each but the last cubie has the freedom to appear in 3 orientations, which gives +the factor 37 (the last cubie is then in a fixed orientation, the other two orientations would +yield illegal cube states). – Each of these raw states has the (ygr)-cubie in any of the +24 possible positions. Or, otherwise speaking, each truly different state appears in 24 +whole-cube rotations. To factor out the whole-cube rotations, we count only the states with +(ygr)-cubie in its default position (DRB) and divide the number of raw states by 24, q.e.d. +God’s number: What is the minimal number of moves needed to solve any cube posi- +tion? – For the 2x2x2 pocket cube, it is 11 in HTM (half-turn metric) and 14 in QTM. +Branching factor: 3 · 3 = 9 in HTM and 3 · 2 = 6 in QTM. +2.2.2 +3x3x3 Cube +The number of distinct states for the 3x3x3 Cube is (Wikipedia, 2022b) +8! · 37 · 12! · 211 +2 += 43, 252, 003, 274, 489, 856, 000 ≈ 4.3 · 1019 +(2) +Why this formula? – We have 8 corner cubies which we can place in 8! ways on the 8 +cube positions. Each but the last cubie has the freedom to appear in 3 orientations, which +gives the factor 37. We have 12 edge cubies which we can place in 12! ways on the edge +positions. Each but the last cubie has the freedom to appear in 2 orientations, which gives +the factor 211. The division by 2 stems from the fact, that neither alone two corner cubies +may be swapped nor alone two edge cubies may be swapped. Instead, the number of such +swaps must be even (factor 2). +God’s Number: What is the minimal number of moves needed to solve any cube +position? — For the 3x3x3 Rubik’s Cube, it is 20 in HTM (half-turn metric) and 26 in QTM. +This is a result from Rokicki et al. (2014), see also http://www.cube20.org/qtm/. +Branching factor: 6 · 3 = 18 in HTM and 6 · 2 = 12 in QTM. +7 + +3 +2 +0 +1 +5 +4 +8 +11 +18 +17 +23 +22 +6 +7 +9 +10 +19 +16 +20 +21 +14 +13 +15 +12 +Figure 3: Sticker numbering for the 2x2x2 cube +2.3 +The Cube State +A cube should be represented by objects in GBG in such a way that +(a) cube states that are equivalent are represented by identical objects +(b) if two cube states are equivalent, it should be easy to check this by comparing their +objects +(c) cube transformations are easy to carry out on these objects. +Condition (a) means that if two twist sequences lead to the same cube state (e.g. U−1 +and UUU), this should result also in identical objects. Condition (b) means, that the equality +should be easy to check, given the objects. That is, a cube should not be represented by +its twist sequence. +A cube state is in GBG represented by abstract class CubeState and has two describ- +ing members +fc[i] += +fcol[i] +(3) +sℓ[i] += +sloc[i] +(4) +fc[i] = fcol[i] denotes the face color at sticker location i. The color is one out of +0,1,2,3,4,5 for the colors w,b,o,y,g,r. +sℓ[i] = sloc[i] contains the sticker location of the sticker which is in position i for the +solved cube d. +Members fc and sℓ are vectors with 24 (2x2x2 cube) or 48 (3x3x3 cube) elements +where i denotes the ith sticker location. +The stickers are numbered in a certain way which is detailed in Figures 3 and 4 for the +flattened representations of the 2x2x2 and 3x3x3 cube, resp. +In principle, one of the two members fc and sℓ would be sufficient to characterize a +state, since the fcol-sloc-relation +fc[sℓ[i]] = d.fc[i] +(5) +holds, where d denotes the default cube. This is because sℓ[i] transports the sticker i +of the default cube d to location sℓ[i], i.e. it has the color d.fc[i]. That is, we can easily +8 + +6 +5 +4 +7 +3 +0 +1 +2 +10 +9 +8 +16 +23 +22 +36 +35 +34 +46 +45 +44 +11 +15 +17 +21 +37 +33 +47 +43 +12 +13 +14 +18 +19 +20 +38 +39 +32 +40 +41 +42 +28 +27 +26 +29 +25 +30 +31 +24 +Figure 4: Sticker numbering for the 3x3x3 cube. We do not number the center cubies, they +stay invariant under twists. +Table 1: The three relevant twists for the 2x2x2 cube +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 11 +12 13 14 15 +16 17 18 19 +20 21 22 23 +U twist +T +1 +2 +3 +0 +11 +8 +6 +7 +18 +9 +10 17 +12 13 14 15 +16 22 23 19 +20 21 +4 +5 +L twist +T +22 +1 +2 21 +5 +6 +7 +4 +3 +0 +10 11 +12 13 +8 +9 +16 17 18 19 +20 14 15 23 +F twist +T +7 +4 +2 +3 +14 +5 +6 13 +9 +10 11 +8 +12 18 19 15 +16 17 +0 +1 +20 21 22 23 +U−1 +T −1 +3 +0 +1 +2 +22 23 6 +7 +5 +9 +10 +4 +12 13 14 15 +16 11 +8 +19 +20 21 17 18 +L−1 +T −1 +9 +1 +2 +8 +7 +4 +5 +6 +14 15 10 11 +12 13 21 22 +16 17 18 19 +20 +3 +0 +23 +F−1 +T −1 +18 19 2 +3 +1 +5 +6 +0 +11 +8 +9 +10 +12 +7 +4 +15 +16 17 13 14 +20 21 22 23 +calculate fc given sℓ. With some more effort, it is also possible to calculate sℓ given fc (see +Appendix A). Although one of these members fc and sℓ would be sufficient, we keep both +because this allows to better perform assertions or cross checks during transformations. +Sometime we need the inverse function s−1 +ℓ [i]: Which sticker is at location i? It is easy +to calculate s−1 +ℓ +given sℓ with the help of the relation: +s−1 +ℓ [sℓ[i]] = i +(6) +(Note that it is not possible to invert fc, because the face coloring function is not bijective.) +2.4 +Transformations +2.4.1 +Twist Transformations +Each basic twist is a counterclockwise4 rotation of a face by 90◦. Table 1 shows the 2x2x2 +transformation functions for three basic twists. Each twist transformation can be coded in +two forms: +1. T[i] (forward transformation): Which is the new location for the sticker being at i +before the twist? +4The rotation is counterclockwise when looking at this face. +9 + +2 +1 +3 +0 +23 +22 +5 +4 +8 +11 +18 +17 +6 +7 +9 +10 +19 +16 +20 +21 +14 +13 +15 +12 +Figure 5: The default 2x2x2 cube after twist U1 +2. T −1[i] (inverse transformation): Which is the (parent) location of the sticker that lands +in i after the twist? +Example (read off from column 0 of Table 1): The L-twist transports sticker at 0 to 22: +T[0] = 22. The (parent) sticker being at location 9 before the L-twist comes to location 0 +after the twist: T −1[0] = 9. Likewise, for the U-twist we have T[0] = 1 and T −1[0] = 3. We +show in Fig. 5 the default cube after twist U1. +How can we apply a twist transformation to a cube state programmatically? – We +denote with f′ +c and s′ +ℓ the new states for fc and sℓ after transformation. The following +relations allow to calculate the transformed cube state: +f′ +c[i] += +fc[T −1[i]] +(7) +s′ +ℓ[s−1 +ℓ [i]] += +T[i] +(8) +Eq. (7) says: The new color for sticker 0 is the color of the sticker which moves into +location 0 (fc[9] in the case of an L-twist). To explain Eq. (8), we first note that s−1 +ℓ [i] is the +sticker being at i before the transformation. Then, Eq. (8) says: „The new location for the +sticker being at i before the transformation is T[i].“ For example, the L-twist transports the +current sticker at location 0 to the new location T[0] = 22, i. e. s′ +ℓ[0] = 22. +For the 2x2x2 cube, these 3 twists U, L, F are sufficient, because D=U−1, R=L−1, +B=F−1. This is because the 2x2x2 cube has no center cubies. For the 3x3x3 cube, we +need all 6 twists U, L, F, D, R, B because this cube has center cubies. +In any case, we will show in Sec. 2.4.2 that only one row in Table 1 or Table 2, say T +for the U-twist, has to be known or established ’by hand’. All other twists and their inverses +can be calculated programmatically with the help of Eqs. (9)-(15) that will be derived in +Sec. 2.4.2. +Table 2: The U twist for the 3x3x3 cube +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 11 +12 13 14 15 +16 17 18 19 +20 21 22 23 +U twist T +2 +3 +4 +5 +6 +7 +0 +1 +22 23 16 11 +12 13 14 15 +36 17 18 19 +20 21 34 35 +24 25 26 27 +28 29 30 31 +32 33 34 35 +36 37 38 39 +40 41 42 43 +44 45 46 47 +U twist T +24 25 26 27 +28 29 30 31 +32 33 44 45 +46 37 38 39 +40 41 42 43 +8 +9 +10 47 +10 + +Table 3: Two basic whole-cube rotations for the 2x2x2 cube +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 11 +12 13 14 15 +16 17 18 19 +20 21 22 23 +u rotation +T +1 +2 +3 +0 +11 +8 +9 +10 +18 19 16 17 +15 12 13 14 +21 22 23 20 +6 +7 +4 +5 +f rotation +T +7 +4 +5 +6 +14 15 12 13 +9 +10 11 +8 +17 18 19 16 +2 +3 +0 +1 +23 20 21 22 +u−1 +T −1 +3 +0 +1 +2 +22 23 20 21 +5 +6 +7 +4 +13 14 15 12 +10 11 +8 +19 +19 16 17 18 +f−1 +T −1 +18 19 16 17 +1 +2 +3 +0 +11 +8 +9 +10 +6 +7 +4 +5 +15 12 13 14 +21 22 23 20 +Normalizing the 2x2x2 Cube +As stated above, the 3 twists U, L, F are sufficient +for the 2x2x2 cube. Therefore, the (DRB)-cubie will never leave its place, whatever the +twist sequence formed by U, L, F is. The (DRB)-cubie has the stickers (12, 16, 20), and we +can check in Table 1 that columns (12, 16, 20) are always invariant. If we have an arbitrary +initial 2x2x2 cube state, we can normalize it by applying a whole-cube rotation such that +the (ygr)-cubie moves to the (DRB)-location. +Normalizing the 3x3x3 Cube +In the case of the 3x3x3 cube, all center cubies +will be not affected by any twist sequence. Therefore, we normalize a 3x3x3 cube state +by applying initially a whole-cube rotation such that the center cubies are in their normal +position (i.e. white up, blue left and so on). +2.4.2 +Whole-Cube Rotations (WCR) +Each basic whole-cube rotation (WCR) is a counterclockwise rotation of the whole cube +around the u, l, f-axis by 90◦. Table 3 shows two of the 2x2x2 transformation functions for +basic whole-cube rotations. Each rotation can be coded in two forms: +1. T[i] (forward transformation): Which is the new location for the sticker being at i +before the twist? +2. T −1[i] (inverse transformation): Which is the (parent) location of the sticker that lands +in i after the twist? +Besides the basic rotation u there is also u2 (180◦) and u3 = u−1 (270◦ = −90◦). +All whole-cube rotations can be generated from these two forward rotations u and f: +First, we calculate the inverse transformations via +T −1[T[i]] = i +(9) +where T is a placeholder for u or f. Next, we calculate the missing base rotation ℓ (counter- +clockwise around the left face) as +ℓ = fuf−1 +(10) +We use here the programm-code-oriented notation „first trafo first“: Eq. (10) reads as +„first f, then u, then f−1“.5 +5In programm code the relation would read cs.fTr(1).uTr().fTr(3). This is „first trafo first“, because +each transformation is applied to the cube state object to the left and returns the transformed cube state object. +11 + +Table 4: All 24 whole-cube rotations (in first-trafo-first notation) +number +first rotation +∗ u0 +∗ u1 +∗ u2 +∗ u3 +00-03 +id (white up) +id +u +u2 +u3 +04-07 +f (green up) +f +fu +fu2 +fu3 +08-11 +f2 (yellow up) +f2 +f2u +f2u2 +f2u3 +12-15 +f−1 (blue up) +f−1 +f−1u +f−1u2 +f−1u3 +16-19 +ℓ (orange up) +ℓ +ℓu +ℓu2 +ℓu3 +20-23 +ℓ−1 (red up) +ℓ−1 +ℓ−1u +ℓ−1u2 +ℓ−1u3 +Table 5: The 24 inverse whole-cube rotations (in first-trafo-first notation) +number +first rotation +∗ u0 +∗ u1 +∗ u2 +∗ u3 +00-03 +id (white up) +id +u3 +u2 +u1 +04-07 +f (green up) +f−1 +ℓu3 +fu2 +ℓ−1u +08-11 +f2 (yellow up) +f2 +f2u +f2u2 +f2u3 +12-15 +f−1 (blue up) +f +ℓ−1u3 +f−1u2 +ℓu +16-19 +ℓ (orange up) +ℓ−1 +f−1u3 +ℓu2 +fu +20-23 +ℓ−1 (red up) +ℓ +fu−1 +ℓ−1u2 +f−1u +The other basic whole-cube rotations d, r, b are not needed, because d = u−1, r = ℓ−1 +and b = f−1. +The basic whole-cube rotations are rotations of the whole cube around just one axis. +But there are also composite whole-cube rotations which consists of a sequence of basic +rotations. +How many different (composite) rotations are there for the cube? – A little thought +reveals that there are 24 of them: To be specific, we consider the default cube where we +have 4 rotations with the white face up, 4 with the blue face up, and so on. In total we have +6 · 4 = 24 rotations since there are 6 faces. Table 4 lists all of them, togehter with the WCR +numbering convention used in GBG. +Sometimes we need the inverse whole-cube rotations which are given in Table 5. In +this table, we read for example from the element with number 5, that the WCR with key 5 +(which is fu according to Table 4) has the inverse WCR ℓu3 such that +fu ℓu3 = id +holds. +For convenience, we list in Table 6 the relation. For example, the +trafo with Key=5 (fu) has the inverse trafo with InverseKey=19 (ℓu3). Note that there are +10 whole-cube rotations which are their own inverse. +Generating all twists from U twist +With the help of WCRs we can generate the other +twists from the U twist only: We simply rotate the face that we want to twist to the up-face, +12 + +Table 6: Whole-cube rotations: relation +key +0 1 2 3 +4 +5 +6 +7 +8 9 10 11 +12 13 14 15 +16 17 18 19 +20 21 22 23 +inv key +0 3 2 1 +12 19 6 21 +8 9 10 11 +4 +23 14 17 +20 15 18 05 +16 +7 +22 13 +apply the U twist and rotate back. This reads in first-trafo-first notation: +L = f−1Uf +(11) +F = ℓUℓ−1 +(12) +D = f2Uf2 +(13) +R = fUf−1 +(14) +B = ℓ−1Uℓ +(15) +Thus, given the U twist from Table 1 or Table 2 and the basic WCRs given in Table 3 and +Eq. (10), we can calculate all other forward transformations with the help of Eqs. (11)–(15). +Then, all inverse transformations are calculable with the help of Eq. (9). +2.4.3 +Color Transformations +Color transformations are special transformations that allow to discover non-trivial symmet- +ric (equivalent) states. +One way to describe a color transformation is to select a valid color permutation and to +paint each sticker with the new color according to this color permutation. This is of course +nothing one can do with a real cube without destroying or altering it, but it is a theoretical +concept leading to an equivalent state. +Another way of looking at it is to record the twist sequence that leads from the default +cube to a certain scrambled cube state. Then we go back to the default cube, make at +first a whole-cube rotation (leading to a color-transformed default cube) and then apply the +recorded twist sequence to the color-transformed default cube. +In any case, the transformed cube will be usually not in its normal position, so we apply +finally a normalizing operation to it. +What are valid color permutations? – These are permutations of the cube colors reach- +able when applying one of the available 24 WCRs (Table 4) to the default cube. For exam- +ple, if we apply WCR f (number 04) to the default cube, we get +g +w +o +y +r +b +Figure 6: The color transformation according to WCR f (number 04) +that is, g (green) is the new color for each up-sticker that was w (white) before and so +on. The colors o and r remain untouched under this color permutation. [However, other +transformations like fu, fu2 and fu3 will change every color.] +13 + +2 +1 +3 +0 +23 +22 +5 +4 +8 +11 +18 +17 +6 +7 +9 +10 +19 +16 +20 +21 +14 +13 +15 +12 +Figure 7: The cube of Fig. 5 before color transformation. +16 +19 +17 +18 +20 +23 +2 +1 +11 +10 +13 +12 +3 +0 +8 +9 +14 +15 +21 +22 +4 +7 +5 +6 +8 +2 +9 +1 +4 +7 +14 +11 +18 +17 +23 +0 +5 +6 +15 +10 +19 +16 +20 +3 +21 +13 +22 +12 +(a) +(b) +Figure 8: The cube of Fig. 7 with color transformation from Fig 6: (a) before normalization, +(b) after normalization. +How can we apply a color transformation to a cube state programmatically? – We +denote with f′ and s′ +ℓ the new states for f and sℓ after transformation. +The following +relations allow to calculate the transformed cube state: +f′ +c[i] += +c[fc[i]] +(16) +s′ +ℓ[s−1 +ℓ [i]] += +T[i] +(17) +where c[] is the 6-element color trafo vector (holding the new colors for current colors 0:w, +1:b, ..., 5:r) and T is the 24- or 48-element vector of the WCR that produces this color +transformation. Eq. (16) is simple: If a certain sticker has color 0 (w, white) before the color +transformation, then it will get the new color c[0], e.g. 4 (g, green), after the transformation. +Eq. (17) looks complicated, but it has a similar meaning as in the twist trafo: Take i = 0 as +example: The new place for the sticker being at 0 before the trafo (and coming from s−1 +ℓ [0]) +is T[0]. Therefore, we write the number T[0] into s′ +ℓ[s−1 +ℓ [0]]. +A color transformation example is shown in Figs. 7 and 8. Fig. 7 is just a replication +of Fig. 5 showing a default cube after U1 twist. The color transformation number 04 applied +to the cube of Fig. 7 is shown in Fig. 8 (a)-(b) in two steps: +(a) The stickers are re-painted and re-numbered (white becomes green, blue becomes +white and so on). The structure of coloring is the same as in Fig. 7. Now the (DRB)- +cubie is no longer the (ygr)-cubie, it does not carry the numbers (12,16,20). +14 + +(b) We apply the proper WCR that brings the (ygr)-cubie back to the (DRB)-location. +Compared to (a), each 4-sticker cube face is just rotated to another face, but not +changed internally. We can check that the (DRB)-location now carries again the num- +bers (12,16,20), as in Fig. 7 and as it should for a normalized cube. +2.5 +Symmetries +Symmetries are transformations of the game state (and the attached action, if applicable) +that lead to equivalent states. +That is, if s is a certain state with value V (s), then all +states ssym being symmetric to s have the same value V (ssym) = V (s) because they are +equivalent. Equivalent means: If s can be solved by a twist sequence of length n, then +ssym can be solved by an equivalent twist sequence of same length n. +In the case of Rubik’s cube, all whole-cube rotations (WCRs) are symmetries because +they do not change the value of a state. But whole-cube rotations are ’trivial’ symmetries +because they are usually factored out by the normalization of the cube: After 2x2x2 cube +normalization, which brings the (ygr)-cubie in a certain position, or after 3x3x3 cube nor- +malization, which brings the center cubies in certain faces, all WCR-symmetric states are +transformed to the same state. +Non-trivial symmetries are all color transformations (Sec. 2.4.3): In general, color trans- +formations transform a state s to a truly different state ssym, even after cube normalization.6 +Since there are 24 color transformations in Rubik’s cube, there are also 24 non-trivial sym- +metries (including self). +Symmetries are useful to learn to solve Rubik’s cube for two reasons: (a) to accelerate +learning and (b) to smooth an otherwise noisy value function. +(a) Accelerated learning: If a state s (or state-action pair) is observed, not only the +weights activated by that state are updated, but also the weights of all symmetric states +ssym, because they have the same V (ssym) = V (s) and thus the same reward. In this +way, a single observed sample is connected with more weight updates (better sample +efficiency). +(b) Smoothed value function: By this we mean that the value function V (s) is replaced +by +V (sym)(s) = +1 +|Fs| +� +s′∈Fs +V (s′) +(18) +where Fs is the set of states being symmetric to s. If V (s) were the ideal value function, +both terms V (s) and V (sym)(s) would be the same.7 But in a real n-tuple network, V (s) +is non-ideal due to n-tuple-noise (cross-talk from other states that activate the same +n-tuple LUT entries). If we average over the symmetric states s′ ∈ Fs, the noise will be +dampened. +6In rare cases – e.g. for the solved cube – the transformed state may be identical to s or to another +symmetry state, but this happens seldom for sufficiently scrambled cubes, see Sec. 6.3. +7because all V (s′) in Eq. (18) are the same for an ideal V +15 + +The downside of symmetries is their computational cost: In the case of Rubik’s cube, +the calculation of color transformations is a costly operation. On the other hand, the number +of necessary training episodes to reach a certain performance may be reduced. In the +end, the use of symmetries may pay off, because the total training time may be reduced as +well. In any case, we will have a better sample efficiency, since we learn more from each +observed state or state-action pair. Secondly, the smoothing effect introduced with Eq. (18) +can lead to better overall performance, because the smoothed value function provides a +better guidance on the path towards the solved cube. +In order to balance computation time, GBG offers the option to select with nSym the +number of symmetries actually used. If we specify for example nSym = 8 in GBG’s Rubik’s +cube implementation, then the state itself and 8 – 1 = 7 random other (non-id) color trans- +formations will be selected. The resulting set Fs of 8 states is then used for weight update +and value function computation. +3 +N-Tuple Systems +N-tuple systems coupled with TD were first applied to game learning by Lucas (2008), al- +though n-tuples were already introduced by Bledsoe and Browning (1959) for character +recognition purposes. The remarkable success of n-tuples in learning to play Othello (Lu- +cas, 2008) motivated other authors to benefit from this approach for a number of other +games. +The main goal of n-tuple systems is to map a highly non-linear function in a low di- +mensional space to a high dimensional space where it is easier to separate ‘good’ and +‘bad’ regions. This can be compared to the kernel trick of support-vector machines. An +n-tuple is defined as a sequence of n cells of the board. Each cell can have m positional +values representing the possible states of that cell.8 Therefore, every n-tuple will have a +(possibly large) look-up table indexed in form of an n-digit number in base m. Each entry +corresponds to a feature and carries a trainable weight. An n-tuple system is a system +consisting of k n-tuples. As an example we show in Fig. 9 an n-tuple system consisting of +four 8-tuples. +Let Θ be the vector of all weights θi of the n-tuple system.9 The length of this vector +may be large number, e.g. mnk, if all k n-tuples have the same length n and each cell +has m positional values. Let Φ(s) be a binary vector of the same length representing the +feature occurences in state s (that is, Φi(s) = 1 if in state s the cell of a specific n-tuple as +indexed by i has the positional value as indexed by i, Φi(s) = 0 else). The value function +of the n-tuple network given state s is +V (s) = σ (Φ(s) · Θ) +(19) +with transfer function σ which may be a sigmoidal function or simply the identity function. +8A typical example is a 2-player board game, where we usually have 3 positional values {0: empty, 1: +player1, 2: player2 }. But other, user-defined values are possible as well. +9The index i indexes three qualities: an n-tuple, a cell in this n-tuple and a positional value for this cell. +16 + +Figure 9: Example n-tuples: We show 4 random-walk 8-tuples on a 6x7 board. The tuples are +selected manually to show that not only snake-like shapes are possible, but also bifurcations +or cross shapes. Tuples may or may not be symmetric. +An agent using this n-tuple system derives a policy from the value function in Eq. (19) +as follows: Given state s and the set A(s) of available actions in state s, it applies with a +forward model f every action a ∈ A(s) to state s, yielding the next state s′ = f(s, a). Then +it selects the action that maximizes V (s′). +Each time a new agent is constructed, all n-tuples are either created in fixed, user- +defined positions and shapes, or they are formed by random walk. In a random walk, all +cells are placed randomly with the constraint that each cell must be adjacent10 to at least +one other cell in the n-tuple. +Agent training proceeds in the TD-n-tuple algorithm as follows: Let s′ be the actual +state generated by the agent and let s be the previous state generated by this agent. TD(0) +learning adapts the value function with model parameters Θ through (Sutton and Barto, +1998) +Θ ← Θ + αδ∇ΘV (s) +(20) +Here, α is the learning rate and V is in our case the n-tuple value function of Eq. (19). δ is +the usual TD error (Sutton and Barto, 1998) after the agent has acted and generated s′: +δ = r + γV (s′) − V (s) +(21) +where the sum of the first two terms, reward r plus the discounted value γV (s′), is the +desirable target for V (s). +10The form of adjacency, e. g. 4- or 8-point neighborhood or any other (might be cell-dependent) form of +adjacency, is user-defined. +17 + +0 +3 +5 +6 +6 +5 +6 +3 +4 +4 +5 +6 +54 +N-Tuple Representions for the Cube +In order to apply n-tuples to cubes, we have to define a board in one way or the other on +which we can place the n-tuples. This is not as straightforward as in other board games, but +we are free to invent abstract boards. Once we have defined a board, we can number the +board cells k = 0, . . . , K−1 and translate a cube state into a BoardVector: A BoardVector +b is a vector of K non-negative integer numbers bk ∈ {0, . . . , Nk − 1}. Each k represents +a board cell and every board cell k has a predefined number Nk of position values.11 +A BoardVector is useful to calculate the feature occurence vector Φ(s) in Eq. (19) for +a given n-tuple set: If an n-tuple contains board cell k, then look into bk to get the position +value for this cell k. Set Φi(s) = 1 for that index i that indexes this n-tuple cell and this +position value. +In the following we present different options for boards and BoardVectors. We do this +mainly for the 2x2x2 cube, because it is somewhat simpler to explain. But the same ideas +apply to the 3x3x3 cube as well, they are just a little bit longer. Therefore, we defer the +lengthy details of the 3x3x3 cube to Appendix B. +4.1 +CUBESTATE +A natural way to translate the cube state into a board is to use the flattened representation +of Fig. 11 as the board and extract from it the 24-element vector b, according to the given +numbering. The kth element bk represents a certain cubie face location and gets a number +from {0, . . . , 5} according to its current face color fc. The solved cube is for example +represented by b = [0000 1111 2222 . . . 5555]. +This representation CUBESTATE is what the BoardVecType CUBESTATE in our GBG- +implementation means: Each board vector is a copy of fcol, the face colors of all cubie +faces. fcol is also the vector that uniquely defines each cube state. An upper bound of +possible combinations for b is 624 = 4.7 · 1018. If we factor out the (DRB)-cubie, which +always stays at its home position, we can reduce this to 21 board cells with 6 positional +values, leading to 621 = 2.1 · 1016 weights. Both numbers are of course way larger than +the true number of distinct states (Sec. 2.2.1) which is 3.6 · 106. This is because most of +the combinations are dead weights in the n-tuple LUTs, they will never be activated during +game play. +The dead weights occur because many combinations are not realizable, e.g. three +white faces in one cubie or any of the 63 − 8 · 3 = 192 cubie-face-color combinations that +are not present in the real cube. The problem is that the dead weights are scattered in a +complicated way among the active weights and it is thus not easy to factor them out. +4.2 +STICKER +McAleer et al. (2019) had the interesting idea for the 3x3x3 cube that 20 stickers (cubie +faces) are enough. To characterize the full 3x3x3 cube, we need only one (not 2 or 3) sticker +11In GBG package ntuple2 (base for agent TDNTuple3Agt), all Nk have to be the same. +In package +ntuple4( base for agent TDNTuple4Agt), numbers Nk may be different for different k. +18 + +(a) Top view +(b) Bottom view +Figure 10: The sticker representation used to reduce dimensionality: Stickers that are used +are shown in white, whereas ignored stickers are dark blue (from McAleer et al. (2019)). +3 +2 +0 +1 +5 +4 +8 +11 +18 +17 +23 +22 +6 +7 +9 +10 +19 +16 +20 +21 +14 +13 +15 +12 +Figure 11: Tracked stickers for the 2x2x2 cube (white), while ignored stickers are blue. +for every of the 20 cubies, as shown in Fig. 10. This is because the location of one sticker +uniquely defines the location and orientation of that cubie. We name this representation +STICKER in GBG. +Translated to the 2x2x2 cube, this means that 8 stickers are enough because we have +only 8 cubies. We may for example track the 4 top stickers 0,1,2,3 plus the 4 bottom +stickers 12,13,14,15 as shown in Fig. 11 and ignore the 16 other stickers. Since we always +normalize the cube such that the (DRB)-cubie with sticker 12 stays in place, we can reduce +this even more to 7 stickers (all but sticker 12). +How to lay out this representation as a board? – McAleer et al. (2019) create a rect- +angular one-hot-encoding board with 7 × 21 = 147 cells (7 rows for the stickers and 21 +columns for the locations) carrying only 0’s and 1’s. This is fine for the approach of McAleer +et al. (2019), where they use this board as input for a DNN, but not so nice for n-tuples. +Without constraints, such a board amounts to 2147 = 1.7 · 1044 combinations, which is +unpleasantly large (much larger than in CUBESTATE).12 +STICKER has more dead weights than CUBESTATE, so it seems like a step back. But +the point is, that the dead weights are better structured: If for example sticker 0 appears at +column 1 then this column and the two other columns for the same cubie are automatically +12A possible STICKER BoardVector for the default cube would read b = [1000000 0100000 0010000 . . . ], +meaning that location 0 has the first sticker, location 1 has the second sticker, and so on. In any STICKER +BoardVector there are only 7 columns carrying exactly one 1, the other carry only 0’s. Every row carries exactly +one 1. +19 + +Table 7: The correspondence corner location ↔ STICKER2 for the solved cube. The yellow +colored cells show the location of the 7 (2x2x2) and 8 (3x3x3) corner stickers that we track. +2x2x2 +location +0 1 2 3 +4 +5 +6 +7 +8 +9 +10 11 +12 13 14 15 +16 17 18 19 +20 21 22 23 +3x3x3 +location +0 2 4 6 +8 10 12 14 +16 18 20 22 +24 26 28 30 +32 34 36 38 +40 42 44 46 +STICKER2 +corner +a b c d +a +d +h +g +a +g +f +b +e +f +g +h +e +c +b +f +e +h +d +c +face ID +1 1 1 1 +2 +3 +2 +3 +3 +2 +3 +2 +1 +1 +1 +1 +2 +2 +3 +2 +3 +3 +2 +3 +forbidden for all other stickers. Likewise, if sticker 1 is placed in another column, another set +of 3 columns is forbidden, and so on. We can use this fact to form a much more compact +representation STICKER2. +4.3 +STICKER2 +As the analysis in the preceding section has shown, the 21 location columns of STICKER +cannot carry the tracked stickers in arbitrary combinations. Each cubie (represented by 3 +columns in STICKER) carries only exactly one sticker. We can make this fact explicit by +choosing another representation for the 21 locations: +corner location = (corner cubie, face ID). +That is, each location is represented by a pair: corner cubie a,b,c,d,f,g,h (we number the +top cubies with letters a,b,c,d and the bottom cubies with letters e,f,g,h and omit e because +it corresponds to the (DRB)-cubie) and a face ID. To number the faces with a face ID, we +follow the convention that we start at the top (bottom) face with face ID 1 and then move +counter-clockwise around the corner cubie to visit the other faces (2,3). Table 7 shows the +explicit numbering in this new representation. +To represent a state as board vector we use now a much smaller board shown in +Table 8: Each cell in the first row has 7 position values (the letters) and each cell in the +second row has 3 position values (the face IDs). We show in Table 8 the board vector for +the default cube, b = [abcdfgh 1111111]. Representation STICKER2 allows for 77 · 37 = +1.8 · 109 combinations in total, which is much smaller than STICKER and CUBESTATE. +Table 8: STICKER2 board representation for the default 2x2x2 cube. For the BoardVector, +cells are numbered row-by-row from 0 to 16. +corner +a +b +c +d +f +g +h +7 positions +face ID +1 +1 +1 +1 +1 +1 +1 +3 positions +STICKER2 has some dead weights remaining, because the combinations can carry +the same letter multiple times, which is not allowed for a real cube state. But this rate of +dead weights is tolerable. +It turns out that STICKER2 is in all aspects better than CUBESTATE or STICKER. +Therefore, we will only report the results for STICKER2 in the following. +20 + +4.4 +Adjacency Sets +To create n-tuples by random walk, we need adjacency sets (sets of neighbors) to be +defined for every board cell k. +For CUBESTATE, the board is the flattened representation of the 2x2x2 cube (Fig. 3). +The adjacency set is defined as the 4-point neighborhood, where two stickers are neigh- +bors if they share a common edge on the cube, i.e. are neighbors on the cube. +For STICKER2, the board consists of 16 cells shown in Table 8. Here, the adjacency +set for cell k contains all other cells different from k. +Again, the details of ideas similar to Sec. 4.1–4.4, but now for the 3x3x3 cube, are +shown in Appendix B.1–B.4. +5 +Learning the Cube +5.1 +McAleer and Agostinelli +The works of McAleer et al. (2018, 2019) and Agostinelli et al. (2019) contain up to now +the most advanced methods for learning to solve the cube from scratch. Agostinelli et al. +(2019) introduces the cost-to-go function for a general Marko decision process +J(s) = min +a∈A(s) +� +s′ +P a(s, s′) +� +ga(s, s′) + γJ(s′) +� +(22) +where P a(s, s′) is the probability of transitioning from state s to s′ by taking action a and +ga(s, s′) is the cost for this transition. In the Rubik’s cube case, we have deterministic tran- +sitions, that is s′ = f(s, a) is deterministically prescribed by a forward model f. Therefore, +the sum reduces to one term and we specialize to γ = 1. Furthermore, we set ga(s, s′) = 1, +because only the length of the solution path counts, so that we get the simpler equation +J(s) = min +a∈A(s) +� +1 + J(s′) +� +with +s′ = f(s, a). +(23) +Here, A(s) is the set of available actions in state s. We additionally set J(s∗) = 0 if s∗ +is the solved cube. To better understand Eq. (23) we look at a few examples: If s1 is a state +one twist away from s∗, Eq. (23) will find this twist and set J(s1) = 1. If s2 is a state two +twists away from s∗ and all one-twist states have already their correct labels J(s1) = 1, +then Eq. (23) will find the twist leading to a s1 state and set J(s2) = 1 + 1 = 2. While +iterations proceed, more and more states (being further away from s∗) will be correctly +labeled, once their preceding states are correctly labeled. In the end we should ideally +have +J(sn) = n. +However, the number of states for Rubik’s cube is too large to store them all in tabular +form. Therefore, McAleer et al. (2019) and Agostinelli et al. (2019) approximate J(s) with +a deep neural network (DNN). To train such a network in the Rubik’s cube case, they +21 + +Algorithm 1 DAVI algorithm (from Agostinelli et al. (2019)). Input: B: batch size, K: +maximum number of twists, M: training iterations, C: how often to check for convergence, +ϵ: error threshold. Output: Θ, the trained neural network parameters. +1: function DAVI(B, K, M, C, ϵ) +2: +Θ ← INITIALIZENETWORKPARAMETERS +3: +ΘC ← Θ +4: +for m = 1, . . . , M do +5: +X ←GENERATESCRAMBLEDSTATES(B, K) +▷ B scrambled cubes +6: +for xi ∈ X do +7: +yi ← mina∈A(s) [1 + jΘC(f(xi, a))] +▷ cost-to-go function, Eq. (23) +8: +(Θ, loss) ← TRAIN(jΘ, X, y) +▷ loss = MSE(jΘ(xi), yi) +9: +if (m mod C = 0 & loss < ϵ) then +10: +ΘC ← Θ +11: +return Θ +introduce Deep Approximate Value Iteration (DAVI)13 shown in Algorithm 1. The network +output jΘ(s) is trained in line 8 to approximate the (unknown) cost-to-go J(s) for every +state s = xi. The main trick of DAVI is, as Agostinelli et al. (2019) write: „For learning to +occur, we must train on a state distribution that allows information to propagate from the +goal state to all the other states seen during training. Our approach for achieving this is +simple: each training state xi is obtained by randomly scrambling the goal state ki times, +where ki is uniformly distributed between 1 and K. During training, the cost-to-go function +first improves for states that are only one move away from the goal state. The cost-to-go +function then improves for states further away as the reward signal is propagated from the +goal state to other states through the cost-to-go function.“ +Agostinelli et al. (2019) use in Algorithm 1 two sets of parameters to train the DNN: the +parameters Θ being trained and the parameters ΘC used to obtain improved estimates +of the cost-to-go function. If they did not use this two separate sets, performance often +„saturated after a certain point and sometimes became unstable. Updating ΘC only after +the error falls below a threshold ϵ yields better, more stable, performance.“ (Agostinelli +et al., 2019) To train the DNN, they used M = 1 000 000 iterations, each with batch size +B = 10 000. Thus, the trained DNN has seen ten billion cubes (1010) during training, which +is still only a small subset of the 4.3 · 1019 possible cube states. +The heuristic function of the trained DNN alone cannot solve 100% of the cube states. +Especially for higher twist numbers ki, an additional solver or search algorithm is needed. +This is in the case of McAleer et al. (2019) a Monte Carlo Tree Search (MCTS), similar to +AlphaZero (Silver et al., 2017), which uses the DNN as the source for prior probabilities. +Agostinelli et al. (2019) use instead a variant of A∗-search, which is found to produce +solutions with a shorter path in a shorter runtime than MCTS. +13More precisely, McAleer et al. (2019) use Autodidactic Iteration (ADI), a precursor to DAVI, very similar to +DAVI, just a bit more complicated to explain. Therefore, we describe here only DAVI. +22 + +Algorithm 2 TD-n-tuple algorithm for Rubik’s cube. Input: pmax: maximum number of +twists, M: training iterations, Etrain: maximum episode length during training, c: nega- +tive cost-to-go, Rpos: positive reward for reaching the solved cube s∗, α: learning rate. +jΘ(s): n-tuple network value prediction for state s. Output: Θ, the trained n-tuple network +parameters. +1: function TDNTUPLE(pmax, M, Etrain, c, Rpos) +2: +Θ ← INITIALIZENETWORKPARAMETERS +3: +for m = 1, . . . , M do +4: +p ∼ U(1, . . . , pmax) +▷ Draw p uniformly random from {1, 2, . . . , pmax} +5: +s ← SCRAMBLESOLVEDCUBE(p) +▷ start state +6: +for k = 1, . . . , Etrain do +7: +snew ← arg max +a∈A(s) +V (s′) +with +s′ = f(s, a) +and +8: +V (s′) = c + +� Rpos +if +s′ = s∗ +jΘ(s′) +if +s′ ̸= s∗ +9: +Train network jΘ with Eq. (20) to bring V (s) closer to target T = V (snew): +V (s) ← V (s) + α(T − V (s)) +10: +s ← snew +11: +if (s = s∗) then +12: +break +▷ break out of k-loop +13: +return Θ +23 + +5.2 +N-Tuple-based TD Learning +To solve the Rubik’s cube in GBG we use an algorithm that is on the one hand inspired by +DAVI, but on the other hand more similar to traditional reinforcement learning schemes like +temporal difference (TD) learning. In fact, we want to use in the end the same TD-FARL +algorithm (Konen and Bagheri, 2021) that we use for all other GBG games. +We show in Algorithm 2 our method, that we will explain in the following, highlighting +also the similarities and dissimilarities to DAVI. +First of all, instead of minimizing the positive cost-to-go as in DAVI, we maximize in +lines 7-8 a value function V (s′) with a negative cost-to-go. This maximization is functionally +equivalent, but more similar to the usual TD-learning scheme. The negative cost-to-go, e.g. +c = −0.1, plays the role of the positive 1 in Eq. (23). +Secondly, we replace the DNN of DAVI by the simpler-to-train n-tuple network jΘ with +STICKER2 representation as described in Sec. 3 and 4. +That is, each time jΘ(s′) is +requested, we first calculate for state s′ the BoardVector in STICKER2 representation, +then the occurence vector Φ(s′) and the value function V (s′) according to Eq. (19). +The central equations for V (s′) in Algorithm 2, lines 7-8, work similar to Eq. (23) in +DAVI: If s = s1 is a state one twist away from s∗, the local search in arg max V (s′) will find +this twist and the training step in line 9 moves V (s) closer to c+Rpos.14 Likewise, neighbors +s2 of s1 will find s1 and thus move V (s2) closer to 2c + Rpos. Similar for s3, s4, . . . under +the assumption that a ’known’ state is in the neighborhood. We have a clear gradient on +the path towards the solved cube s∗. If there are no ’known’ states in the neighborhood +of sn, we get for V (sn) what the net maximally estimates for all those neighbors. We pick +the neighbor with the highest estimate, wander around randomly until we hit a state with a +’known’ neighbor or until we reach the limit Etrain of too many steps. +Note that Algorithm 2 is different from DAVI insofar that it follows the path s → s′ → . . . +as prescribed by the current V , which may lead to a state sequence ’wandering in the +unknown’ until Etrain is reached. In contrast to that, DAVI generates many start states s0 +drawn from the distribution of training set states and trains the network just on pairs (s0, T), +i.e. they do just one step on the path. We instead follow the full path, because we want +the training method for Rubik’s cube to be as similar as possible to the training method for +other GBG games.15 +Algorithm 2 is basically the same algorithm as GBG uses for other games. The only +differences are (i) the cube-specific start state selection borrowed from DAVI (a 1-twist start +state has the same probability as a 10-twist start state) and (ii) the cube-specific reward in +line 8 of Algorithm 2 with its negative cost-go-go c which is however a common element of +many RL rewards. +Algorithm 2 currently learns with only one parameter vector Θ. However, it could be +extended as in DAVI to two parameter vectors Θ and ΘC. The weight training step in line +14It is relevant, that Rpos is a positive number, e.g. 1.0 (and not 0, as it was for DAVI). This is because we +start with an initial n-tuple network with all weights set to 0, so the initial response of the network to any state +is 0.0. Thus, if Rpos were 0, a one-twist state would see all its neighbors (including s∗) initially as responding +0.0 and would not learn the right transition to s∗. With Rpos = 1.0 it will quickly find s∗. +15We note in passing that we tested the DAVI variant with Etrain = 1 for our TD-n-tuple method as well. +However, we found that this method gave much worse results, so we stick with our GBG method here. +24 + +9 is done with the help of Eq. (20) for Θ using the error signal δ of Eq. (21). +There are two extra elements, TCL and MCTS, that complete our n-tuple-based TD +learning. They are described in the next two subsections. +5.2.1 +Temporal Coherence Learning (TCL) +The TCL algorithm developed by Beal and Smith Beal and Smith (1999) is an extension +of TD learning. It replaces the global learning rate α with the weight-individual product +ααi for every weight θi. Here, the adjustable learning rate αi is a free parameter set by a +pretty simple procedure: For each weight θi, two counters Ni and Ai accumulate the sum +of weight changes and the sum of absolute weight changes. If all weight changes have the +same sign, then αi = |Ni|/Ai = 1, and the learning rate stays at its upper bound. If weight +changes have alternating signs, then the global learning rate is probably too large. In this +case, αi = |Ni|/Ai → 0 for t → ∞, and the effective learning rate will be largely reduced +for this weight. +In our previous work (Bagheri et al., 2015) we extended TCL to αi = g(|Ni|/Ai) where +g is a transfer function being either the identity function (standard TCL) or an exponential +function g(x) = eβ(x−1). It was shown in Bagheri et al. (2015) that TCL with this exponential +transfer function leads to faster learning and higher win rates for the game ConnectFour. +5.2.2 +MCTS +We use Monte Carlo Tree Search (MCTS) (Browne et al., 2012) to augment our trained +network during testing and evaluation. This is the method also used by McAleer et al. +(2019) and by AlphaGo Zero (Silver et al., 2017), but they use it also during training. +MCTS builds iteratively a search tree starting with a tree containing only the start state +s0 as the root node. Until the iteration budget is exhausted, MCTS does the following: In +every iteration we start from the root node and select actions following the tree policy until +we reach a yet unexpanded leaf node sℓ. The tree policy is implemented in our MCTS +wrapper according to the UCB formula (Silver et al., 2017): +anew += +arg max +a∈A(s) +�W(s, a) +N(s, a) + U(s, a) +� +(24) +U(s, a) += +cpuctP(s, a) +� +ε + � +b∈A(s) N(s, b) +1 + N(s, a) +(25) +Here, W(s, a) is the accumulator for all backpropagated values that arrive along branch +a of the node that carries state s. Likewise, N(s, a) is the visit counter and P(s, a) the prior +probability. A(s) is the set of actions available in state s. ε is a small positive constant for +the special case � +b N(s, b) = 0: It guarantees that in this special case the maximum of +U(s, a) is given by the maximum of P(s, a). The prior probabilities P(s, a) are obtained +25 + +Algorithm 3 TD-n-tuple training algorithm. Input: see Algorithm 2. Output: Θ: trained +n-tuple network parameters. +1: function TDNTUPLETRAIN(pmax, M, Etrain, c, Rpos) +2: +Θ ← INITIALIZENETWORKPARAMETERS +3: +INITIALIZETCLPARAMETERS +▷ Set TCL-accumulators Ni = Ai = 0, αi = 1 ∀i +4: +for m = 1, . . . , M do +5: +Perform one m-iteration of Algorithm 2 with learning rates ααi instead of α +6: +Ni ← Ni + ∆θi and Ai ← Ai + |∆θi| +▷ Update TCL-accumulators +7: +▷ where ∆θi is the last term in Eq. (20) +8: +αi ← |Ni|/Ai +∀i with Ai ̸= 0 +9: +return Θ +Algorithm 4 Evaluation algorithm with MCTS solver. Input: trained n-tuple network jΘ, +p: number of scrambling twists, B: batch size, Eeval: maximum episode length during +evaluation, I: number of MCTS-iterations, cPUCT : relative weight for U(s, a) in Eq. (24), +dmax: maximum MCTS tree depth. Output: solved rate. +1: function TDNTUPLEEVAL(jΘ, p, B, Eeval, I, cPUCT , dmax) +2: +X ←GENERATESCRAMBLEDCUBES(B, p) +▷ B scrambled cubes +3: +Csolved ← 0 +4: +for xi ∈ X do +5: +s ← xi +6: +for k = 1, . . . , Eeval do +7: +T ← PERFORMMCTSSEARCH(s, I, cPUCT , dmax, jΘ) +8: +a ← SELECTMOSTVISITEDACTION +9: +s ← f(s, a) +10: +if (s = s∗) then +11: +Csolved ← Csolved + 1 +12: +break +▷ break out of k-loop +13: +return Csolved/B +▷ percentage solved +by sending the trained network’s values of all follow-up states s′ = f(s, a) with a ∈ A(s) +through a softmax function (see Sec. 3).16 +Once an unexpanded leaf node sℓ is reached, the node is expanded by initializing +its accumulators: W(s, a) = N(s, a) = 0 and P(s, a) = ps′ where ps′ is the softmax- +squashed output jΘ(s′) of our n-tuple network for each state s′ = f(s, a). The value of +the node is the network output of the best state jΘ(sbest) = maxs′ jΘ(s′) and this value is +backpropagated up the tree. +More details on our MCTS wrapper can be found in Scheiermann and Konen (2022). +16Note that the prior probabilities and the MCTS iteration are only needed at test time, so that we – different +to AlphaZero – do not need MCTS during self-play training. +26 + +5.2.3 +Method Summary +We summarize the different ingredients of our n-tuple-based TD learning method in Algo- +rithm 3 (training) and Algorithm 4 (evaluation). +In line 5 of Algorithm 3 we perform one m-iteration of Algorithm 2 which does an update +step for weight vector Θ, see Eq. (20). All weights of activated n-tuple entries get a weight +change ∆θi equal to the last term in Eq. (20) where the global α is replaced by ααi. +Line 2 in Algorithm 4 generates a set X of B scrambled cube states. Line 7 builds for +each xi ∈ X an MCTS tree (see Sec. 5.2.2) starting from root node xi and line 8 selects +the most visited action of the root node. If the goal state s∗ is not found during Eeval k-loop +trials, this xi is considered as not being solved. +6 +Results +6.1 +Experimental setup +We use for all our GBG experiments the same RL method based on n-tuple systems and +TCL. Only its hyperparameters are tuned to the specific game, as shown below. We refer +to this method/agent as TCL-base whenever it alone is used for game playing. If we wrap +such an agent by an MCTS wrapper with a given number of iterations, then we refer to this +as TCL-wrap. +We investigate two variants of Rubik’s Cube: 2x2x2 and 3x3x3. We trained all TCL +agents by presenting them M = 3 000 000 cubes scrambled with p random twists, where +p is chosen uniformly at random from {1, . . . , pmax}. Here, pmax = 13 [16] for 2x2x2 and +pmax = 9 [13] for 3x3x3, where the first number is for HTM, while the second number +in square brackets is for QTM. With these pmax cube twists we cover the complete cube +space for 2x2x2, where God’s number (Sec. 2.2) is known to be 11 [14]. But we cover only +a small subset in the 3x3x3 case, where God’s number is known to be 20 [26] (Rokicki +et al., 2014).17 We train 3 agents for each cube variant { 2x2x2, 3x3x3 } × { HTM, QTM } +to assess the variability of training. +The hyperparameters of the agent for each cube variant were found by manual fine- +tuning. For brevity, we defer the exact explanation and setting of all parameters to Ap- +pendix C. +We evaluate the trained agents for each p on 200 scrambled cubes that are created by +applying the given number p of random scrambling twists to a solved cube. The agent now +tries to solve each scrambled cube. A cube is said to be unsolved during evaluation if the +agent cannot reach the solved cube in Eeval = 50 steps.18 +17We limit ourselves to pmax = 9 [13] in the 3x3x3 HTM [QTM ] case, because our network has not enough +capacity to learn all states of the 3x3x3 Rubik’s cube. Experiments with higher twist numbers during training +did not improve the solved-rates. +18During training, we use lower maximum episode lengths Etrain (see Appendix C) than Eeval = 50 in +order to reduce computation time (in the beginning, many episodes cannot be solved, and 50 would waste a +lot of computation time). But Etrain is always at least pmax + 3 in order to ensure that the agent has a fair +chance to solve the cube and collect the reward. +27 + +HTM +0% +25% +50% +75% +100% +1 +3 +5 +7 +9 +11 +13 +scrambling twists +percentage solved +cubeWidth +2x2x2 +3x3x3 +iterMWrap +0 +100 +800 +QTM +0% +25% +50% +75% +100% +1 +3 +5 +7 +9 +11 +13 +15 +scrambling twists +percentage solved +cubeWidth +2x2x2 +3x3x3 +iterMWrap +0 +100 +800 +Figure 12: Percentage of solved cubes as a function of scrambling twists p for the trained +TD-N-tuple agent wrapped by MCTS wrapper with different numbers of iterations. The red +curves are TCL-base without wrapper, the other colors show different forms of TCL-wrap. +Twist type is HTM (left) and QTM (right). Each point is the average of 3 independently trained +agents. +6.2 +Cube Solving with MCTS Wrapper, without Symmetries +The trained TD-N-tuple agents learn to solve the cubes to some extent, as the red curves +TCL-base in Fig. 12 show, but they are in many cases (i.e. p > pmax/2) far from being +perfect. These are the results from training each agent for 3 million episodes, but the +results would not change considerably, if 10 million training episodes were used. +Scheiermann and Konen (2022) have shown, that the performance of agents, namely +TD-N-tuple agents, is largely improved, if the trained agents are wrapped during test, play +and evaluation by an MCTS wrapper. This holds for Rubik’s cube as well, as Fig. 12 shows: +For the 2x2x2 cube, the non-wrapped agent TCL-base (red curve) is already quite good, +but with wrapping it becomes almost perfect. For the 3x3x3 cube, the red curves are not +satisfactorily: the solved-rates are below 20% for p = 9 [13] in the HTM [QTM ] case. But +at least MCTS wrapping boosts the solved-rates by a factor of 3 [QTM: from 16% to 48%] +or 4.5 [HTM: from 10% to 45%]. +All these results are without incorporating symmetries. +How symmetries affect the +solved-rates will be investigated in Sec. 6.4. But before this, we look in the next section at +the number of symmetries that effectively exist in a cube state. +6.3 +Number of Symmetric States +Not every cube state has 24 truly different symmetric states (24 = number of color sym- +metries). For example in the solved cube, all color-symmetric states are the same (after +normalization). Thus, we have here only one truly different symmetric state. +However, we show in this section that for the majority of cube states the number of +truly different symmetric states is close to 24. Two states are truly different if they are +not the same after the normalizing operation. We generate a cube state by applying p +random scrambling twists to the default cube. Now we apply all 24 color transformations +(Sec. 2.4.3) to it and count the truly different states. The results are shown in Fig. 13 for +28 + +twistType: HTM +twistType: QTM +0 +4 +8 +12 +16 +0 +4 +8 +12 +16 +0 +5 +10 +15 +20 +25 +scrambling twists +NsymmetricStates +cubeWidth +2x2x2 +3x3x3 +Figure 13: Count of truly different symmetric states for cube states generated by p random +scrambling twists. Each point is an average over 500 such states. +both cube sizes and both twist types. For the 3x3x3 cube, the number of states quickly (for +p > 5) approaches the maximum N = 24, while for the 2x2x2 cube it is a bit slower: p > 4 +or p > 8 is needed to surpass N = 20. +As a consequence, it makes sense to use 16 or even 24 symmetries when training +and evaluating cube agents. Especially for scrambled states with higher p, the 24 color +transformations used to construct symmetric states will usually lead to 24 different states. +6.4 +The Benefit of Symmetries +In order to investigate the benefits of symmetries, we first train a TCL agent with dif- +ferent numbers of symmetries. As described in Sec. 2.5, we select in each step nSym += 0, 8, 16, 24 symmetric states. Which symmetric states are chosen is selected randomly. +Symmetries are used (a) to update the weights for each symmetric state and (b) to build +with Eq. (18) a smoothed value function which is used to decide about the next action dur- +ing training. For 0, 8, 16, 24 symmetries, we train 3 agents each (3x3x3 cube, STICKER2, +QTM). The 3 agents differ due to their differently created random-walk n-tuple sets. +Fig. 14 shows the learning curves for different nSym = 0, 8, 16, 24. It is found that agents +with nSym > 0 learn faster and achieve a higher asymptotic solved rate. +Next, we evaluate each of the trained agents by trying to solve for each p ∈ {1, . . . , 15} +(scrambling twists) 200 different scrambled cubes. During evaluation, we use again the +same nSym as in training to form a smoothed value function. We compare in Fig. 15 different +29 + +50% +60% +70% +80% +90% +0e+00 +1e+06 +2e+06 +3e+06 +episodes +percentage solved +nSym +24 +16 +8 +0 +Figure 14: Learning curves for different numbers nSym = 0, 8, 16, 24 of symmetries. Shown is +the solved rate of (3x3x3, QTM) cubes. The solved rate is the average over all twist numbers +p = 1, . . . , 13 with 200 testing cubes for each p and over 3 agents with different random-walk +n-tuple sets. +symmetry results, both without wrapping (TCL-base, red curves) and with MCTS-wrapped +agents using 100 (green) or 800 (blue) iterations. It is clearly visible that MCTS wrapping +has a large effect, as it was also the case in Fig 12. But in addition to that, the use of +symmetries leads for each agent, wrapped or not, to a substantial increase in solved-rates +(a surplus of 10-20%). It is remarkable, that even for p=14 or 15 a solved rate above or +near 50% can be reached19 by the combination (nSym=16, 800 MCTS iterations). +Surprisingly, it seems that with wrapping it is only important whether we use symme- +tries, not how many, since the difference between nSym = 8, 16, 24 is only marginal. For +800 MCTS iterations, the solved rate for nSym = 24 is in most cases even smaller than that +for nSym = 8, 16. This is surprising because it would have been expected that also with +wrapping a larger nSym should lead to a smoother value function and thus should in theory +produce larger solved rates. – Note that this is not a contradiction to Fig. 14, because the +learning curves were obtained without wrapping and the red TCL-base curves in Fig. 15 +(again without wrapping) show the same positive trend with increasing nSym20. The red +curves in Fig. 15 show approximately the same average solved rates as the asymptotic +values in Fig. 14. +6.5 +Computational Costs +Table 9 shows the computational costs when training and testing with symmetries. All +computations were done on a single CPU Intel i7-9850H @ 2.60GHz. If we subtract the +19p is above pmax=13, the maximum twist number used during training. +20i.e. nSym= 24 is for every p clearly better than nSym= 16 +30 + +QTM +3x3x3 +0% +25% +50% +75% +100% +1 +3 +5 +7 +9 +11 +13 +15 +scrambling twists +percentage solved +nSym +0 +8 +16 +24 +iterMWrap +0 +100 +800 +Figure 15: With symmetries: Percentage of solved cubes (3x3x3, QTM) as a function of +scrambling twists p for TD-N-tuple agents trained and evaluated with different numbers of +symmetries nSym and wrapped by MCTS wrappers with different iterations. The red curves +are TCL-base (without wrapper), the other colors show different forms of TCL-wrap. The +solved rates are the average over 200 testing cubes for each p and over 3 agents with differ- +ent random-walk n-tuple sets. +computational costs for nsym= 0, computation time increases more or less linearly with +iter and roughly linearly with nSym. Computation times for nSym= 24 are approximately +10x larger than those for nSym= 0. +Computation times are dependent on the solved rate: If a cube with p = 13 is solved, +the episode takes normally 12-15 steps. If the cube is not solved, the episode needs 50 +steps, i.e. a factor of 3-4 more. Thus, the numbers in Table 9 should be taken only as +rough indication of the trend. +Bottom line: Training time through symmetries increases by a factor of 13/0.5 = 26 +(nSym= 24) and testing time increases through 800 MCTS iterations by a factor of about +3130/8 ≈ 400. +Training with symmetries takes between 5.4h and 13h on a normal CPU, depending +on the number of symmetries. This is much less than the 44h on a 32-core server with 3 +GPUs that were used by McAleer et al. (2019). But it also does not reach the same quality +as McAleer et al. (2019). +31 + +Table 9: Computation times with symmetries. All numbers are for 3x3x3 cube, STICKER2 +and QTM. Training: 3 million self-play episodes, w/o MCTS in the training loop. Testing: 200 +scrambled cubes with p = 13, agents wrapped by MCTS wrapper with iter iterations. +nSym +training +testing +[hours] +[seconds] +iter +0 +100 +400 +800 +0 +0.5 +0.5 +48 +196 +390 +8 +5.4 +4.0 +241 +877 +1400 +16 +9.5 +7.3 +464 +1380 +2330 +24 +13.0 +8.0 +550 +1760 +3130 +7 +Related Work +Ernö Rubik invented Rubik’s cube in 1974. Rubik’s cube has gained worldwide popularity +with many human-oriented algorithms being developed to solve the cube from arbitrary +scrambled start states. By ’human-oriented’ we mean algorithms that are simple to mem- +orize for humans. They usually will find long, suboptimal solutions. For a long time it was +an unsolved question what is the minimal number of moves (God’s Number) needed to +solve any given cube state. The early work of Thistlethwaite (1981) put an upper bound on +this number with his 52-move algorithm. This was one of the first works to systematically +use group theory as an aid to solve Rubik’s cube. Later, several authors have gradually +reduced the upper bound 52 (Joyner, 2014), until Rokicki et al. (2014) could prove in 2014 +for the 3x3x3 cube that God’s Number is 20 in HTM and 26 in QTM. +Computer algorithms to solve Rubik’s cube rely often on hand-engineered features and +group theory. One popular solver for Rubik’s cube is the two-phase algorithm of Kociemba +(2015). A variant of A∗ heuristic search was used by Korf (1991), along with a pattern +database heuristic, to find the shortest possible solutions. +The problem of letting a computer learn to solve Rubik’s cube turned out to be much +harder: Irpan (2016) experimented with different neural net baseline architectures (LSTM +gave for him reportedly best results) and tried to boost them with AdaBoost. However, he +had only for scrambling twist ≤ 7 solved rates of better than 50% and the baseline turned +out to be better than the boosted variants. Brunetto and Trunda (2017) found somewhat +better results with a DNN, they could solve cube states with 18 twists with a rate above +50%. But they did not learn from scratch because they used an optimal solver based +on Kociemba (2015) to generate training examples for the DNN. Smith et al. (2016) tried +to learn Rubik’s cube by genetic programming. However, their learned solver could only +reliably solve cubes with up to 5 scrambling twists. +A breakthrough in learning to solve Rubik’s cube are the works of McAleer et al. (2018, +2019) and Agostinelli et al. (2019): With Autodidactic Iteration (ADI) and Deep Approxi- +mate Value Iteration (DAVI) they were able to learn from scratch to solve Rubik’s cube in +QTM for arbitrary scrambling twists. Their method has been explained in detail already +in Sec. 5.1, so we highlight here only their important findings: McAleer et al. (2019) only +needs to inspect less than 4000 cubes with its trained network DeepCube when solving +32 + +for a particular cube, while the optimal solver of Korf (1991) inspects 122 billion different +nodes, so Korf’s method is much slower. +Agostinelli et al. (2019) extended the work of McAleer et al. (2019) by replacing the +MCTS solver with a batch-weighted A∗ solver which is found to produce shorter solution +paths and have shorter run times. At the same time, Agostinelli et al. (2019) applied their +agent DeepCubeA successfully to other puzzles like LightsOut, Sokoban, and the 15-, 24-, +35- and 48-puzzle21. DeepCubeA could solve all of them. +The deep network used by McAleer et al. (2019) and Agostinelli et al. (2019) were +trained without human knowledge or supervised input from computerized solvers. The +network of McAleer et al. (2019) had over 12 million weights and was trained for 44 hours +on a 32-core server with 3 GPUs. The network of McAleer et al. (2019) has seen 8 billion +cubes during training. – Our approach started from scratch as well. It required much less +computational effort (e.g. 5.4h training time on a single standard CPU for nSym=8, see +Table 9). It can solve the 2x2x2 cube completely, but the 3x3x3 cube only partly (up to 15 +scrambling twists). Each trained agent for the 3x3x3 cube has seen 48 million scrambled +cubes22 during training. +8 +Summary and Outlook +We have presented new work on how to solve Rubik’s cube with n-tuple systems, reinforce- +ment learning and an MCTS solver. The main ideas were already presented in Scheier- +mann and Konen (2022) but only for HTM and up to p = 9 twists. Here we extended +this work to QTM as well and presented all the details of cube representation and n-tuple +learning algorithms necessary to reproduce our Rubik’s cube results. As a new aspect, +we added cube symmetries and studied their effect on solution quality. We found that the +use of symmetries boosts the solved rates by 10-20%. Based on this, we could increase +for QTM the number of scrambling twists where at least 45% of the cubes are solved from +p = 13 without symmetries to p = 15 with symmetries. +We cannot solve the 3x3x3 cube completely, as McAleer et al. (2019) and Agostinelli +et al. (2019) do. But our solution is much less computational demanding than their ap- +proach. +Further work might be to look into larger or differently structured n-tuple systems, per- +haps utilizing the staging principle that Ja´skowski (2018) used to produce world-record +results in the game 2048. +21a set of 15, 24, ... numbers has to be ordered on a 4 × 4, 5 × 5, ... square with one empty field +223 · 106 × 16 = training episodes × episode length Etrain. This is an upper bound: some episodes may +have shorter length, but each unsolved episode has length Etrain. +33 + +References +F. Agostinelli, S. McAleer, A. Shmakov, and P. Baldi. Solving the Rubik’s cube with deep +reinforcement learning and search. Nature Machine Intelligence, 1(8):356–363, 2019. +1, 4, 5, 21, 22, 32, 33 +S. Bagheri, M. Thill, P. Koch, and W. Konen. Online adaptable learning rates for the game +Connect-4. IEEE Transactions on Computational Intelligence and AI in Games, 8(1): +33–42, 2015. 25 +D. F. Beal and M. C. Smith. Temporal coherence and prediction decay in TD learning. In +T. Dean, editor, Int. Joint Conf. on Artificial Intelligence (IJCAI), pages 564–569. Morgan +Kaufmann, 1999. ISBN 1-55860-613-0. 25 +W. W. Bledsoe and I. Browning. Pattern recognition and reading by machine. In Proceed- +ings of the Eastern Joint Computer Conference, pages 225–232, 1959. 16 +C. B. Browne, E. Powley, D. Whitehouse, S. M. Lucas, P. I. Cowling, P. Rohlfshagen, +S. Tavener, D. 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(4))? +This problem seems ill-posed at first sight, because a certain face color, e.g. white, +appears multiple times in fc and it is not possible to tell from the appearance of white alone +to which sticker location sℓ it corresponds. But with a little more effort, i.e. by looking at the +neighbors of the white sticker, we can solve the problem, as we show in the following. +A.1 +2x2x2 cube +All cubies of the 2x2x2 cube are corner cubies. We track for each cubie exactly one sticker. +This can be for example the set +B = {0, 1, 2, 3, 12, 13, 14, 15} +of 8 stickers, which is the same as the set of tracked stickers shown in Fig. 11. +For each s ∈ B: +1. Build the cubie that contains s as the first sticker.23 +2. Locate the cubie in fc. That is, find a location in fc with the same color as the 1st +cubie face. If found, check if the neighbor to the right24 has the color of the 2nd cubie +face. If yes, check if its neighbor to the right has the color of the 3rd cubie face. If +yes, we have located the cubie in fc and we return it, i.e. its three sticker locations +C = [a, b, c]. +3. Having located the cubie, we can infer three elements of sℓ: +sℓ[s] += +C[0] +(26) +sℓ[R[s]] += +C[1] +(27) +sℓ[R[R[s]]] += +C[2] +(28) +Here R[s] is the right neighbor of sticker s. R[R[s]]] is the left neighbor. +In total, we have located 8 × 3 = 24 stickers, i.e. the whole transformation for sℓ.25 +23We know for example from looking at the default cube in Fig. 11 that sticker s = 0 is part of the 0-8-4-cubie. +24By neighbor to the right we mean the next sticker when we march in clockwise orientation around the +actual cubie. +25The relevant GBG source code is in CubeState.locate and CubeState2x2.apply_sloc_slow. +37 + +A.2 +3x3x3 cube +The 3x3x3 cube has 8 corner cubies and 12 edge cubies. We track for each cubie exactly +one sticker. This can be for the corners the set +B = {0, 2, 4, 6, 24, 26, 28, 30} +and for the edges the set +E = {1, 3, 5, 7, 25, 27, 29, 31, 11, 15, 21, 33}. +We do for the corner set B the same as we did for the 2x2x2 cube. +For each element s ∈ E of the edge set: +1. Build the edge cubie cE that contains s as the first sticker. +2. Locate the cubie in fc. That is, find an edge location in fc with the same color as the +1st cubie face. If found, check if the other sticker of that cubie has the same color as +the other sticker of cE. If yes, we have located the edge cubie in fc and we return it, +i.e. its two stickers C = [a, b]. +3. Having located the cubie, we can infer two elements of sℓ: +sℓ[s] += +C[0] +(29) +sℓ[O[s]] += +C[1] +(30) +Here O[s] is the other sticker of the edge cubie that has sticker s as first sticker. +In total, we have located +8 × 3 + 12 × 2 = 48 +stickers, i.e. the whole transformation for sℓ.26 +B +N-Tuple Representations for the 3x3x3 Cube +In this appendix we describe the n-tuple representations of the cube, analogously to the +2x2x2 cube Sec. 4, but now for the 3x3x3 cube. +B.1 +CUBESTATE +A natural way to translate the cube state into a board is to use the flattened representation +of Fig. 4 as the board and extract from it the 48-element vector b, according to the given +numbering. The kth element bk represents a certain cubie face location and gets a number +from {0, . . . , 5} according to its current face color fc. The solved cube is for example +represented by b = [00000000 11111111 . . . 55555555]. +This representation CUBESTATE is what the BoardVecType CUBESTATE in our GBG- +implementation means: Each board vector is a copy of fcol, the face colors of all cubie +faces. An upper bound of possible combinations for b is 648 = 2.2 · 1032. This is much +larger than the true number of distinct states (Sec. 2.2.2) which is 4.3 · 1019. +26The relevant GBG source code is in CubeState.locate, CubeState3x3.locate_edge and CubeState3x3.apply_sloc_slow. +38 + +Table 10: The correspondence edge location ↔ STICKER2 for the solved cube. The yellow +colored cells show the location of the 12 edge stickers that we track. +3x3x3 +location +1 3 5 +7 +9 11 13 15 +17 19 21 23 +25 27 29 31 +33 35 37 39 +41 43 45 47 +STICKER2 +edge +A B C D +D +G +K +E +E +J +F +A +I +J +K +L +H +B +F +I +L +G +C +H +face ID +1 1 1 +1 +2 +2 +2 +2 +1 +1 +1 +1 +1 +1 +1 +1 +2 +2 +2 +2 +1 +1 +1 +1 +B.2 +STICKER +McAleer et al. (2019) had the interesting idea for the 3x3x3 cube that 20 stickers (cubie +faces) are enough. To characterize the 3x3x3 cube, we need according to McAleer et al. +(2019) only one (not 2 or 3) sticker for every of the 20 cubies, as shown in Fig. 10. This +is because the location of one sticker uniquely defines the location and orientation of that +cubie. We name this representation STICKER in GBG. +We track the 4 top corner stickers 0,2,4,6 plus the 4 bottom corner stickers 24,26,28,30 +plus one sticker for each of the 12 edge stickes as shown in Fig. 10, in total 20 stickers and +ignore the 28 other stickers. +How to lay out this representation as a board? – McAleer et al. (2019) create a rect- +angular one-hot-encoding board with 20 × 24 = 480 cells (20 rows for the stickers and +24 columns for the locations27) carrying only 0’s and 1’s. This is fine for the approach of +McAleer et al. (2019), where they use this board as input for a DNN, but not so nice for +n-tuples. Without constraints, such a board amounts to 2480 ≈ 10145 combinations, which +is unpleasantly large (much larger than in CUBESTATE).28 +Another possibility to lay out the board: Specify 20 board cells (the stickers) with 24 +position values each. This amounts to 2420 = 4.0 · 1027 combinations. +B.3 +STICKER2 +Analogously to Sec. 4.3, we represent the 24 corner locations and 24 edge locations as: +corner location = (corner cubie, face ID), +edge location = (edge cubie, face ID). +That is, each corner location is represented by a corner cubie a,b,c,d,e,f,g,h and by a face +ID 1,2,3. Table 7 shows the explicit numbering in this new representation. Additionally, +each edge location is represented by an edge cubie A,B,C,D,E,F,G,H,I,J,K,L29 and by a +face ID 1,2. Convention for face ID numbering of edge cubies: For top- and bottom-layer +edge cubies, it is 1 for U and D stickers, 2 else. The face ID for middle-layer edge cubies is +1 for F and B stickers, 2 else. Table 10 shows the explicit numbering in this representation. +The corresponding board consists of 8 + 8 + 12 +12 = 40 cells shown in Table 11. +The 8 cell pairs in the first two rows code the locations of the tracked corner stickers +278 · 3 for the corner stickers and 12 · 2 for the edge stickers +28McAleer et al. (2019) do not need a weight for every of the 2480 possible states, as the n-tuple network +would need. Instead they need only 480 · 4096 = 2 · 106 weights to the first hidden layer having 4096 neurons. +294 U-stickers, 4 D-sticker, 4 middle-layer stickers (2F, 2B) +39 + +0,2,4,6,24,26,28,30, see Table 7 in Sec. 4.3. The 12 cell pairs in the last two rows code the +location of the tracked edge stickers 1,3,5,7,17,21,43,47,25,27,29,31, see Table 10. This +n-tuple coding requires tuple cells with varying number of position values and leads to +88 · 38 · 1212 · 212 = 4.0 · 1027 +combinations in representation STICKER2.30 +Table 11: STICKER2 board representation for the default 3x3x3 cube. For the BoardVector, +cells are numbered row-by-row from 0 to 39. +corner +a +b +c +d +e +f +g +h +8 positions +face ID +1 +1 +1 +1 +1 +1 +1 +1 +3 positions +edge +A +B +C +D +E +F +G +H +I +J +K +L +12 positions +face ID +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +2 positions +B.4 +Adjacency Sets +To create n-tuples by random walk, we need to define adjacency sets (sets of neighbors) +for every board cell k. +For CUBESTATE, the board is the flattened representation of the 3x3x3 cube (Fig. 4). +The adjacency set is defined as the 4-point neighborhood, where two stickers are neigh- +bors if they are neighbors (share a common edge) on the cube. +For STICKER2, the board consists of 40 cells shown in Table 11. Since it matters for +the corner stickers mostly where the other corner stickers are and for the edge stickers +mostly where the other edge stickers are, it is reasonable to form two adjacency subsets +S1 = {00, . . . , 15} and S2 = {16, . . . , 39} and to define the adjacency set +Adj(k) = Si \ {k} +for each k ∈ Si, i = 1, 2. +C +Hyperparameters +In this appendix we list all parameter settings for the GBG agents used in this paper. Pa- +rameters were manually tuned with two goals in mind: (a) to reach high-quality results and +(b) to reach stable (robust) performance when conducting multiple training runs with differ- +ent random seeds. The agents listed further down are the best-so-far agents found (best +among all agents that learn from scratch by self-play). +The detailed meaning of RL parameters is explained in Konen and Bagheri (2021): +30This is, by the way, identical to (8·3)8 ·(12·2)12 = 24(8+12) = 2420 = 4.0·1027, the same number we had +above in the second mode of STICKER. But STICKER2 has the advantage that the combinations are spread +over more board cells (40) than in STICKER (20). By having more board cells with fewer position values, the +n-tuples can better represent the relationships between cube states. +40 + +• Algorithms 2, 5 and 7 in Konen and Bagheri (2021) explain parameters α (learning +rate), γ (discount factor), ϵ (exploration rate) and output sigmoid σ (either identity or +tanh). +• Appendix A.3 explains our eligibility method, parameters are: eligibility trace factor λ, +horizon cut ch, eligibility trace type ET (normal) or RESET (reset on random move). +If not otherwise stated, we use in this paper λ = 0 (no eligibility traces). For λ = 0, +horizon cut ch and eligibity trace type are irrelevant. If λ > 0, their defaults ch = 0.1 +and trace type ET apply. +• Appendix A.5 explains our TCL method (also summarized in Sec. 5.2.1). Parameters +of TCL are: TC-Init (initialization constant for counters), TC transfer function (TC-id +or TC-EXP), β (exponential factor in case of TC-EXP), TC accumulation type (delta +or recommended weight-change). +Another branch of our algorithm is the MCTS wrapper, which can be used to wrap +TD-N-tuple agents during evaluation and testing. MCTS wrapping is briefly explained in +Sec. 5.2.2. The precise algorithm for MCTS wrapping is explained in detail in (Scheiermann +and Konen, 2022, Sec. II-B).31 Parameters of MCTS are: +• cPUCT : relative weight for the prior probabilities of the wrapped agent in relation to +the value that the wrapper estimates +• dmax: maximum depth of the MCTS tree, if -1: no maximum depth +• UseSoftMax: boolean, whether to use SoftMax normalization for the priors or not +• UseLastMCTS: boolean, whether to re-use the MCTS from the previous move within +an episode or not +Further parameter explanations: +• Sec. 4 in this document explains n-tuples, parameters are: number of n-tuples, length +of n-tuples, and n-tuple creation mode (fixed, random walk, random points). +• Sec. 2.5 in this document explains symmetries. If parameter nSym = 0, do not use +symmetries. If nSym > 0, use this number nSym of symmetries. In the Rubik’s cube +case, nSym is a number between 0 and 24. +• LearnFromRM: whether to learn from random moves or not. (Does not apply here, +because we use in Rubiks’s cube always ϵ = 0, i.e. we have no random moves.) +• ChooseStart-01: whether to start episodes from different 1-ply start states or always +from the default start state. (Does not apply here, because we start in Rubik’s cube +never from the default cube, but always from the p-twisted cube.) +31As (Scheiermann and Konen, 2022, Sec. IV-E) shows, the MCTS wrapper may be used as well during +training, but due to large computation times needed for this, we do not follow that route in this paper. +41 + +• Etrain: maximum episode length during training, if -1: no maximum length. +• Eeval: maximum episode length during evaluation and play, if -1: no maximum length. +All agents were trained with no MCTS wrapper inside the training loop. The hyper- +parameters of the agent for each cube variant were found by manual fine-tuning. See +also (Konen, 2022). +In the following, we list the precise settings for all agents used in this paper. If not stated +otherwise, these common settings apply to all agents: sigmoid σ = id, LearnFromRM = +false, ChooseStart-01 = false. Wrapper settings during test and evaluation: MCTS wrapper +with cPUCT = 1.0, dmax = 50, UseSoftMax = true, UseLastMCTS = true. +Common parameters of Algorithm 2 in Sec. 5.2 are: cost-to-go c = −0.1 and positive +reward Rpos = 1.0. +The parameters for training without symmetries (nSym = 0) in Sec. 6.2 are: +• 2x2x2 cube, HTM: α = 0.25, γ = 1.0, ϵ = 0.0, λ = 0.0, no output sigmoid. N-tuples: +60 7-tuples created by random walk. +TCL activated with transfer function TC-id, +TC-Init= 10−4 and rec-weight-change accumulation. 3,000,000 training episodes. +pmax = 13, Etrain = 16, Eeval = 50. +Agent filename in GBG: 2x2x2_STICKER2_AT/TCL4-p13-ET16-3000k-60-7t-stub.agt.zip +• 2x2x2 cube, QTM: same as 2x2x2 cube, HTM, but with pmax = 16, Etrain = 20. +Agent filename in GBG: 2x2x2_STICKER2_QT/TCL4-p16-ET20-3000k-60-7t-stub.agt.zip +• 3x3x3 cube, HTM: same as 2x2x2 cube, HTM, but with 120 7-tuples created by +random walk, pmax = 9, Etrain = 13. +Agent filename in GBG: 3x3x3_STICKER2_AT/TCL4-p9-ET13-3000k-120-7t-stub.agt.zip +• 3x3x3 cube, QTM: same as 3x3x3 cube, HTM, but with pmax = 13, Etrain = 16. +Agent filename in GBG: 3x3x3_STICKER2_QT/TCL4-p13-ET16-3000k-120-7t-stub.agt.zip +The agent files given in the list above are just stubs, i.e. agents that are initialized with +the correct parameters but not yet trained. This is because a trained agent can require up +to 80 MB disk space, which is too much for GitHub. Instead, a user of GBG may load such +a stub agent, train it (takes between 10-40 minutes) and save it to local disk. +When evaluating in Sec. 6.2 the trained agents with different MCTS wrappers, we test +in each case whether cPUCT = 1.0 or 10 is better. In most cases, cPUCT = 1.0 is better, +but for (2x2x2, QTM, 800 iterations) and for (3x3x3, HTM, 100 iterations) cPUCT = 10.0 is +the better choice. +The parameters for training with symmetries (nSym = 8, 16, 24) in Sec. 6.4 are: +• 3x3x3 cube, QTM: same as 3x3x3 cube, QTM in Sec. 6.2, but with nsym = 8, 16, 24. +Agent filename in GBG: 3x3x3_STICKER2_QT/TCL4-p13-ET16-3000k-120-7t-nsym08-stub.agt.zip, +3x3x3_STICKER2_QT/TCL4-p13-ET16-3000k-120-7t-nsym16-stub.agt.zip, +3x3x3_STICKER2_QT/TCL4-p13-ET16-3000k-120-7t-nsym24-stub.agt.zip. +42 + +Again, the agent filenames are just stubs, i.e. agents that are initialized with the correct +parameters but not yet trained. As above, a user of GBG may load such a stub agent, train +it (which takes in the symmetry case between 5.4h and 13h, see Table 9) and save it to +local disk. +For further details and experiment shell scripts, see also the associated Papers-with- +Code repository https://github.com/WolfgangKonen/PapersWithCodeRubiks. +43 + diff --git a/AtFLT4oBgHgl3EQfxTCi/content/tmp_files/load_file.txt b/AtFLT4oBgHgl3EQfxTCi/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fbb2f084cb6348652ae268c75946e045a8614c56 --- /dev/null +++ b/AtFLT4oBgHgl3EQfxTCi/content/tmp_files/load_file.txt @@ -0,0 +1,2815 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf,len=2814 +page_content='e-print http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='gm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='fh-koeln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='de/ciopwebpub/Konen22b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='d/TR-Rubiks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='pdf Towards Learning Rubik’s Cube with N-tuple-based Reinforcement Learning Wolfgang Konen Technical Report, Computer Science Institute, TH Köln, University of Applied Sciences, Germany wolfgang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='konen@th-koeln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='de Sep 2022, last update Jan 2023 Abstract This work describes in detail how to learn and solve the Rubik’s cube game (or puzzle) in the General Board Game (GBG) learning and playing framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We cover the cube sizes 2x2x2 and 3x3x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We describe in detail the cube’s state representation, how to transform it with twists, whole-cube rotations and color transformations and explain the use of symmetries in Rubik’s cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Next, we discuss different n-tuple representations for the cube, how we train the agents by reinforcement learning and how we improve the trained agents during evaluation by MCTS wrapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We present results for agents that learn Rubik’s cube from scratch, with and without MCTS wrapping, with and without symmetries and show that both, MCTS wrapping and symmetries, increase computational costs, but lead at the same time to much better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We can solve the 2x2x2 cube completely, and the 3x3x3 cube in the majority of the cases for scrambled cubes up to p = 15 (QTM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We cannot yet reliably solve 3x3x3 cubes with more than 15 scrambling twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Although our computational costs are higher with MCTS wrapping and with sym- metries than without, they are still considerably lower than in the approaches of McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2018, 2019) and Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019) who provide the best Rubik’s cube learning agents so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='12167v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='LG] 28 Jan 2023 Contents 1 Introduction 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1 Motivation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3 Method Summary .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 28 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3 Number of Symmetric States .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' 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fcol 37 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1 2x2x2 cube .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='4 Adjacency Sets .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 40 C Hyperparameters 40 3 1 Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1 Motivation Game learning and game playing is an interesting test bed for strategic decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Games usually have large state spaces, and they often require complex pattern recognition and strategic planning capabilities to decide which move is the best in a certain situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' If algorithms learn a game (or, even better, a variety of different games) just by self-play, given no other knowledge than the game rules, it is likely that they perform also well on other problems of strategic decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In recent years, reinforcement learning (RL) and deep neural networks (DNN) achieved superhuman capabilities in a number of competitive games (Mnih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Silver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This success has been a product of the combination of reinforcement learning, deep learning and Monte Carlo Tree Search (MCTS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' However, current deep reinforcement learning (DRL) methods struggle in environments with a high number of states and a small number of reward states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (a) (b) Figure 1: (a) Scrambled 3x3x3 Rubik’s Cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (b) 2x2x2 cube in the middle of a twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The Rubik’s cube puzzle is an example of such an environment since the classical 3x3x3 cube has 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3 · 1019 states and only one state (the solved cube) has a reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' A somewhat simpler puzzle is the 2x2x2 cube with 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='6 · 106 state and again only one reward state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Both cubes are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The difficult task to learn from scratch how to solve arbitrary scrambled cubes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' without being taught by expert knowledge, whether from humans or from computerized solvers) was not achievable with DRL methods for a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Recently, the works of McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2018, 2019) and Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019) provided a breakthrough in that direction (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1 and 7 for details): Their approach DAVI (Deep Approximate Value Iteration) learned from scratch to solve arbitrary scrambled 3x3x3 cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This work investigates whether TD-n-tuple learning with much lower computational de- mands can solve (or partially solve) Rubik’s cube as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2 Overview The General Board Game (GBG) learning and playing framework (Konen, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Konen and Bagheri, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Konen, 2022) was developed for education and research in AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' GBG allows applying the new algorithm easily to a variety of games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' GBG is open source and available on GitHub1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The main contribution of this paper is to take the TD-n-tuple approach from GBG (Scheiermann and Konen, 2022) that was also successful on other games (Othello, ConnectFour) and to investigate this algorithm on various cube puzzles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We will show that it can solve the 2x2x2 cube perfectly and the 3x3x3 cube partly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' At the same time it has drastically reduced computational requirements compared to McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We will show that wrapping the base agent with an MCTS wrapper, as it was done by McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019) and Scheiermann and Konen (2022), is essential to reach this success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This work is at the same time an in-depth tutorial how to represent a cube and its transformations within a computer program such that all types of cube operations can be computed efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' As another important contribution we will show how symmetries (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='5, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='4) applied to cube puzzles can greatly increase sample efficiency and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The rest of this paper is organized as follows: Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 2 lays the foundation for Rubik’s cube, its state representation, its transformations and its symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 3 we in- troduce n-tuple systems and how they can be used to derive policies for game-playing agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 4 defines and discusses several n-tuple representations for the cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 5 presents algorithms for learning the cube: first the DAVI algorithm of McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019) and then our n-tuple-based TD learning (with extensions TCL and MCTS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 6 we present the results when applying our n-tuple-based TD learning method to the 2x2x2 and the 3x3x3 cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 7 discusses related work and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 8 con- cludes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='com/WolfgangKonen/GBG 5 2 Foundations 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1 Conventions and Symbols We consider in this paper two well-known cube types, namely the 2x2x2 cube (pocket cube) and the 3x3x3 cube (Rubik’s cube).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1 Color arrangement Each cube consists of smaller cubies: 8 corner cubies for the 2x2x2 cube and 8 corner, 12 edge and 6 center cubies for the 3x3x3 cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' A corner cubie has 3 stickers of different color on its 3 faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' An edge cubie has two, a center cubie has one sticker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We enumerate the 6 cube faces with (ULF) = (Up, Left, Front) and (DRB) = (Down, Right, Back).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We number the 6 colors with 0,1,2,3,4,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' My cube has these six colors 012 = wbo = (white,blue,orange) in the (ULF)-cubie2 and 345 = ygr = (yellow,green,red) in the opposing (DRB)-cubie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The solved cube in default position has colors (012345) for the faces (ULFDRB), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' the white color is at the Up face, blue at Left, orange as Front and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We can cut the cube such that up- and bottom-face can be folded away and have a flattened representation as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' w b o g r y Figure 2: The face colors of the default cube in flattened representation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2 Twist and Rotation Symbols Twists of cube faces are denoted by uppercase letters U, L, F, D, R, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Each of these twists means a 90◦ counterclockwise rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3 If U = U1 is a 90◦ rotation, then U2 is a 180◦ rotation and U3=U−1 is a 270◦ rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Whole-cube rotations are denoted by lowercase letters u, ℓ, f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (We do not need d, r, b here, because d = u−1, r = ℓ−1 and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=') Further symbols like fc[i], sℓ[i] that characterize a cube state will be explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3 Twist Types Cube puzzles can have different twist types or twist metrics: 2We run through the faces of a cubie in counter-clockwise orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 3The rotation is counterclockwise when looking at the respective face 6 QTM (quarter turn metric): only quarter twists are allowed: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' U1 and U−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' HTM (half turn metric): quarter and half turns (twists) are allowed: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' U1, U2, U3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' By allowed we mean what counts as one move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In QTM we can realize U2 via U U as well, but it costs us 2 moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In HTM, U2 counts as one move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The twist type influences God’s number and the branching factor of the game, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2 Facts about Cubes 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1 2x2x2 Cube The number of distinct states for the 2x2x2 pocket cube is (Wikipedia, 2022a) 8!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' · 37 24 = 7!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' · 36 = 3, 674, 160 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='6 · 106 (1) Why this formula?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' — We have 8 cubies which we can place in 8!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' ways on the 8 cube positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Each but the last cubie has the freedom to appear in 3 orientations, which gives the factor 37 (the last cubie is then in a fixed orientation, the other two orientations would yield illegal cube states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' – Each of these raw states has the (ygr)-cubie in any of the 24 possible positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Or, otherwise speaking, each truly different state appears in 24 whole-cube rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' To factor out the whole-cube rotations, we count only the states with (ygr)-cubie in its default position (DRB) and divide the number of raw states by 24, q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' God’s number: What is the minimal number of moves needed to solve any cube posi- tion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' – For the 2x2x2 pocket cube, it is 11 in HTM (half-turn metric) and 14 in QTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Branching factor: 3 · 3 = 9 in HTM and 3 · 2 = 6 in QTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2 3x3x3 Cube The number of distinct states for the 3x3x3 Cube is (Wikipedia, 2022b) 8!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' · 37 · 12!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' · 211 2 = 43, 252, 003, 274, 489, 856, 000 ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3 · 1019 (2) Why this formula?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' – We have 8 corner cubies which we can place in 8!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' ways on the 8 cube positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Each but the last cubie has the freedom to appear in 3 orientations, which gives the factor 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We have 12 edge cubies which we can place in 12!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' ways on the edge positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Each but the last cubie has the freedom to appear in 2 orientations, which gives the factor 211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The division by 2 stems from the fact, that neither alone two corner cubies may be swapped nor alone two edge cubies may be swapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Instead, the number of such swaps must be even (factor 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' God’s Number: What is the minimal number of moves needed to solve any cube position?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' — For the 3x3x3 Rubik’s Cube, it is 20 in HTM (half-turn metric) and 26 in QTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This is a result from Rokicki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2014), see also http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='cube20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='org/qtm/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Branching factor: 6 · 3 = 18 in HTM and 6 · 2 = 12 in QTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 7 3 2 0 1 5 4 8 11 18 17 23 22 6 7 9 10 19 16 20 21 14 13 15 12 Figure 3: Sticker numbering for the 2x2x2 cube 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3 The Cube State A cube should be represented by objects in GBG in such a way that (a) cube states that are equivalent are represented by identical objects (b) if two cube states are equivalent, it should be easy to check this by comparing their objects (c) cube transformations are easy to carry out on these objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Condition (a) means that if two twist sequences lead to the same cube state (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' U−1 and UUU), this should result also in identical objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Condition (b) means, that the equality should be easy to check, given the objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' That is, a cube should not be represented by its twist sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' A cube state is in GBG represented by abstract class CubeState and has two describ- ing members fc[i] = fcol[i] (3) sℓ[i] = sloc[i] (4) fc[i] = fcol[i] denotes the face color at sticker location i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The color is one out of 0,1,2,3,4,5 for the colors w,b,o,y,g,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' sℓ[i] = sloc[i] contains the sticker location of the sticker which is in position i for the solved cube d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Members fc and sℓ are vectors with 24 (2x2x2 cube) or 48 (3x3x3 cube) elements where i denotes the ith sticker location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The stickers are numbered in a certain way which is detailed in Figures 3 and 4 for the flattened representations of the 2x2x2 and 3x3x3 cube, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In principle, one of the two members fc and sℓ would be sufficient to characterize a state, since the fcol-sloc-relation fc[sℓ[i]] = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='fc[i] (5) holds, where d denotes the default cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This is because sℓ[i] transports the sticker i of the default cube d to location sℓ[i], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' it has the color d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='fc[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' That is, we can easily 8 6 5 4 7 3 0 1 2 10 9 8 16 23 22 36 35 34 46 45 44 11 15 17 21 37 33 47 43 12 13 14 18 19 20 38 39 32 40 41 42 28 27 26 29 25 30 31 24 Figure 4: Sticker numbering for the 3x3x3 cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We do not number the center cubies, they stay invariant under twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='Table 1: The three relevant twists for the 2x2x2 cube ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='6 ' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='U twist ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='11 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='F−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='T −1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='18 19 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='16 17 13 14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='20 21 22 23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='calculate fc given sℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' With some more effort, it is also possible to calculate sℓ given fc (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Although one of these members fc and sℓ would be sufficient, we keep both because this allows to better perform assertions or cross checks during transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Sometime we need the inverse function s−1 ℓ [i]: Which sticker is at location i?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' It is easy to calculate s−1 ℓ given sℓ with the help of the relation: s−1 ℓ [sℓ[i]] = i (6) (Note that it is not possible to invert fc, because the face coloring function is not bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=') 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='4 Transformations 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1 Twist Transformations Each basic twist is a counterclockwise4 rotation of a face by 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Table 1 shows the 2x2x2 transformation functions for three basic twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Each twist transformation can be coded in two forms: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' T[i] (forward transformation): Which is the new location for the sticker being at i before the twist?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 4The rotation is counterclockwise when looking at this face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 9 2 1 3 0 23 22 5 4 8 11 18 17 6 7 9 10 19 16 20 21 14 13 15 12 Figure 5: The default 2x2x2 cube after twist U1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' T −1[i] (inverse transformation): Which is the (parent) location of the sticker that lands in i after the twist?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Example (read off from column 0 of Table 1): The L-twist transports sticker at 0 to 22: T[0] = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The (parent) sticker being at location 9 before the L-twist comes to location 0 after the twist: T −1[0] = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Likewise, for the U-twist we have T[0] = 1 and T −1[0] = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 5 the default cube after twist U1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' How can we apply a twist transformation to a cube state programmatically?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' – We denote with f′ c and s′ ℓ the new states for fc and sℓ after transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The following relations allow to calculate the transformed cube state: f′ c[i] = fc[T −1[i]] (7) s′ ℓ[s−1 ℓ [i]] = T[i] (8) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (7) says: The new color for sticker 0 is the color of the sticker which moves into location 0 (fc[9] in the case of an L-twist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' To explain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (8), we first note that s−1 ℓ [i] is the sticker being at i before the transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Then, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (8) says: „The new location for the sticker being at i before the transformation is T[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='“ For example, the L-twist transports the current sticker at location 0 to the new location T[0] = 22, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' s′ ℓ[0] = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' For the 2x2x2 cube, these 3 twists U, L, F are sufficient, because D=U−1, R=L−1, B=F−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This is because the 2x2x2 cube has no center cubies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' For the 3x3x3 cube, we need all 6 twists U, L, F, D, R, B because this cube has center cubies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In any case, we will show in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2 that only one row in Table 1 or Table 2, say T for the U-twist, has to be known or established ’by hand’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' All other twists and their inverses can be calculated programmatically with the help of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (9)-(15) that will be derived in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='Table 2: The U twist for the 3x3x3 cube ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='7 ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='32 33 34 35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='36 37 38 39 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='40 41 42 43 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='44 45 46 47 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='U twist T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='24 25 26 27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='28 29 30 31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='32 33 44 45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='46 37 38 39 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='40 41 42 43 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='10 47 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='Table 3: Two basic whole-cube rotations for the 2x2x2 cube ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='10 11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='12 13 14 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='16 17 18 19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='20 21 22 23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='u rotation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='18 19 16 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='15 12 13 14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='21 22 23 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='f rotation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='14 15 12 13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='10 11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='17 18 19 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='23 20 21 22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='u−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='T −1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='22 23 20 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='13 14 15 12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='10 11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='19 16 17 18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='f−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='T −1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='18 19 16 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='15 12 13 14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='21 22 23 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='Normalizing the 2x2x2 Cube ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='As stated above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' the 3 twists U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' F are sufficient for the 2x2x2 cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Therefore, the (DRB)-cubie will never leave its place, whatever the twist sequence formed by U, L, F is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The (DRB)-cubie has the stickers (12, 16, 20), and we can check in Table 1 that columns (12, 16, 20) are always invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' If we have an arbitrary initial 2x2x2 cube state, we can normalize it by applying a whole-cube rotation such that the (ygr)-cubie moves to the (DRB)-location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Normalizing the 3x3x3 Cube In the case of the 3x3x3 cube, all center cubies will be not affected by any twist sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Therefore, we normalize a 3x3x3 cube state by applying initially a whole-cube rotation such that the center cubies are in their normal position (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' white up, blue left and so on).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2 Whole-Cube Rotations (WCR) Each basic whole-cube rotation (WCR) is a counterclockwise rotation of the whole cube around the u, l, f-axis by 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Table 3 shows two of the 2x2x2 transformation functions for basic whole-cube rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Each rotation can be coded in two forms: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' T[i] (forward transformation): Which is the new location for the sticker being at i before the twist?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' T −1[i] (inverse transformation): Which is the (parent) location of the sticker that lands in i after the twist?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Besides the basic rotation u there is also u2 (180◦) and u3 = u−1 (270◦ = −90◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' All whole-cube rotations can be generated from these two forward rotations u and f: First, we calculate the inverse transformations via T −1[T[i]] = i (9) where T is a placeholder for u or f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Next, we calculate the missing base rotation ℓ (counter- clockwise around the left face) as ℓ = fuf−1 (10) We use here the programm-code-oriented notation „first trafo first“: Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (10) reads as „first f, then u, then f−1“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='5 5In programm code the relation would read cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='fTr(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='uTr().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='fTr(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This is „first trafo first“, because each transformation is applied to the cube state object to the left and returns the transformed cube state object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='Table 4: All 24 whole-cube rotations (in first-trafo-first notation) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='number ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='first rotation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='∗ u0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='∗ u1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='∗ u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='∗ u3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='00-03 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='id (white up) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='id ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='u3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='04-07 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='f (green up) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='fu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='fu2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='fu3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='08-11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='f2 (yellow up) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='f2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='f2u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='f2u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='f2u3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='12-15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='f−1 (blue up) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='f−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='f−1u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='f−1u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='f−1u3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='16-19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='ℓ (orange up) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='ℓu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='ℓu2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='ℓu3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='20-23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='ℓ−1 (red up) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='ℓ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='ℓ−1u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='ℓ−1u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='ℓ−1u3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='Table 5: The 24 inverse whole-cube rotations (in first-trafo-first notation) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='number ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='first rotation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='∗ u0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='∗ u1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='∗ u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='∗ u3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='00-03 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='id (white up) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='id ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='u3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='u1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='04-07 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='f (green up) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='f−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='ℓu3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='fu2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='ℓ−1u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='08-11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='f2 (yellow up) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='f2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='f2u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='f2u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='f2u3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='12-15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='f−1 (blue up) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='ℓ−1u3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='f−1u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='ℓu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='16-19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='ℓ (orange up) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='ℓ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='f−1u3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='ℓu2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='fu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='20-23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='ℓ−1 (red up) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='fu−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='ℓ−1u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='f−1u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='The other basic whole-cube rotations d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' b are not needed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' because d = u−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' r = ℓ−1 and b = f−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The basic whole-cube rotations are rotations of the whole cube around just one axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' But there are also composite whole-cube rotations which consists of a sequence of basic rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' How many different (composite) rotations are there for the cube?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' – A little thought reveals that there are 24 of them: To be specific, we consider the default cube where we have 4 rotations with the white face up, 4 with the blue face up, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In total we have 6 · 4 = 24 rotations since there are 6 faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Table 4 lists all of them, togehter with the WCR numbering convention used in GBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Sometimes we need the inverse whole-cube rotations which are given in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In this table, we read for example from the element with number 5, that the WCR with key 5 (which is fu according to Table 4) has the inverse WCR ℓu3 such that fu ℓu3 = id holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' For convenience, we list in Table 6 the relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' For example, the trafo with Key=5 (fu) has the inverse trafo with InverseKey=19 (ℓu3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Note that there are 10 whole-cube rotations which are their own inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Generating all twists from U twist With the help of WCRs we can generate the other twists from the U twist only: We simply rotate the face that we want to twist to the up-face, 12 Table 6: Whole-cube rotations: relation key 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 inv key 0 3 2 1 12 19 6 21 8 9 10 11 4 23 14 17 20 15 18 05 16 7 22 13 apply the U twist and rotate back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This reads in first-trafo-first notation: L = f−1Uf (11) F = ℓUℓ−1 (12) D = f2Uf2 (13) R = fUf−1 (14) B = ℓ−1Uℓ (15) Thus, given the U twist from Table 1 or Table 2 and the basic WCRs given in Table 3 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (10), we can calculate all other forward transformations with the help of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (11)–(15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Then, all inverse transformations are calculable with the help of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3 Color Transformations Color transformations are special transformations that allow to discover non-trivial symmet- ric (equivalent) states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' One way to describe a color transformation is to select a valid color permutation and to paint each sticker with the new color according to this color permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This is of course nothing one can do with a real cube without destroying or altering it, but it is a theoretical concept leading to an equivalent state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Another way of looking at it is to record the twist sequence that leads from the default cube to a certain scrambled cube state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Then we go back to the default cube, make at first a whole-cube rotation (leading to a color-transformed default cube) and then apply the recorded twist sequence to the color-transformed default cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In any case, the transformed cube will be usually not in its normal position, so we apply finally a normalizing operation to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' What are valid color permutations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' – These are permutations of the cube colors reach- able when applying one of the available 24 WCRs (Table 4) to the default cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' For exam- ple, if we apply WCR f (number 04) to the default cube, we get g w o y r b Figure 6: The color transformation according to WCR f (number 04) that is, g (green) is the new color for each up-sticker that was w (white) before and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The colors o and r remain untouched under this color permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' [However, other transformations like fu, fu2 and fu3 will change every color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='] 13 2 1 3 0 23 22 5 4 8 11 18 17 6 7 9 10 19 16 20 21 14 13 15 12 Figure 7: The cube of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 5 before color transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 16 19 17 18 20 23 2 1 11 10 13 12 3 0 8 9 14 15 21 22 4 7 5 6 8 2 9 1 4 7 14 11 18 17 23 0 5 6 15 10 19 16 20 3 21 13 22 12 (a) (b) Figure 8: The cube of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 7 with color transformation from Fig 6: (a) before normalization, (b) after normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' How can we apply a color transformation to a cube state programmatically?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' – We denote with f′ and s′ ℓ the new states for f and sℓ after transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The following relations allow to calculate the transformed cube state: f′ c[i] = c[fc[i]] (16) s′ ℓ[s−1 ℓ [i]] = T[i] (17) where c[] is the 6-element color trafo vector (holding the new colors for current colors 0:w, 1:b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=', 5:r) and T is the 24- or 48-element vector of the WCR that produces this color transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (16) is simple: If a certain sticker has color 0 (w, white) before the color transformation, then it will get the new color c[0], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 4 (g, green), after the transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (17) looks complicated, but it has a similar meaning as in the twist trafo: Take i = 0 as example: The new place for the sticker being at 0 before the trafo (and coming from s−1 ℓ [0]) is T[0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Therefore, we write the number T[0] into s′ ℓ[s−1 ℓ [0]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' A color transformation example is shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 7 is just a replication of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 5 showing a default cube after U1 twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The color transformation number 04 applied to the cube of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 7 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 8 (a)-(b) in two steps: (a) The stickers are re-painted and re-numbered (white becomes green, blue becomes white and so on).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The structure of coloring is the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Now the (DRB)- cubie is no longer the (ygr)-cubie, it does not carry the numbers (12,16,20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 14 (b) We apply the proper WCR that brings the (ygr)-cubie back to the (DRB)-location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Compared to (a), each 4-sticker cube face is just rotated to another face, but not changed internally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We can check that the (DRB)-location now carries again the num- bers (12,16,20), as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 7 and as it should for a normalized cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='5 Symmetries Symmetries are transformations of the game state (and the attached action, if applicable) that lead to equivalent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' That is, if s is a certain state with value V (s), then all states ssym being symmetric to s have the same value V (ssym) = V (s) because they are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Equivalent means: If s can be solved by a twist sequence of length n, then ssym can be solved by an equivalent twist sequence of same length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In the case of Rubik’s cube, all whole-cube rotations (WCRs) are symmetries because they do not change the value of a state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' But whole-cube rotations are ’trivial’ symmetries because they are usually factored out by the normalization of the cube: After 2x2x2 cube normalization, which brings the (ygr)-cubie in a certain position, or after 3x3x3 cube nor- malization, which brings the center cubies in certain faces, all WCR-symmetric states are transformed to the same state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Non-trivial symmetries are all color transformations (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3): In general, color trans- formations transform a state s to a truly different state ssym, even after cube normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='6 Since there are 24 color transformations in Rubik’s cube, there are also 24 non-trivial sym- metries (including self).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Symmetries are useful to learn to solve Rubik’s cube for two reasons: (a) to accelerate learning and (b) to smooth an otherwise noisy value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (a) Accelerated learning: If a state s (or state-action pair) is observed, not only the weights activated by that state are updated, but also the weights of all symmetric states ssym, because they have the same V (ssym) = V (s) and thus the same reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In this way, a single observed sample is connected with more weight updates (better sample efficiency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (b) Smoothed value function: By this we mean that the value function V (s) is replaced by V (sym)(s) = 1 |Fs| � s′∈Fs V (s′) (18) where Fs is the set of states being symmetric to s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' If V (s) were the ideal value function, both terms V (s) and V (sym)(s) would be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='7 But in a real n-tuple network, V (s) is non-ideal due to n-tuple-noise (cross-talk from other states that activate the same n-tuple LUT entries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' If we average over the symmetric states s′ ∈ Fs, the noise will be dampened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 6In rare cases – e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' for the solved cube – the transformed state may be identical to s or to another symmetry state, but this happens seldom for sufficiently scrambled cubes, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 7because all V (s′) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (18) are the same for an ideal V 15 The downside of symmetries is their computational cost: In the case of Rubik’s cube, the calculation of color transformations is a costly operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' On the other hand, the number of necessary training episodes to reach a certain performance may be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In the end, the use of symmetries may pay off, because the total training time may be reduced as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In any case, we will have a better sample efficiency, since we learn more from each observed state or state-action pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Secondly, the smoothing effect introduced with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (18) can lead to better overall performance, because the smoothed value function provides a better guidance on the path towards the solved cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In order to balance computation time, GBG offers the option to select with nSym the number of symmetries actually used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' If we specify for example nSym = 8 in GBG’s Rubik’s cube implementation, then the state itself and 8 – 1 = 7 random other (non-id) color trans- formations will be selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The resulting set Fs of 8 states is then used for weight update and value function computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 3 N-Tuple Systems N-tuple systems coupled with TD were first applied to game learning by Lucas (2008), al- though n-tuples were already introduced by Bledsoe and Browning (1959) for character recognition purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The remarkable success of n-tuples in learning to play Othello (Lu- cas, 2008) motivated other authors to benefit from this approach for a number of other games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The main goal of n-tuple systems is to map a highly non-linear function in a low di- mensional space to a high dimensional space where it is easier to separate ‘good’ and ‘bad’ regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This can be compared to the kernel trick of support-vector machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' An n-tuple is defined as a sequence of n cells of the board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Each cell can have m positional values representing the possible states of that cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='8 Therefore, every n-tuple will have a (possibly large) look-up table indexed in form of an n-digit number in base m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Each entry corresponds to a feature and carries a trainable weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' An n-tuple system is a system consisting of k n-tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' As an example we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 9 an n-tuple system consisting of four 8-tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Let Θ be the vector of all weights θi of the n-tuple system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='9 The length of this vector may be large number, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' mnk, if all k n-tuples have the same length n and each cell has m positional values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Let Φ(s) be a binary vector of the same length representing the feature occurences in state s (that is, Φi(s) = 1 if in state s the cell of a specific n-tuple as indexed by i has the positional value as indexed by i, Φi(s) = 0 else).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The value function of the n-tuple network given state s is V (s) = σ (Φ(s) · Θ) (19) with transfer function σ which may be a sigmoidal function or simply the identity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 8A typical example is a 2-player board game, where we usually have 3 positional values {0: empty, 1: player1, 2: player2 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' But other, user-defined values are possible as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 9The index i indexes three qualities: an n-tuple, a cell in this n-tuple and a positional value for this cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 16 Figure 9: Example n-tuples: We show 4 random-walk 8-tuples on a 6x7 board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The tuples are selected manually to show that not only snake-like shapes are possible, but also bifurcations or cross shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Tuples may or may not be symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' An agent using this n-tuple system derives a policy from the value function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (19) as follows: Given state s and the set A(s) of available actions in state s, it applies with a forward model f every action a ∈ A(s) to state s, yielding the next state s′ = f(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Then it selects the action that maximizes V (s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Each time a new agent is constructed, all n-tuples are either created in fixed, user- defined positions and shapes, or they are formed by random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In a random walk, all cells are placed randomly with the constraint that each cell must be adjacent10 to at least one other cell in the n-tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Agent training proceeds in the TD-n-tuple algorithm as follows: Let s′ be the actual state generated by the agent and let s be the previous state generated by this agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' TD(0) learning adapts the value function with model parameters Θ through (Sutton and Barto, 1998) Θ ← Θ + αδ∇ΘV (s) (20) Here, α is the learning rate and V is in our case the n-tuple value function of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' δ is the usual TD error (Sutton and Barto, 1998) after the agent has acted and generated s′: δ = r + γV (s′) − V (s) (21) where the sum of the first two terms, reward r plus the discounted value γV (s′), is the desirable target for V (s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 10The form of adjacency, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 4- or 8-point neighborhood or any other (might be cell-dependent) form of adjacency, is user-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 17 0 3 5 6 6 5 6 3 4 4 5 6 54 N-Tuple Representions for the Cube In order to apply n-tuples to cubes, we have to define a board in one way or the other on which we can place the n-tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This is not as straightforward as in other board games, but we are free to invent abstract boards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Once we have defined a board, we can number the board cells k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' , K−1 and translate a cube state into a BoardVector: A BoardVector b is a vector of K non-negative integer numbers bk ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' , Nk − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Each k represents a board cell and every board cell k has a predefined number Nk of position values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='11 A BoardVector is useful to calculate the feature occurence vector Φ(s) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (19) for a given n-tuple set: If an n-tuple contains board cell k, then look into bk to get the position value for this cell k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Set Φi(s) = 1 for that index i that indexes this n-tuple cell and this position value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In the following we present different options for boards and BoardVectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We do this mainly for the 2x2x2 cube, because it is somewhat simpler to explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' But the same ideas apply to the 3x3x3 cube as well, they are just a little bit longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Therefore, we defer the lengthy details of the 3x3x3 cube to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1 CUBESTATE A natural way to translate the cube state into a board is to use the flattened representation of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 11 as the board and extract from it the 24-element vector b, according to the given numbering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The kth element bk represents a certain cubie face location and gets a number from {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' , 5} according to its current face color fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The solved cube is for example represented by b = [0000 1111 2222 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 5555].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This representation CUBESTATE is what the BoardVecType CUBESTATE in our GBG- implementation means: Each board vector is a copy of fcol, the face colors of all cubie faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' fcol is also the vector that uniquely defines each cube state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' An upper bound of possible combinations for b is 624 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='7 · 1018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' If we factor out the (DRB)-cubie, which always stays at its home position, we can reduce this to 21 board cells with 6 positional values, leading to 621 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1 · 1016 weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Both numbers are of course way larger than the true number of distinct states (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1) which is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='6 · 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This is because most of the combinations are dead weights in the n-tuple LUTs, they will never be activated during game play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The dead weights occur because many combinations are not realizable, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' three white faces in one cubie or any of the 63 − 8 · 3 = 192 cubie-face-color combinations that are not present in the real cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The problem is that the dead weights are scattered in a complicated way among the active weights and it is thus not easy to factor them out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2 STICKER McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019) had the interesting idea for the 3x3x3 cube that 20 stickers (cubie faces) are enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' To characterize the full 3x3x3 cube, we need only one (not 2 or 3) sticker 11In GBG package ntuple2 (base for agent TDNTuple3Agt), all Nk have to be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In package ntuple4( base for agent TDNTuple4Agt), numbers Nk may be different for different k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 18 (a) Top view (b) Bottom view Figure 10: The sticker representation used to reduce dimensionality: Stickers that are used are shown in white, whereas ignored stickers are dark blue (from McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 3 2 0 1 5 4 8 11 18 17 23 22 6 7 9 10 19 16 20 21 14 13 15 12 Figure 11: Tracked stickers for the 2x2x2 cube (white), while ignored stickers are blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' for every of the 20 cubies, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This is because the location of one sticker uniquely defines the location and orientation of that cubie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We name this representation STICKER in GBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Translated to the 2x2x2 cube, this means that 8 stickers are enough because we have only 8 cubies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We may for example track the 4 top stickers 0,1,2,3 plus the 4 bottom stickers 12,13,14,15 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 11 and ignore the 16 other stickers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Since we always normalize the cube such that the (DRB)-cubie with sticker 12 stays in place, we can reduce this even more to 7 stickers (all but sticker 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' How to lay out this representation as a board?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' – McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019) create a rect- angular one-hot-encoding board with 7 × 21 = 147 cells (7 rows for the stickers and 21 columns for the locations) carrying only 0’s and 1’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This is fine for the approach of McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019), where they use this board as input for a DNN, but not so nice for n-tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Without constraints, such a board amounts to 2147 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='7 · 1044 combinations, which is unpleasantly large (much larger than in CUBESTATE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='12 STICKER has more dead weights than CUBESTATE, so it seems like a step back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' But the point is, that the dead weights are better structured: If for example sticker 0 appears at column 1 then this column and the two other columns for the same cubie are automatically 12A possible STICKER BoardVector for the default cube would read b = [1000000 0100000 0010000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' ], meaning that location 0 has the first sticker, location 1 has the second sticker, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In any STICKER BoardVector there are only 7 columns carrying exactly one 1, the other carry only 0’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Every row carries exactly one 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 19 Table 7: The correspondence corner location ↔ STICKER2 for the solved cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The yellow colored cells show the location of the 7 (2x2x2) and 8 (3x3x3) corner stickers that we track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 2x2x2 location 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 3x3x3 location 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 STICKER2 corner a b c d a d h g a g f b e f g h e c b f e h d c face ID 1 1 1 1 2 3 2 3 3 2 3 2 1 1 1 1 2 2 3 2 3 3 2 3 forbidden for all other stickers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Likewise, if sticker 1 is placed in another column, another set of 3 columns is forbidden, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We can use this fact to form a much more compact representation STICKER2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3 STICKER2 As the analysis in the preceding section has shown, the 21 location columns of STICKER cannot carry the tracked stickers in arbitrary combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Each cubie (represented by 3 columns in STICKER) carries only exactly one sticker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We can make this fact explicit by choosing another representation for the 21 locations: corner location = (corner cubie, face ID).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' That is, each location is represented by a pair: corner cubie a,b,c,d,f,g,h (we number the top cubies with letters a,b,c,d and the bottom cubies with letters e,f,g,h and omit e because it corresponds to the (DRB)-cubie) and a face ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' To number the faces with a face ID, we follow the convention that we start at the top (bottom) face with face ID 1 and then move counter-clockwise around the corner cubie to visit the other faces (2,3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Table 7 shows the explicit numbering in this new representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' To represent a state as board vector we use now a much smaller board shown in Table 8: Each cell in the first row has 7 position values (the letters) and each cell in the second row has 3 position values (the face IDs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We show in Table 8 the board vector for the default cube, b = [abcdfgh 1111111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Representation STICKER2 allows for 77 · 37 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='8 · 109 combinations in total, which is much smaller than STICKER and CUBESTATE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Table 8: STICKER2 board representation for the default 2x2x2 cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' For the BoardVector, cells are numbered row-by-row from 0 to 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' corner a b c d f g h 7 positions face ID 1 1 1 1 1 1 1 3 positions STICKER2 has some dead weights remaining, because the combinations can carry the same letter multiple times, which is not allowed for a real cube state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' But this rate of dead weights is tolerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' It turns out that STICKER2 is in all aspects better than CUBESTATE or STICKER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Therefore, we will only report the results for STICKER2 in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='4 Adjacency Sets To create n-tuples by random walk, we need adjacency sets (sets of neighbors) to be defined for every board cell k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' For CUBESTATE, the board is the flattened representation of the 2x2x2 cube (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The adjacency set is defined as the 4-point neighborhood, where two stickers are neigh- bors if they share a common edge on the cube, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' are neighbors on the cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' For STICKER2, the board consists of 16 cells shown in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Here, the adjacency set for cell k contains all other cells different from k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Again, the details of ideas similar to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='4, but now for the 3x3x3 cube, are shown in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1–B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 5 Learning the Cube 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1 McAleer and Agostinelli The works of McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2018, 2019) and Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019) contain up to now the most advanced methods for learning to solve the cube from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019) introduces the cost-to-go function for a general Marko decision process J(s) = min a∈A(s) � s′ P a(s, s′) � ga(s, s′) + γJ(s′) � (22) where P a(s, s′) is the probability of transitioning from state s to s′ by taking action a and ga(s, s′) is the cost for this transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In the Rubik’s cube case, we have deterministic tran- sitions, that is s′ = f(s, a) is deterministically prescribed by a forward model f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Therefore, the sum reduces to one term and we specialize to γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Furthermore, we set ga(s, s′) = 1, because only the length of the solution path counts, so that we get the simpler equation J(s) = min a∈A(s) � 1 + J(s′) � with s′ = f(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (23) Here, A(s) is the set of available actions in state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We additionally set J(s∗) = 0 if s∗ is the solved cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' To better understand Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (23) we look at a few examples: If s1 is a state one twist away from s∗, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (23) will find this twist and set J(s1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' If s2 is a state two twists away from s∗ and all one-twist states have already their correct labels J(s1) = 1, then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (23) will find the twist leading to a s1 state and set J(s2) = 1 + 1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' While iterations proceed, more and more states (being further away from s∗) will be correctly labeled, once their preceding states are correctly labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In the end we should ideally have J(sn) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' However, the number of states for Rubik’s cube is too large to store them all in tabular form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Therefore, McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019) and Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019) approximate J(s) with a deep neural network (DNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' To train such a network in the Rubik’s cube case, they 21 Algorithm 1 DAVI algorithm (from Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Input: B: batch size, K: maximum number of twists, M: training iterations, C: how often to check for convergence, ϵ: error threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Output: Θ, the trained neural network parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 1: function DAVI(B, K, M, C, ϵ) 2: Θ ← INITIALIZENETWORKPARAMETERS 3: ΘC ← Θ 4: for m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' , M do 5: X ←GENERATESCRAMBLEDSTATES(B, K) ▷ B scrambled cubes 6: for xi ∈ X do 7: yi ← mina∈A(s) [1 + jΘC(f(xi, a))] ▷ cost-to-go function, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (23) 8: (Θ, loss) ← TRAIN(jΘ, X, y) ▷ loss = MSE(jΘ(xi), yi) 9: if (m mod C = 0 & loss < ϵ) then 10: ΘC ← Θ 11: return Θ introduce Deep Approximate Value Iteration (DAVI)13 shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The network output jΘ(s) is trained in line 8 to approximate the (unknown) cost-to-go J(s) for every state s = xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The main trick of DAVI is, as Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019) write: „For learning to occur, we must train on a state distribution that allows information to propagate from the goal state to all the other states seen during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Our approach for achieving this is simple: each training state xi is obtained by randomly scrambling the goal state ki times, where ki is uniformly distributed between 1 and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' During training, the cost-to-go function first improves for states that are only one move away from the goal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The cost-to-go function then improves for states further away as the reward signal is propagated from the goal state to other states through the cost-to-go function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='“ Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019) use in Algorithm 1 two sets of parameters to train the DNN: the parameters Θ being trained and the parameters ΘC used to obtain improved estimates of the cost-to-go function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' If they did not use this two separate sets, performance often „saturated after a certain point and sometimes became unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Updating ΘC only after the error falls below a threshold ϵ yields better, more stable, performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='“ (Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=', 2019) To train the DNN, they used M = 1 000 000 iterations, each with batch size B = 10 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Thus, the trained DNN has seen ten billion cubes (1010) during training, which is still only a small subset of the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3 · 1019 possible cube states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The heuristic function of the trained DNN alone cannot solve 100% of the cube states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Especially for higher twist numbers ki, an additional solver or search algorithm is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This is in the case of McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019) a Monte Carlo Tree Search (MCTS), similar to AlphaZero (Silver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=', 2017), which uses the DNN as the source for prior probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019) use instead a variant of A∗-search, which is found to produce solutions with a shorter path in a shorter runtime than MCTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 13More precisely, McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019) use Autodidactic Iteration (ADI), a precursor to DAVI, very similar to DAVI, just a bit more complicated to explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Therefore, we describe here only DAVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 22 Algorithm 2 TD-n-tuple algorithm for Rubik’s cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Input: pmax: maximum number of twists, M: training iterations, Etrain: maximum episode length during training, c: nega- tive cost-to-go, Rpos: positive reward for reaching the solved cube s∗, α: learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' jΘ(s): n-tuple network value prediction for state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Output: Θ, the trained n-tuple network parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 1: function TDNTUPLE(pmax, M, Etrain, c, Rpos) 2: Θ ← INITIALIZENETWORKPARAMETERS 3: for m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' , M do 4: p ∼ U(1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' , pmax) ▷ Draw p uniformly random from {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' , pmax} 5: s ← SCRAMBLESOLVEDCUBE(p) ▷ start state 6: for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' , Etrain do 7: snew ← arg max a∈A(s) V (s′) with s′ = f(s, a) and 8: V (s′) = c + � Rpos if s′ = s∗ jΘ(s′) if s′ ̸= s∗ 9: Train network jΘ with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (20) to bring V (s) closer to target T = V (snew): V (s) ← V (s) + α(T − V (s)) 10: s ← snew 11: if (s = s∗) then 12: break ▷ break out of k-loop 13: return Θ 23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2 N-Tuple-based TD Learning To solve the Rubik’s cube in GBG we use an algorithm that is on the one hand inspired by DAVI, but on the other hand more similar to traditional reinforcement learning schemes like temporal difference (TD) learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In fact, we want to use in the end the same TD-FARL algorithm (Konen and Bagheri, 2021) that we use for all other GBG games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We show in Algorithm 2 our method, that we will explain in the following, highlighting also the similarities and dissimilarities to DAVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' First of all, instead of minimizing the positive cost-to-go as in DAVI, we maximize in lines 7-8 a value function V (s′) with a negative cost-to-go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This maximization is functionally equivalent, but more similar to the usual TD-learning scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The negative cost-to-go, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' c = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1, plays the role of the positive 1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Secondly, we replace the DNN of DAVI by the simpler-to-train n-tuple network jΘ with STICKER2 representation as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' That is, each time jΘ(s′) is requested, we first calculate for state s′ the BoardVector in STICKER2 representation, then the occurence vector Φ(s′) and the value function V (s′) according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The central equations for V (s′) in Algorithm 2, lines 7-8, work similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (23) in DAVI: If s = s1 is a state one twist away from s∗, the local search in arg max V (s′) will find this twist and the training step in line 9 moves V (s) closer to c+Rpos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='14 Likewise, neighbors s2 of s1 will find s1 and thus move V (s2) closer to 2c + Rpos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Similar for s3, s4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' under the assumption that a ’known’ state is in the neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We have a clear gradient on the path towards the solved cube s∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' If there are no ’known’ states in the neighborhood of sn, we get for V (sn) what the net maximally estimates for all those neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We pick the neighbor with the highest estimate, wander around randomly until we hit a state with a ’known’ neighbor or until we reach the limit Etrain of too many steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Note that Algorithm 2 is different from DAVI insofar that it follows the path s → s′ → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' as prescribed by the current V , which may lead to a state sequence ’wandering in the unknown’ until Etrain is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In contrast to that, DAVI generates many start states s0 drawn from the distribution of training set states and trains the network just on pairs (s0, T), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' they do just one step on the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We instead follow the full path, because we want the training method for Rubik’s cube to be as similar as possible to the training method for other GBG games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='15 Algorithm 2 is basically the same algorithm as GBG uses for other games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The only differences are (i) the cube-specific start state selection borrowed from DAVI (a 1-twist start state has the same probability as a 10-twist start state) and (ii) the cube-specific reward in line 8 of Algorithm 2 with its negative cost-go-go c which is however a common element of many RL rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Algorithm 2 currently learns with only one parameter vector Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' However, it could be extended as in DAVI to two parameter vectors Θ and ΘC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The weight training step in line 14It is relevant, that Rpos is a positive number, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='0 (and not 0, as it was for DAVI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This is because we start with an initial n-tuple network with all weights set to 0, so the initial response of the network to any state is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Thus, if Rpos were 0, a one-twist state would see all its neighbors (including s∗) initially as responding 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='0 and would not learn the right transition to s∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' With Rpos = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='0 it will quickly find s∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 15We note in passing that we tested the DAVI variant with Etrain = 1 for our TD-n-tuple method as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' However, we found that this method gave much worse results, so we stick with our GBG method here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 24 9 is done with the help of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (20) for Θ using the error signal δ of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' There are two extra elements, TCL and MCTS, that complete our n-tuple-based TD learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' They are described in the next two subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1 Temporal Coherence Learning (TCL) The TCL algorithm developed by Beal and Smith Beal and Smith (1999) is an extension of TD learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' It replaces the global learning rate α with the weight-individual product ααi for every weight θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Here, the adjustable learning rate αi is a free parameter set by a pretty simple procedure: For each weight θi, two counters Ni and Ai accumulate the sum of weight changes and the sum of absolute weight changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' If all weight changes have the same sign, then αi = |Ni|/Ai = 1, and the learning rate stays at its upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' If weight changes have alternating signs, then the global learning rate is probably too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In this case, αi = |Ni|/Ai → 0 for t → ∞, and the effective learning rate will be largely reduced for this weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In our previous work (Bagheri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=', 2015) we extended TCL to αi = g(|Ni|/Ai) where g is a transfer function being either the identity function (standard TCL) or an exponential function g(x) = eβ(x−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' It was shown in Bagheri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2015) that TCL with this exponential transfer function leads to faster learning and higher win rates for the game ConnectFour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2 MCTS We use Monte Carlo Tree Search (MCTS) (Browne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=', 2012) to augment our trained network during testing and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This is the method also used by McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019) and by AlphaGo Zero (Silver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=', 2017), but they use it also during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' MCTS builds iteratively a search tree starting with a tree containing only the start state s0 as the root node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Until the iteration budget is exhausted, MCTS does the following: In every iteration we start from the root node and select actions following the tree policy until we reach a yet unexpanded leaf node sℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The tree policy is implemented in our MCTS wrapper according to the UCB formula (Silver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=', 2017): anew = arg max a∈A(s) �W(s, a) N(s, a) + U(s, a) � (24) U(s, a) = cpuctP(s, a) � ε + � b∈A(s) N(s, b) 1 + N(s, a) (25) Here, W(s, a) is the accumulator for all backpropagated values that arrive along branch a of the node that carries state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Likewise, N(s, a) is the visit counter and P(s, a) the prior probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' A(s) is the set of actions available in state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' ε is a small positive constant for the special case � b N(s, b) = 0: It guarantees that in this special case the maximum of U(s, a) is given by the maximum of P(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The prior probabilities P(s, a) are obtained 25 Algorithm 3 TD-n-tuple training algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Input: see Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Output: Θ: trained n-tuple network parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 1: function TDNTUPLETRAIN(pmax, M, Etrain, c, Rpos) 2: Θ ← INITIALIZENETWORKPARAMETERS 3: INITIALIZETCLPARAMETERS ▷ Set TCL-accumulators Ni = Ai = 0, αi = 1 ∀i 4: for m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' , M do 5: Perform one m-iteration of Algorithm 2 with learning rates ααi instead of α 6: Ni ← Ni + ∆θi and Ai ← Ai + |∆θi| ▷ Update TCL-accumulators 7: ▷ where ∆θi is the last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (20) 8: αi ← |Ni|/Ai ∀i with Ai ̸= 0 9: return Θ Algorithm 4 Evaluation algorithm with MCTS solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Input: trained n-tuple network jΘ, p: number of scrambling twists, B: batch size, Eeval: maximum episode length during evaluation, I: number of MCTS-iterations, cPUCT : relative weight for U(s, a) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (24), dmax: maximum MCTS tree depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Output: solved rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 1: function TDNTUPLEEVAL(jΘ, p, B, Eeval, I, cPUCT , dmax) 2: X ←GENERATESCRAMBLEDCUBES(B, p) ▷ B scrambled cubes 3: Csolved ← 0 4: for xi ∈ X do 5: s ← xi 6: for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' , Eeval do 7: T ← PERFORMMCTSSEARCH(s, I, cPUCT , dmax, jΘ) 8: a ← SELECTMOSTVISITEDACTION 9: s ← f(s, a) 10: if (s = s∗) then 11: Csolved ← Csolved + 1 12: break ▷ break out of k-loop 13: return Csolved/B ▷ percentage solved by sending the trained network’s values of all follow-up states s′ = f(s, a) with a ∈ A(s) through a softmax function (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='16 Once an unexpanded leaf node sℓ is reached, the node is expanded by initializing its accumulators: W(s, a) = N(s, a) = 0 and P(s, a) = ps′ where ps′ is the softmax- squashed output jΘ(s′) of our n-tuple network for each state s′ = f(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The value of the node is the network output of the best state jΘ(sbest) = maxs′ jΘ(s′) and this value is backpropagated up the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' More details on our MCTS wrapper can be found in Scheiermann and Konen (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 16Note that the prior probabilities and the MCTS iteration are only needed at test time, so that we – different to AlphaZero – do not need MCTS during self-play training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 26 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3 Method Summary We summarize the different ingredients of our n-tuple-based TD learning method in Algo- rithm 3 (training) and Algorithm 4 (evaluation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In line 5 of Algorithm 3 we perform one m-iteration of Algorithm 2 which does an update step for weight vector Θ, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' All weights of activated n-tuple entries get a weight change ∆θi equal to the last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (20) where the global α is replaced by ααi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Line 2 in Algorithm 4 generates a set X of B scrambled cube states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Line 7 builds for each xi ∈ X an MCTS tree (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2) starting from root node xi and line 8 selects the most visited action of the root node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' If the goal state s∗ is not found during Eeval k-loop trials, this xi is considered as not being solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 6 Results 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1 Experimental setup We use for all our GBG experiments the same RL method based on n-tuple systems and TCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Only its hyperparameters are tuned to the specific game, as shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We refer to this method/agent as TCL-base whenever it alone is used for game playing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' If we wrap such an agent by an MCTS wrapper with a given number of iterations, then we refer to this as TCL-wrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We investigate two variants of Rubik’s Cube: 2x2x2 and 3x3x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We trained all TCL agents by presenting them M = 3 000 000 cubes scrambled with p random twists, where p is chosen uniformly at random from {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' , pmax}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Here, pmax = 13 [16] for 2x2x2 and pmax = 9 [13] for 3x3x3, where the first number is for HTM, while the second number in square brackets is for QTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' With these pmax cube twists we cover the complete cube space for 2x2x2, where God’s number (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2) is known to be 11 [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' But we cover only a small subset in the 3x3x3 case, where God’s number is known to be 20 [26] (Rokicki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='17 We train 3 agents for each cube variant { 2x2x2, 3x3x3 } × { HTM, QTM } to assess the variability of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The hyperparameters of the agent for each cube variant were found by manual fine- tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' For brevity, we defer the exact explanation and setting of all parameters to Ap- pendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We evaluate the trained agents for each p on 200 scrambled cubes that are created by applying the given number p of random scrambling twists to a solved cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The agent now tries to solve each scrambled cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' A cube is said to be unsolved during evaluation if the agent cannot reach the solved cube in Eeval = 50 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='18 17We limit ourselves to pmax = 9 [13] in the 3x3x3 HTM [QTM ] case, because our network has not enough capacity to learn all states of the 3x3x3 Rubik’s cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Experiments with higher twist numbers during training did not improve the solved-rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 18During training, we use lower maximum episode lengths Etrain (see Appendix C) than Eeval = 50 in order to reduce computation time (in the beginning, many episodes cannot be solved, and 50 would waste a lot of computation time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' But Etrain is always at least pmax + 3 in order to ensure that the agent has a fair chance to solve the cube and collect the reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 27 HTM 0% 25% 50% 75% 100% 1 3 5 7 9 11 13 scrambling twists percentage solved cubeWidth 2x2x2 3x3x3 iterMWrap 0 100 800 QTM 0% 25% 50% 75% 100% 1 3 5 7 9 11 13 15 scrambling twists percentage solved cubeWidth 2x2x2 3x3x3 iterMWrap 0 100 800 Figure 12: Percentage of solved cubes as a function of scrambling twists p for the trained TD-N-tuple agent wrapped by MCTS wrapper with different numbers of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The red curves are TCL-base without wrapper, the other colors show different forms of TCL-wrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Twist type is HTM (left) and QTM (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Each point is the average of 3 independently trained agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2 Cube Solving with MCTS Wrapper, without Symmetries The trained TD-N-tuple agents learn to solve the cubes to some extent, as the red curves TCL-base in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 12 show, but they are in many cases (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' p > pmax/2) far from being perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' These are the results from training each agent for 3 million episodes, but the results would not change considerably, if 10 million training episodes were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Scheiermann and Konen (2022) have shown, that the performance of agents, namely TD-N-tuple agents, is largely improved, if the trained agents are wrapped during test, play and evaluation by an MCTS wrapper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This holds for Rubik’s cube as well, as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 12 shows: For the 2x2x2 cube, the non-wrapped agent TCL-base (red curve) is already quite good, but with wrapping it becomes almost perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' For the 3x3x3 cube, the red curves are not satisfactorily: the solved-rates are below 20% for p = 9 [13] in the HTM [QTM ] case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' But at least MCTS wrapping boosts the solved-rates by a factor of 3 [QTM: from 16% to 48%] or 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='5 [HTM: from 10% to 45%].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' All these results are without incorporating symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' How symmetries affect the solved-rates will be investigated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' But before this, we look in the next section at the number of symmetries that effectively exist in a cube state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3 Number of Symmetric States Not every cube state has 24 truly different symmetric states (24 = number of color sym- metries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' For example in the solved cube, all color-symmetric states are the same (after normalization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Thus, we have here only one truly different symmetric state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' However, we show in this section that for the majority of cube states the number of truly different symmetric states is close to 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Two states are truly different if they are not the same after the normalizing operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We generate a cube state by applying p random scrambling twists to the default cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Now we apply all 24 color transformations (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3) to it and count the truly different states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 13 for 28 twistType: HTM twistType: QTM 0 4 8 12 16 0 4 8 12 16 0 5 10 15 20 25 scrambling twists NsymmetricStates cubeWidth 2x2x2 3x3x3 Figure 13: Count of truly different symmetric states for cube states generated by p random scrambling twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Each point is an average over 500 such states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' both cube sizes and both twist types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' For the 3x3x3 cube, the number of states quickly (for p > 5) approaches the maximum N = 24, while for the 2x2x2 cube it is a bit slower: p > 4 or p > 8 is needed to surpass N = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' As a consequence, it makes sense to use 16 or even 24 symmetries when training and evaluating cube agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Especially for scrambled states with higher p, the 24 color transformations used to construct symmetric states will usually lead to 24 different states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='4 The Benefit of Symmetries In order to investigate the benefits of symmetries, we first train a TCL agent with dif- ferent numbers of symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' As described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='5, we select in each step nSym = 0, 8, 16, 24 symmetric states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Which symmetric states are chosen is selected randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Symmetries are used (a) to update the weights for each symmetric state and (b) to build with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (18) a smoothed value function which is used to decide about the next action dur- ing training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' For 0, 8, 16, 24 symmetries, we train 3 agents each (3x3x3 cube, STICKER2, QTM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The 3 agents differ due to their differently created random-walk n-tuple sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 14 shows the learning curves for different nSym = 0, 8, 16, 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' It is found that agents with nSym > 0 learn faster and achieve a higher asymptotic solved rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Next, we evaluate each of the trained agents by trying to solve for each p ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' , 15} (scrambling twists) 200 different scrambled cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' During evaluation, we use again the same nSym as in training to form a smoothed value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We compare in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 15 different 29 50% 60% 70% 80% 90% 0e+00 1e+06 2e+06 3e+06 episodes percentage solved nSym 24 16 8 0 Figure 14: Learning curves for different numbers nSym = 0, 8, 16, 24 of symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Shown is the solved rate of (3x3x3, QTM) cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The solved rate is the average over all twist numbers p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' , 13 with 200 testing cubes for each p and over 3 agents with different random-walk n-tuple sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' symmetry results, both without wrapping (TCL-base, red curves) and with MCTS-wrapped agents using 100 (green) or 800 (blue) iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' It is clearly visible that MCTS wrapping has a large effect, as it was also the case in Fig 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' But in addition to that, the use of symmetries leads for each agent, wrapped or not, to a substantial increase in solved-rates (a surplus of 10-20%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' It is remarkable, that even for p=14 or 15 a solved rate above or near 50% can be reached19 by the combination (nSym=16, 800 MCTS iterations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Surprisingly, it seems that with wrapping it is only important whether we use symme- tries, not how many, since the difference between nSym = 8, 16, 24 is only marginal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' For 800 MCTS iterations, the solved rate for nSym = 24 is in most cases even smaller than that for nSym = 8, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This is surprising because it would have been expected that also with wrapping a larger nSym should lead to a smoother value function and thus should in theory produce larger solved rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' – Note that this is not a contradiction to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 14, because the learning curves were obtained without wrapping and the red TCL-base curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 15 (again without wrapping) show the same positive trend with increasing nSym20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The red curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 15 show approximately the same average solved rates as the asymptotic values in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='5 Computational Costs Table 9 shows the computational costs when training and testing with symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' All computations were done on a single CPU Intel i7-9850H @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='60GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' If we subtract the 19p is above pmax=13, the maximum twist number used during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 20i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' nSym= 24 is for every p clearly better than nSym= 16 30 QTM 3x3x3 0% 25% 50% 75% 100% 1 3 5 7 9 11 13 15 scrambling twists percentage solved nSym 0 8 16 24 iterMWrap 0 100 800 Figure 15: With symmetries: Percentage of solved cubes (3x3x3, QTM) as a function of scrambling twists p for TD-N-tuple agents trained and evaluated with different numbers of symmetries nSym and wrapped by MCTS wrappers with different iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The red curves are TCL-base (without wrapper), the other colors show different forms of TCL-wrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The solved rates are the average over 200 testing cubes for each p and over 3 agents with differ- ent random-walk n-tuple sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' computational costs for nsym= 0, computation time increases more or less linearly with iter and roughly linearly with nSym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Computation times for nSym= 24 are approximately 10x larger than those for nSym= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Computation times are dependent on the solved rate: If a cube with p = 13 is solved, the episode takes normally 12-15 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' If the cube is not solved, the episode needs 50 steps, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' a factor of 3-4 more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Thus, the numbers in Table 9 should be taken only as rough indication of the trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Bottom line: Training time through symmetries increases by a factor of 13/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='5 = 26 (nSym= 24) and testing time increases through 800 MCTS iterations by a factor of about 3130/8 ≈ 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Training with symmetries takes between 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='4h and 13h on a normal CPU, depending on the number of symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This is much less than the 44h on a 32-core server with 3 GPUs that were used by McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' But it also does not reach the same quality as McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 31 Table 9: Computation times with symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' All numbers are for 3x3x3 cube, STICKER2 and QTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Training: 3 million self-play episodes, w/o MCTS in the training loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Testing: 200 scrambled cubes with p = 13, agents wrapped by MCTS wrapper with iter iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' nSym training testing [hours] [seconds] iter 0 100 400 800 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='5 48 196 390 8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='0 241 877 1400 16 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3 464 1380 2330 24 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='0 550 1760 3130 7 Related Work Ernö Rubik invented Rubik’s cube in 1974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Rubik’s cube has gained worldwide popularity with many human-oriented algorithms being developed to solve the cube from arbitrary scrambled start states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' By ’human-oriented’ we mean algorithms that are simple to mem- orize for humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' They usually will find long, suboptimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' For a long time it was an unsolved question what is the minimal number of moves (God’s Number) needed to solve any given cube state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The early work of Thistlethwaite (1981) put an upper bound on this number with his 52-move algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This was one of the first works to systematically use group theory as an aid to solve Rubik’s cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Later, several authors have gradually reduced the upper bound 52 (Joyner, 2014), until Rokicki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2014) could prove in 2014 for the 3x3x3 cube that God’s Number is 20 in HTM and 26 in QTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Computer algorithms to solve Rubik’s cube rely often on hand-engineered features and group theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' One popular solver for Rubik’s cube is the two-phase algorithm of Kociemba (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' A variant of A∗ heuristic search was used by Korf (1991), along with a pattern database heuristic, to find the shortest possible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The problem of letting a computer learn to solve Rubik’s cube turned out to be much harder: Irpan (2016) experimented with different neural net baseline architectures (LSTM gave for him reportedly best results) and tried to boost them with AdaBoost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' However, he had only for scrambling twist ≤ 7 solved rates of better than 50% and the baseline turned out to be better than the boosted variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Brunetto and Trunda (2017) found somewhat better results with a DNN, they could solve cube states with 18 twists with a rate above 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' But they did not learn from scratch because they used an optimal solver based on Kociemba (2015) to generate training examples for the DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2016) tried to learn Rubik’s cube by genetic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' However, their learned solver could only reliably solve cubes with up to 5 scrambling twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' A breakthrough in learning to solve Rubik’s cube are the works of McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2018, 2019) and Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019): With Autodidactic Iteration (ADI) and Deep Approxi- mate Value Iteration (DAVI) they were able to learn from scratch to solve Rubik’s cube in QTM for arbitrary scrambling twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Their method has been explained in detail already in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1, so we highlight here only their important findings: McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019) only needs to inspect less than 4000 cubes with its trained network DeepCube when solving 32 for a particular cube, while the optimal solver of Korf (1991) inspects 122 billion different nodes, so Korf’s method is much slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019) extended the work of McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019) by replacing the MCTS solver with a batch-weighted A∗ solver which is found to produce shorter solution paths and have shorter run times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' At the same time, Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019) applied their agent DeepCubeA successfully to other puzzles like LightsOut, Sokoban, and the 15-, 24-, 35- and 48-puzzle21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' DeepCubeA could solve all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The deep network used by McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019) and Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019) were trained without human knowledge or supervised input from computerized solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The network of McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019) had over 12 million weights and was trained for 44 hours on a 32-core server with 3 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The network of McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019) has seen 8 billion cubes during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' – Our approach started from scratch as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' It required much less computational effort (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='4h training time on a single standard CPU for nSym=8, see Table 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' It can solve the 2x2x2 cube completely, but the 3x3x3 cube only partly (up to 15 scrambling twists).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Each trained agent for the 3x3x3 cube has seen 48 million scrambled cubes22 during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 8 Summary and Outlook We have presented new work on how to solve Rubik’s cube with n-tuple systems, reinforce- ment learning and an MCTS solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The main ideas were already presented in Scheier- mann and Konen (2022) but only for HTM and up to p = 9 twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Here we extended this work to QTM as well and presented all the details of cube representation and n-tuple learning algorithms necessary to reproduce our Rubik’s cube results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' As a new aspect, we added cube symmetries and studied their effect on solution quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We found that the use of symmetries boosts the solved rates by 10-20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Based on this, we could increase for QTM the number of scrambling twists where at least 45% of the cubes are solved from p = 13 without symmetries to p = 15 with symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We cannot solve the 3x3x3 cube completely, as McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019) and Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019) do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' But our solution is much less computational demanding than their ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Further work might be to look into larger or differently structured n-tuple systems, per- haps utilizing the staging principle that Ja´skowski (2018) used to produce world-record results in the game 2048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 21a set of 15, 24, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' numbers has to be ordered on a 4 × 4, 5 × 5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' square with one empty field 223 · 106 × 16 = training episodes × episode length Etrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This is an upper bound: some episodes may have shorter length, but each unsolved episode has length Etrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 33 References F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Agostinelli, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' McAleer, A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Heywood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Discovering Rubik’s cube subgroups using coevo- lutionary GP: A five twist experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In Proceedings of the Genetic and Evolutionary Computation Conference (GECCO), pages 789–796, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 32 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Sutton and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Barto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Reinforcement Learning: An Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' MIT Press, Cam- bridge, MA, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 17 35 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Thistlethwaite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Thistlethwaites’s 52-move algorithm, 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' jaapsch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='net/puzzles/thistle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='htm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Reconstructed by Jaap Scherphuis, retrieved Sep-01-2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 32 Wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Pocket Cube, 2022a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' URL https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='org/wiki/Pocket_Cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' retrieved Aug-17-2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 7 Wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Rubik’s Cube, 2022b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' URL https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='org/wiki/Rubik’s_ Cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' retrieved Aug-17-2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 7 36 Appendix A Calculating sloc from fcol Given the face colors fc (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (3)) of a transformed cube, how can we calculate the trans- formed sticker locations sℓ (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (4))?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This problem seems ill-posed at first sight, because a certain face color, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' white, appears multiple times in fc and it is not possible to tell from the appearance of white alone to which sticker location sℓ it corresponds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' But with a little more effort, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' by looking at the neighbors of the white sticker, we can solve the problem, as we show in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1 2x2x2 cube All cubies of the 2x2x2 cube are corner cubies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We track for each cubie exactly one sticker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This can be for example the set B = {0, 1, 2, 3, 12, 13, 14, 15} of 8 stickers, which is the same as the set of tracked stickers shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' For each s ∈ B: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Build the cubie that contains s as the first sticker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='23 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Locate the cubie in fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' That is, find a location in fc with the same color as the 1st cubie face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' If found, check if the neighbor to the right24 has the color of the 2nd cubie face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' If yes, check if its neighbor to the right has the color of the 3rd cubie face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' If yes, we have located the cubie in fc and we return it, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' its three sticker locations C = [a, b, c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Having located the cubie, we can infer three elements of sℓ: sℓ[s] = C[0] (26) sℓ[R[s]] = C[1] (27) sℓ[R[R[s]]] = C[2] (28) Here R[s] is the right neighbor of sticker s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' R[R[s]]] is the left neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In total, we have located 8 × 3 = 24 stickers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' the whole transformation for sℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='25 23We know for example from looking at the default cube in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 11 that sticker s = 0 is part of the 0-8-4-cubie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 24By neighbor to the right we mean the next sticker when we march in clockwise orientation around the actual cubie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 25The relevant GBG source code is in CubeState.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='locate and CubeState2x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='apply_sloc_slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 37 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2 3x3x3 cube The 3x3x3 cube has 8 corner cubies and 12 edge cubies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We track for each cubie exactly one sticker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This can be for the corners the set B = {0, 2, 4, 6, 24, 26, 28, 30} and for the edges the set E = {1, 3, 5, 7, 25, 27, 29, 31, 11, 15, 21, 33}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We do for the corner set B the same as we did for the 2x2x2 cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' For each element s ∈ E of the edge set: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Build the edge cubie cE that contains s as the first sticker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Locate the cubie in fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' That is, find an edge location in fc with the same color as the 1st cubie face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' If found, check if the other sticker of that cubie has the same color as the other sticker of cE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' If yes, we have located the edge cubie in fc and we return it, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' its two stickers C = [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Having located the cubie, we can infer two elements of sℓ: sℓ[s] = C[0] (29) sℓ[O[s]] = C[1] (30) Here O[s] is the other sticker of the edge cubie that has sticker s as first sticker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In total, we have located 8 × 3 + 12 × 2 = 48 stickers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' the whole transformation for sℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='26 B N-Tuple Representations for the 3x3x3 Cube In this appendix we describe the n-tuple representations of the cube, analogously to the 2x2x2 cube Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 4, but now for the 3x3x3 cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1 CUBESTATE A natural way to translate the cube state into a board is to use the flattened representation of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 4 as the board and extract from it the 48-element vector b, according to the given numbering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The kth element bk represents a certain cubie face location and gets a number from {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' , 5} according to its current face color fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The solved cube is for example represented by b = [00000000 11111111 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 55555555].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This representation CUBESTATE is what the BoardVecType CUBESTATE in our GBG- implementation means: Each board vector is a copy of fcol, the face colors of all cubie faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' An upper bound of possible combinations for b is 648 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2 · 1032.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This is much larger than the true number of distinct states (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2) which is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3 · 1019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 26The relevant GBG source code is in CubeState.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='locate, CubeState3x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='locate_edge and CubeState3x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='apply_sloc_slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 38 Table 10: The correspondence edge location ↔ STICKER2 for the solved cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The yellow colored cells show the location of the 12 edge stickers that we track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 3x3x3 location 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 STICKER2 edge A B C D D G K E E J F A I J K L H B F I L G C H face ID 1 1 1 1 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 1 1 1 1 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2 STICKER McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019) had the interesting idea for the 3x3x3 cube that 20 stickers (cubie faces) are enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' To characterize the 3x3x3 cube, we need according to McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019) only one (not 2 or 3) sticker for every of the 20 cubies, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This is because the location of one sticker uniquely defines the location and orientation of that cubie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We name this representation STICKER in GBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' We track the 4 top corner stickers 0,2,4,6 plus the 4 bottom corner stickers 24,26,28,30 plus one sticker for each of the 12 edge stickes as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 10, in total 20 stickers and ignore the 28 other stickers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' How to lay out this representation as a board?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' – McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019) create a rect- angular one-hot-encoding board with 20 × 24 = 480 cells (20 rows for the stickers and 24 columns for the locations27) carrying only 0’s and 1’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This is fine for the approach of McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019), where they use this board as input for a DNN, but not so nice for n-tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Without constraints, such a board amounts to 2480 ≈ 10145 combinations, which is unpleasantly large (much larger than in CUBESTATE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='28 Another possibility to lay out the board: Specify 20 board cells (the stickers) with 24 position values each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This amounts to 2420 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='0 · 1027 combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3 STICKER2 Analogously to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3, we represent the 24 corner locations and 24 edge locations as: corner location = (corner cubie, face ID), edge location = (edge cubie, face ID).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' That is, each corner location is represented by a corner cubie a,b,c,d,e,f,g,h and by a face ID 1,2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Table 7 shows the explicit numbering in this new representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Additionally, each edge location is represented by an edge cubie A,B,C,D,E,F,G,H,I,J,K,L29 and by a face ID 1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Convention for face ID numbering of edge cubies: For top- and bottom-layer edge cubies, it is 1 for U and D stickers, 2 else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The face ID for middle-layer edge cubies is 1 for F and B stickers, 2 else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Table 10 shows the explicit numbering in this representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The corresponding board consists of 8 + 8 + 12 +12 = 40 cells shown in Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The 8 cell pairs in the first two rows code the locations of the tracked corner stickers 278 · 3 for the corner stickers and 12 · 2 for the edge stickers 28McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (2019) do not need a weight for every of the 2480 possible states, as the n-tuple network would need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Instead they need only 480 · 4096 = 2 · 106 weights to the first hidden layer having 4096 neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 294 U-stickers, 4 D-sticker, 4 middle-layer stickers (2F, 2B) 39 0,2,4,6,24,26,28,30, see Table 7 in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The 12 cell pairs in the last two rows code the location of the tracked edge stickers 1,3,5,7,17,21,43,47,25,27,29,31, see Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This n-tuple coding requires tuple cells with varying number of position values and leads to 88 · 38 · 1212 · 212 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='0 · 1027 combinations in representation STICKER2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='30 Table 11: STICKER2 board representation for the default 3x3x3 cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' For the BoardVector, cells are numbered row-by-row from 0 to 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' corner a b c d e f g h 8 positions face ID 1 1 1 1 1 1 1 1 3 positions edge A B C D E F G H I J K L 12 positions face ID 1 1 1 1 1 1 1 1 1 1 1 1 2 positions B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='4 Adjacency Sets To create n-tuples by random walk, we need to define adjacency sets (sets of neighbors) for every board cell k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' For CUBESTATE, the board is the flattened representation of the 3x3x3 cube (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The adjacency set is defined as the 4-point neighborhood, where two stickers are neigh- bors if they are neighbors (share a common edge) on the cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' For STICKER2, the board consists of 40 cells shown in Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Since it matters for the corner stickers mostly where the other corner stickers are and for the edge stickers mostly where the other edge stickers are, it is reasonable to form two adjacency subsets S1 = {00, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' , 15} and S2 = {16, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' , 39} and to define the adjacency set Adj(k) = Si \\ {k} for each k ∈ Si, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' C Hyperparameters In this appendix we list all parameter settings for the GBG agents used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Pa- rameters were manually tuned with two goals in mind: (a) to reach high-quality results and (b) to reach stable (robust) performance when conducting multiple training runs with differ- ent random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The agents listed further down are the best-so-far agents found (best among all agents that learn from scratch by self-play).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The detailed meaning of RL parameters is explained in Konen and Bagheri (2021): 30This is, by the way, identical to (8·3)8 ·(12·2)12 = 24(8+12) = 2420 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='0·1027, the same number we had above in the second mode of STICKER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' But STICKER2 has the advantage that the combinations are spread over more board cells (40) than in STICKER (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' By having more board cells with fewer position values, the n-tuples can better represent the relationships between cube states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 40 Algorithms 2, 5 and 7 in Konen and Bagheri (2021) explain parameters α (learning rate), γ (discount factor), ϵ (exploration rate) and output sigmoid σ (either identity or tanh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='3 explains our eligibility method, parameters are: eligibility trace factor λ, horizon cut ch, eligibility trace type ET (normal) or RESET (reset on random move).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' If not otherwise stated, we use in this paper λ = 0 (no eligibility traces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' For λ = 0, horizon cut ch and eligibity trace type are irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' If λ > 0, their defaults ch = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1 and trace type ET apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='5 explains our TCL method (also summarized in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Parameters of TCL are: TC-Init (initialization constant for counters), TC transfer function (TC-id or TC-EXP), β (exponential factor in case of TC-EXP), TC accumulation type (delta or recommended weight-change).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Another branch of our algorithm is the MCTS wrapper, which can be used to wrap TD-N-tuple agents during evaluation and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' MCTS wrapping is briefly explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The precise algorithm for MCTS wrapping is explained in detail in (Scheiermann and Konen, 2022, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' II-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='31 Parameters of MCTS are: cPUCT : relative weight for the prior probabilities of the wrapped agent in relation to the value that the wrapper estimates dmax: maximum depth of the MCTS tree, if -1: no maximum depth UseSoftMax: boolean, whether to use SoftMax normalization for the priors or not UseLastMCTS: boolean, whether to re-use the MCTS from the previous move within an episode or not Further parameter explanations: Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 4 in this document explains n-tuples, parameters are: number of n-tuples, length of n-tuples, and n-tuple creation mode (fixed, random walk, random points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='5 in this document explains symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' If parameter nSym = 0, do not use symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' If nSym > 0, use this number nSym of symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In the Rubik’s cube case, nSym is a number between 0 and 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' LearnFromRM: whether to learn from random moves or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (Does not apply here, because we use in Rubiks’s cube always ϵ = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' we have no random moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=') ChooseStart-01: whether to start episodes from different 1-ply start states or always from the default start state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' (Does not apply here, because we start in Rubik’s cube never from the default cube, but always from the p-twisted cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=') 31As (Scheiermann and Konen, 2022, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' IV-E) shows, the MCTS wrapper may be used as well during training, but due to large computation times needed for this, we do not follow that route in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 41 Etrain: maximum episode length during training, if -1: no maximum length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Eeval: maximum episode length during evaluation and play, if -1: no maximum length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' All agents were trained with no MCTS wrapper inside the training loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The hyper- parameters of the agent for each cube variant were found by manual fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' See also (Konen, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In the following, we list the precise settings for all agents used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' If not stated otherwise, these common settings apply to all agents: sigmoid σ = id, LearnFromRM = false, ChooseStart-01 = false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Wrapper settings during test and evaluation: MCTS wrapper with cPUCT = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='0, dmax = 50, UseSoftMax = true, UseLastMCTS = true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Common parameters of Algorithm 2 in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2 are: cost-to-go c = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='1 and positive reward Rpos = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The parameters for training without symmetries (nSym = 0) in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2 are: 2x2x2 cube, HTM: α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='25, γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='0, ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='0, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='0, no output sigmoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' N-tuples: 60 7-tuples created by random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' TCL activated with transfer function TC-id, TC-Init= 10−4 and rec-weight-change accumulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 3,000,000 training episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' pmax = 13, Etrain = 16, Eeval = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Agent filename in GBG: 2x2x2_STICKER2_AT/TCL4-p13-ET16-3000k-60-7t-stub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='agt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='zip 2x2x2 cube, QTM: same as 2x2x2 cube, HTM, but with pmax = 16, Etrain = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Agent filename in GBG: 2x2x2_STICKER2_QT/TCL4-p16-ET20-3000k-60-7t-stub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='agt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='zip 3x3x3 cube, HTM: same as 2x2x2 cube, HTM, but with 120 7-tuples created by random walk, pmax = 9, Etrain = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Agent filename in GBG: 3x3x3_STICKER2_AT/TCL4-p9-ET13-3000k-120-7t-stub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='agt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='zip 3x3x3 cube, QTM: same as 3x3x3 cube, HTM, but with pmax = 13, Etrain = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Agent filename in GBG: 3x3x3_STICKER2_QT/TCL4-p13-ET16-3000k-120-7t-stub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='agt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='zip The agent files given in the list above are just stubs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' agents that are initialized with the correct parameters but not yet trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' This is because a trained agent can require up to 80 MB disk space, which is too much for GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Instead, a user of GBG may load such a stub agent, train it (takes between 10-40 minutes) and save it to local disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' When evaluating in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2 the trained agents with different MCTS wrappers, we test in each case whether cPUCT = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='0 or 10 is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' In most cases, cPUCT = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='0 is better, but for (2x2x2, QTM, 800 iterations) and for (3x3x3, HTM, 100 iterations) cPUCT = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='0 is the better choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' The parameters for training with symmetries (nSym = 8, 16, 24) in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='4 are: 3x3x3 cube, QTM: same as 3x3x3 cube, QTM in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='2, but with nsym = 8, 16, 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' Agent filename in GBG: 3x3x3_STICKER2_QT/TCL4-p13-ET16-3000k-120-7t-nsym08-stub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='agt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='zip, 3x3x3_STICKER2_QT/TCL4-p13-ET16-3000k-120-7t-nsym16-stub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='agt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='zip, 3x3x3_STICKER2_QT/TCL4-p13-ET16-3000k-120-7t-nsym24-stub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='agt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='zip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 42 Again, the agent filenames are just stubs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' agents that are initialized with the correct parameters but not yet trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' As above, a user of GBG may load such a stub agent, train it (which takes in the symmetry case between 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='4h and 13h, see Table 9) and save it to local disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' For further details and experiment shell scripts, see also the associated Papers-with- Code repository https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content='com/WolfgangKonen/PapersWithCodeRubiks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} +page_content=' 43' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'} diff --git a/C9E1T4oBgHgl3EQfEANP/content/2301.02884v1.pdf b/C9E1T4oBgHgl3EQfEANP/content/2301.02884v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..a9c6427c63a3ccdda2bb7743544b77c93364a45e --- /dev/null +++ b/C9E1T4oBgHgl3EQfEANP/content/2301.02884v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e053822fc5562983f6046e8449603f132a61d69287e9c08733130c3ed4fbfbc3 +size 768320 diff --git a/C9E1T4oBgHgl3EQfEANP/vector_store/index.faiss b/C9E1T4oBgHgl3EQfEANP/vector_store/index.faiss new file mode 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RYAN, AND MATTHEW P. YOUNG +Abstract. Let E be an elliptic curve over Q. +We conjecture +asymptotic estimates for the number of vanishings of L(E, 1, χ) as +χ varies over all primitive Dirichlet characters of orders 4 and 6. +Our conjectures about these families come from conjectures about +random unitary matrices as predicted by the philosophy of Katz- +Sarnak. We support our conjectures with numerical evidence. +Earlier work by David, Fearnley and Kisilevsky formulates anal- +ogous conjectures for characters of any odd prime order. In the +composite order case, however, we need to justify our use of random +matrix theory heuristics by analyzing the equidistribution of the +squares of normalized Gauss sums. Along the way we introduce the +notion of totally order ℓ characters to quantify how quickly quartic +and sextic Gauss sums become equidistributed. Surprisingly, the +rate of equidistribution in the full family of quartic (sextic, resp.) +characters is much slower than in the sub-family of totally quar- +tic (sextic, resp.) characters. A conceptual explanation for this +phenomenon is that the full family of order ℓ twisted elliptic curve +L-functions, with ℓ even and composite, is a mixed family with +both unitary and orthogonal aspects. +1. Introduction +Vanishings of elliptic curve L-functions at the value s = 1 (normalized +so that the functional equation relates s and 2−s) is central to a great +deal of modern number theory. For instance, if an L-function associated +to an elliptic curve vanishes at s = 1, then the BSD conjecture predicts +that the curve will have infinitely many rational points. +Additionally, statistical questions about how often L-functions within +a family vanish at the central value have also been of broad interest. +For example, it is expected (as first conjectured by Chowla [Cho87]) +that, for all primitive Dirichlet characters χ, we have L(χ, 1/2) ̸= 0. +A fruitful way of studying such questions has been to model L-functions +using random matrices. +For example, in [CKRS00] Conrey, Keat- +ing, Rubinstein and Snaith consider the family of twisted L-functions +1 +arXiv:2301.05329v1 [math.NT] 12 Jan 2023 + +2 +JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG +L(f, s, χd) associated to a modular form f of weight k and quadratic +characters χd. They show that the random matrix theory model pre- +dicts that infinitely many values L(f, s, χd) are zero when the weight +of f is 2 or 4, but that only finitely many of the values are zero when +the weight is at least 6. +Another example, due to David, Fearnley and Kisilevsky [DFK04, +DFK07], instead uses the random matrix model to give conjectural +asymptotics for the number of vanishings of elliptic curve L-functions +twisted by families of Dirichlet characters of a fixed order. In particu- +lar, they predict that for an elliptic curve E, the values L(E, 1, χ) are +zero infinitely often if χ has order 3 or 5, but for characters χ with a +fixed prime order ℓ ≥ 7, only finitely many values L(E, 1, χ) are zero. +In recent work, inspired by the conjectures of [DFK04, DFK07], Mazur +and Rubin [MR21] use statistical properties of modular symbols to +heuristically estimate the probability that L(E, 1, χ) vanishes. Their +Conjecture 11.1 implies that for an elliptic curve E over Q, there +should be only finitely many characters χ of a fixed order ℓ such that +L(E, 1, χ) = 0 and ϕ(ℓ) > 4. This further implies the following: Let E +be an elliptic curve over Q and let be F/Q an infinite abelian exten- +sion such that Gal(F/Q) has only finitely many characters of orders 2, +3 and 5. Then E(F) is finitely generated. Finally, for an elliptic curve +E defined over Q, their Proposition 3.2 relates the (order of) vanishing +of L(E, 1, χ) to the growth in rank of E over a finite abelian extension +F/Q. In particular, if BSD holds for E over both Q and F, then +rank(E(F)) = rank(E(Q)) + +� +χ:Gal(F/Q)→C× +ords=1L(E, s, χ). +1.1. Notation and statement of the Main Conjecture. We fix +the following notation. See Definition 3.1 for the definition of totally +order ℓ characters but, roughly speaking, these are order ℓ characters +that, when factored, have all their factors also of order ℓ. Set +Ψℓ = {primitive Dirichlet characters χ of order ℓ} +Ψtot +ℓ += {χ ∈ Ψℓ that are totally order ℓ} +Ψ′ +ℓ = {χ ∈ Ψℓ with cond(χ) prime}. +Note that Ψ′ +ℓ ⊆ Ψtot +ℓ +⊆ Ψℓ. + +VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS +3 +Along the way we will need to estimate the number of characters in +each family and so we define: +Ψℓ(X) = {χ ∈ Ψℓ : cond(χ) ≤ X} +Ψtot +ℓ (X) = {χ ∈ Ψtot +ℓ +: cond(χ) ≤ X} +Ψ′ +ℓ(X) = {χ ∈ Ψ′ +ℓ : cond(χ) ≤ X}. +For an elliptic curve E over Q we also define: +FΨℓ,E = {L(E, s, χ) : χ ∈ Ψℓ} +FΨℓ,E(X) = {L(E, s, χ) ∈ FΨℓ,E : χ ∈ Ψℓ(X)}. +We also define FΨtot +ℓ +,E and FΨtot +ℓ +,E(X) analogously for Ψtot +ℓ +in place of +Ψℓ; we do the same with Ψ′ +ℓ, as well. Finally, let +VΨℓ,E(X) = {L(E, s, χ) ∈ FΨℓ,E(X) : L(E, 1, χ) = 0} +VΨtot +ℓ +,E(X) = {L(E, s, χ) ∈ FΨtot +ℓ +,E(X) : L(E, 1, χ) = 0} +VΨ′ +ℓ,E(X) = {L(E, s, χ) ∈ FΨ′ +ℓ,E(X) : L(E, 1, χ) = 0}. +With this notation, we make the following conjecture. +Conjecture 1.1. Let E be an elliptic curve. Then, there exist con- +stants bE,4 and bE,6 so that +|VΨ4,E(X)| ∼ bE,4X1/2 log5/4 X +and +|VΨ6,E(X)| ∼ bE,6X1/2 log9/4 X +as X → ∞. +Moreover, if we restrict only to those twists by totally quartic or totally +sextic characters, then there exist constants btot +E,4 and btot +E,6 such that +|VΨtot +4 ,E(X)| ∼ btot +E,4X1/2 log1/4 X +and +|VΨtot +6 ,E(X)| ∼ btot +E,6X1/2 log1/4 X +as X → ∞. +Finally, if we restrict only to those twists by characters of prime con- +ductor, then there exist constants b′ +E,4 and b′ +E,6 such that +|VΨ′ +4,E(X)| ∼ b′ +E,4X1/2 log−3/4 X +and +|VΨ′ +6,E(X)| ∼ b′ +E,6X1/2 log−3/4 X +as X → ∞. + +4 +JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG +In particular, we conjecture that families of elliptic curve L-functions +twisted by quartic and sextic characters vanish infinitely often at the +central value. +To assist the reader in comparing the powers of log X in the above +asymptotics, we point out here that for ℓ = 4, |Ψ4(X)| is roughly log X +times as large as |Ψtot +4 (X)|, which in turn is roughly log X times as +large as |Ψ′ +4(X)|. For ℓ = 6, then |Ψ6(X)|/|Ψtot +6 (X)| ≍ (log X)2, and +|Ψtot +6 (X)|/|Ψ′ +6(X)| ≍ log X. Hence, in each of the three families with +a given value of ℓ, the proportion of vanishing twists has the same +order of magnitude. See Proposition 3.6, Lemma 3.7, Proposition 3.8, +and Lemma 3.9 below for asymptotics of the underlying families of +characters. +1.2. Outline of the paper. There are two main ingredients needed +to be able to apply random matrix theory predictions to our families of +twists. The first is a discretization for the central values. As described +in Section 2.1 this can be done for curves E satisfying certain technical +conditions as described in [WW20]. We need this discretization in order +to approximate the probability that L(E, 1, χ) vanishes. +The second ingredient is a proper identification of the symmetry type +of the family, which is largely governed by the distribution of the sign of +the functional equation within the family (see Section 4 of [CFK+05]). +This directly leads to an investigation around the equidistribution of +squares of Gauss sums of quartic and sextic characters, which has con- +nections to the theory of metaplectic automorphic forms [Pat87]. +See Section 3.1 for a thorough discussion. +It is a subtle feature that the families of twists of elliptic curve L- +functions by the characters in Ψtot +ℓ +and Ψ′ +ℓ have unitary symmetry +type, but for composite even values of ℓ, the twists by Ψℓ should be +viewed as a mixed family. To elaborate on this point, consider the case +that ℓ = 4, and first note that a character χ ∈ Ψ4 factors uniquely +as a totally quartic character times a quadratic character of relatively +prime conductors. The totally quartic family has a unitary symmetry, +but the family of twists of an elliptic curve by quadratic characters has +orthogonal symmetry. This tension between the totally quartic aspect +and the quadratic aspect is what leads to the mixed symmetry type. +The situation is analogous to the family L(E, 1 + it, χd); if t = 0 and +d varies then one has an orthogonal family, while if d is fixed and t +varies, then one has a unitary family. See [SY10] for more discussion +on this family. + +VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS +5 +Another interesting feature of these families is that Ψℓ(X) is larger +than Ψtot +ℓ (X) by a logarithmic factor. For instance, when ℓ = 4, then +Ψtot +4 (X) grows linearly in X (see Proposition 3.6 below), and of course +Ψ2(X) also grows linearly in X. Similarly to how the average size of the +divisor function is log X, this indicates that |Ψ4(X)| grows like X log X +(see Lemma 3.7 below). +The rest of the paper is organized as follows. +In the next section +we give the necessary background and notation for L-functions and +their central values and discuss the discretization we use in the paper. +In the subsequent section we estimate some sums involving quartic +and sextic characters and discuss totally quartic and sextic characters +in more detail. In the final section, we motivate the asymptotics in +Conjecture 1.1 and provide numerical evidence that supports them. +Acknowledgments. We thank David Farmer and Brian Conrey for +helpful conversations. This research was done using services provided +by the OSG Consortium [PPK+07, SBH+09], which is supported by the +National Science Foundation awards #2030508 and #1836650. This +material is based upon work supported by the National Science Foun- +dation under agreement No. DMS-2001306 (M.Y.). Any opinions, find- +ings and conclusions or recommendations expressed in this material are +those of the authors and do not necessarily reflect the views of the Na- +tional Science Foundation. +2. L-functions and central values +Let E be an elliptic curve defined over Q of conductor NE. The L- +function of E is given by the Euler product +L(E, s) = +� +p∤NE +� +1 − ap +ps + +1 +p2s−1 +�−1 � +p|NE +� +1 − ap +ps +�−1 += +� +n≥1 +an +ns . +The modularity theorem [BCDT01, TW95, Wil95] implies that L(E, s) +has an analytic continuation to all of C and satisfies the functional +equation +Λ(E, s) = +� √NE +2π +�s +Γ(s)L(E, s) = wEΛ(E, 2 − s) +where the sign of the functional equation is wE = ±1 and is the eigen- +value of the Fricke involution. Let χ be a primitive character and let +cond(χ) be its conductor and suppose that cond(χ) is coprime to the + +6 +JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG +conductor NE of the curve. The twisted L-function has Dirichlet series +L(E, s, χ) = +� +n≥1 +anχ(n) +ns +and the functional equation (cf. [IK04, Prop. 14.20]) +Λ(E, s, χ) = +� +cond(χ)√NE +2π +�s +Γ(s)L(E, s, χ) += wEχ(NE)τ(χ)2 +cond(χ) +Λ(E, 2 − s, χ), +(2.1) +where τ(χ) = � +r∈Z/mZ χ(r)e2πir/m is the Gauss sum and m = cond(χ). +2.1. Discretization. To justify our Conjecture 1.1, we need a condi- +tion that allows us to deduce that L(E, 1, χ) = 0, for a given E and +χ of order ℓ. In particular, we show that L(E, 1, χ) is discretized (see +Lemma 4.2) and so there exists a constant cE,ℓ such that |L(E, 1, χ)| < +cE,ℓ/ +� +cond(χ) implies L(E, 1, χ) = 0. In this section we prove the +results necessary for the discretization. +Let E be an elliptic curve over Q with conductor NE. +Let χ be a +nontrivial primitive Dirichlet character of conductor m and order ℓ. +Set ϵ = {±1} = χ(−1) depending on whether χ is an even or odd +character. Let Ω+(E) and Ω−(E) denote the real and imaginary periods +of E, respectively, with Ω+(E) > 0 and Ω−(E) ∈ iR>0. +The algebraic L-value is defined by +(2.2) +Lalg(E, 1, χ) := L(E, 1, χ) · m +τ(χ)Ωϵ(E) += ϵ · L(E, 1, χ)τ(χ) +Ωϵ(E) +While it has been known for some time that algebraic L-values are +algebraic numbers, recent work of Weirsema and Wuthrich [WW20] +characterizes conditions on E and χ which guarantee integrality. In +particular, under the assumption that the Manin constant c0(E) = 1, +if the conductor m is not divisible by any prime of additive reduction for +E, then Lalg(E, 1, χ) ∈ Z[ζℓ] is an algebraic integer [WW20, Theorem +2]. For a given curve E, we will avoid the finitely many characters χ +for which Lalg(E, 1, χ) fails to be integral. +Proposition 2.1. Let χ be a primitive Dirichlet character of odd order +ℓ and conductor m. Then +Lalg(E, 1, χ) = +� +χ(NE)(ℓ+1)/2 nE(χ), +if wE = 1, +(ζℓ − ζ−1 +ℓ )−1 χ(NE)(ℓ+1)/2 nE(χ) +if wE = −1, +for some algebraic integer nE(χ) ∈ Z[ζℓ + ζ−1 +ℓ ] = Z[ζℓ] ∩ R. + +VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS +7 +Proposition 2.2. Let χ be a primitive Dirichlet character of even +order ℓ and conductor m. Then Lalg(E, 1, χ) = kE nE(χ) where nE(χ) +is some algebraic integer in Z[ζℓ +ζ−1 +ℓ ] = Z[ζℓ]∩R and kE is a constant +depending only on the curve E. In particular, when wE = 1 we have +kE = +� +� +� +� +� +(1 + χ(NE)) +if χ(NE) ̸= −1 +ζℓ/4 +ℓ +, +if 4 | ℓ and χ(NE) = −1 +(ζℓ − ζ−1 +ℓ ) +if 4 ∤ ℓ and χ(NE) = −1. +Proof of Prop 2.1 and Prop 2.2. Since E is defined over Q, we have +L(E, 1, χ) = L(E, 1, χ). Using the functional equation, we obtain +Lalg(E, 1, χ) = ϵ · L(E, 1, χ)τ(χ) +Ωϵ(E) += ϵ · wE χ(NE) τ(χ)τ(χ)2 +m · Ωϵ(E) +L(E, 1, χ) += wE χ(NE) τ(χ) +Ωϵ(E) +L(E, 1, χ) += wEχ(NE) ϵ · τ(χ)L(E, 1, χ) +Ωϵ(E) += wEχ(NE) Lalg(E, 1, χ) +Thus Lalg(E, 1, χ) is a solution to the equation z = wEχ(NE)z. Note +that if z1, z2 ∈ Z[ζℓ] are two distinct solutions to this equation, then +z1/z1 = z2/z2 so that z1/z2 = z1/z2 = (z1/z2), hence z1/z2 ∈ R. Thus +Lalg(E, 1, χ) = αz with α ∈ Z[ζℓ] ∩ R = Z[ζℓ + ζ−1 +ℓ ] and z ∈ Z[ζℓ]. +Suppose that wE = 1. When ℓ is odd, we can take z = χ(NE) +ℓ+1 +2 . Now +suppose that ℓ is even. If χ(NE) ̸= −1, since χ(NE) = ζr +ℓ for some +1 ≤ r ≤ ℓ, we may take z = (1+χ(NE)). Indeed, we have wEχ(NE)z = +ζr +ℓ (1 + ζℓ−r +ℓ +) = ζr +ℓ + 1 = z. If 4 | ℓ and χ(NE) = −1 = ζℓ/2 +ℓ +, we take +z = ζℓ/4 +ℓ +. Finally, if 4 ∤ ℓ and χ(NE) = −1 take z = ζℓ−ζ−1 +ℓ += 2i Im(ζℓ). +When wE = −1 and ℓ is odd, we may take z = (ζℓ − ζ−1 +ℓ )−1χ(NE) +ℓ+1 +2 . +When ℓ is even, if χ(NE) = −1 then we may take z = ζℓ + ζ−1 +ℓ += +2 Re(ζℓ), and if χ(NE) ̸= −1 then we make take z = 1 − χ(NE). +□ +Remark 2.3. We note that for ℓ even, |kE| ≤ 2. +It is clear that +|ζℓ/4 +ℓ +| = 1 and |2i Im(ζℓ)| ≤ 2. +Observe |(1 + χ(NE)| ≤ 2, by the +triangle inequality. + +8 +JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG +Note that since L(E, 1, χ) vanishes if and only if nE(χ) does, we may +interpret the integers nE(χ) as a discretization of the special values +L(E, 1, χ). This is similar to the case of cubic characters considered in +[DFK04] since Q(ζ3)+ = Q, as opposed to characters of prime order ℓ ≥ +5 where further steps were needed to find an appropriate discretization +[DFK07]. +3. Estimates for Dirichlet characters +In this section we discuss various aspects of Dirichlet characters of +order 4 and 6. A necessary condition for a family of L-functions to +be modeled by the family of unitary matrices is that the signs must +be uniformly distributed on the unit circle. From (2.1), L(E, s, χ) has +sign wEχ(NE) τ(χ)2 +cond(χ); we will largely focus on the distribution of the +square of the Gauss sums, viewing the extra factor χ(NE) as a minor +perturbation. To obtain our estimates for the number of vanishings +|VΨℓ,E(X)| (respectively, |VΨ′ +ℓ,E(X)| and |VΨtot +ℓ +,E(X)|) we must estimate +the size of Ψℓ(X) (respectively, Ψ′ +ℓ(X) and Ψtot +ℓ (X)) as well as the size +of an associated sum. We also discuss the family of totally quartic +and sextic characters to explain some phenomena we observed in our +computations. +3.1. Distributions of Gauss sums. Patterson [Pat87], building on +work of Heath-Brown and Patterson [HBP79] on the cubic case, showed +that the normalized Gauss sum τ(χ)/ +� +cond(χ) is uniformly distributed +on the circle for χ varying in each of Ψtot +ℓ +and Ψ′ +ℓ. This result was +first announced in [PHH81]; see [BE81] for an excellent summary of +this and other work related to the distributions of Gauss sums. Patter- +son’s method moreover shows that the argument of τ(χ)χ(k) is equidis- +tributed for any fixed nonzero integer k, and hence so is the argument +of τ(χ)2χ(k). +For the case of quartic and sextic characters with arbitrary conductors, +there do not appear to be any results in the literature that imply their +Gauss sums are uniformly distributed. In Figure 1 we see the distri- +butions of Gauss sums of characters of orders 3 through 9 of arbitrary +conductor up to 200000. We included characters of order 4 and 6 since +those examples are the focus of the paper; we included characters of or- +ders 3, 5, and 7 as consistency checks (in [DFK04, DFK07] the authors +rely on them being uniformly distributed); and we included composite +orders 8 and 9 to see if something similar happens in those cases as +happens in the quartic case. In all cases but the quartic case, we see +that the distributions of the angles of the signs appear to be uniformly + +VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS +9 +Figure 1. Each histogram represents the distribution +of the argument of the τ(χ)2/cond(χ) for characters of +order 3 through 9, from top left to bottom right. Each +histogram is made by calculating the Gauss sums of char- +acters in Ψℓ of each conductor up to 200000. +distributed. The quartic distribution has two obvious peaks that we +discuss below, in Remark 3.17. +The images in Figure 1 suggest that the family of matrices that best +models the vanishing of L(E, 1, χ) is unitary in every case except possi- +bly the case of quartic characters. Nevertheless, in Section 3.4 we show +that the squares of the quartic Gauss sums are indeed equidistributed, +despite what the data suggest. Indeed, we prove that the squares of +the sextic and quartic Gauss sums are equidistributed, allowing us to +apply the heuristics from random matrix theory as in Section 4. +3.2. Totally quartic and sextic characters. Much of the back- +ground material in this section can be found with proofs in [IR90, +Ch. 9]. +Definition 3.1. Let χ be a primitive Dirichlet character of conductor +q and order ℓ. For prime p, let vp be the p-adic valuation, so that +q = � +p pvp(q). We correspondingly factor χ = � +p χ(p), where χ(p) has +conductor pvp(q). We say that χ is totally order ℓ if each χp is exact + +5+00 +5000 +4000 +DORE +2400 +1400 +E- +-1 +0ADODE +24000 +E- +-2 +-1 +0 +1DOS +4400 +3000 +DOZ +1400 +2 +-1 +0 +112000 +40000 +E- +-2 +-1 +0 +34400 +DOSE +DODE +2500 +DOZ +1500 +1400 +500 +0 +E- +-1 +i +2 +350000 +40000 +ADODE +DO +4000 +E- +-2 +-1 +0 +i217500 +15000 +12500 +14000 +DOSr +5000 +2500 +E- +i210 +JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG +order ℓ. +By convention we also consider the trivial character to be +totally order ℓ for every ℓ. +3.2.1. Quartic characters. The construction of quartic characters uses +the arithmetic in Z[i]. The ring Z[i] has class number 1, unit group +{±1, ±i}, and discriminant −4. We say α ∈ Z[i] with (α, 2) = 1 is +primary if α ≡ 1 (mod (1+i)3). Any odd element in Z[i] has a unique +primary associate, which comes from the fact that the unit group in +the ring Z[i]/(1+i)3 may be identified with {±1, ±i}. An odd prime p +splits as p = ππ if and only if p ≡ 1 (mod 4). Given π with N(π) = p, +define the quartic residue symbol [ α +π] for α ∈ Z[i] with (α, π) = 1, +by [ α +π] ∈ {±1, ±i} and [ α +π] ≡ α +p−1 +4 +(mod π). The map χπ(α) = [ α +π] +from (Z[i]/(π))× to {±1, ±i} is a character of order 4. +If α ∈ Z, +then [ α +π]2 ≡ α +p−1 +2 +≡ ( α +p) (mod π). Therefore, χ2 +π(α) = (α +p), showing +in particular that the restriction of the quartic residue symbol to Z +defines a primitive quartic Dirichlet character of conductor p. +Lemma 3.2. Every primitive totally quartic character of odd conductor +is of the form χβ, where β = π1 . . . πk is a product of distinct primary +primes, (β, 2β) = 1, and where +(3.1) +χβ(α) = +�α +β +� += +k +� +i=1 +� α +πi +� +. +The totally quartic primitive characters of even conductor are of the +form χ2χβ where χ2 is one of four quartic characters of conductor 24, +and χβ is totally quartic of odd conductor. +Proof. We begin by classifying the quartic characters of odd prime- +power conductor. If p ≡ 3 (mod 4), there is no quartic character of +conductor pa, since φ(pa) = pa−1(p − 1) ̸≡ 0 (mod 4). Since φ(p) = +p − 1, if p ≡ 1 (mod 4), there are two distinct quartic characters of +conductor p, namely, χπ and χπ, where p = ππ. There are no primitive +quartic characters modulo pj for j ≥ 2. +To see this, suppose χ is +a character of conductor pj, and note that χ(1 + pj−1) ̸= 1, while +χ(1 + pj−1)p = χ(1 + pj) = 1, so χ(1 + pj−1) is a nontrivial pth root of +unity. Since p is odd, χ(1+pj−1) is not a 4th root of unity, so χ cannot +be quartic and primitive. +By the above classification, a primitive totally quartic character χ of +odd conductor must factor over distinct primes pi ≡ 1 (mod 4), and +the p-part of χ must be χπ or χπ, where ππ = p. We may assume that + +VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS +11 +π and π are primary primes. Hence χ factors as � +i χπi. The property +that β := π1 . . . πk is squarefree is equivalent to the condition that the +πi are distinct. Moreover, the property (β, β) = 1 is equivalent to that +πiπi = pi ≡ 1 (mod 4), for all i. Hence, every quartic character of odd +conductor arises uniquely in the form (3.1). +Next we treat p = 2. There are four primitive quartic characters of +conductor 24, since (Z/(24))× ≃ Z/(2) × Z/(4). We claim there are no +primitive quartic characters of conductor 2j, with j ̸= 4. For j ≤ 3 or +j = 5 this is a simple finite computation. For j ≥ 6, one can show this +as follows. First, χ(1 + 2j−1) = −1, since χ2(1 + 2j−1) = χ(1 + 2j) = 1, +and primitivity shows χ(1+2j−1) ̸= 1. By a similar idea, χ(1+2j−2)2 = +χ(1 + 2j−1) = −1, so χ(1 + 2j−2) = ±i. We finish the claim by noting +χ2(1 + 2j−3) = χ(1 + 2j−2) = ±i, so χ(1 + 2j−3) is a square-root of +±i, and hence χ is not quartic. With the claim established, we easily +obtain the final sentence of the lemma. +□ +Example 3.3. The first totally quartic primitive character of compos- +ite conductor has conductor 65. While there are 8 quartic primitive +characters of conductor 65, the LMFDB labels of the totally quartic +ones are 65.18, 65.47, 65.8, and 65.57. +3.2.2. Sextic characters. The construction of sextic characters uses the +arithmetic in the Eisenstein integers Z[ω], where ω = e2πi/3. The ring +Z[ω] has class number 1, unit group {±1, ±ω, ±ω2}, and discriminant +−3. We say α ∈ Z[ω] with (α, 3) = 1 is primary1 if α ≡ 1 (mod 3). +Warning: our usage of primary is consistent with [HBP79], but conflicts +with the definition of [IR90]. However, it is easy to translate since α is +primary in our sense if and only if −α is primary in the sense of [IR90]. +Any element in Z[ω] coprime to 3 has a unique primary associate, which +comes from the fact that the unit group in the ring Z[ω]/(3) may be +identified with {±1, ±ω, ±ω2}. An unramified prime p ∈ Z splits as +p = ππ if and only if p ≡ 1 (mod 3). Given π with N(π) = p, define +the cubic residue symbol ( α +π)3 for α ∈ Z[ω] by ( α +π)3 ∈ {1, ω, ω2} and +( α +π)3 ≡ α +p−1 +3 +(mod π). The map χπ(α) = ( α +π)3 from (Z[ω]/(π))× to +{1, ω, ω2} is a character of order 3. The restriction of χπ to Z induces a +primitive cubic Dirichlet character of conductor p. Note that χπ = χ−π. +Motivated by the fact that a sextic character factors as a cubic times +a quadratic, we next discuss the classification of cubic characters. +1We remark that the usage of primary is context-dependent, and that since we +do not mix quartic and sextic characters, we hope there will not be any ambiguity + +12 +JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG +Lemma 3.4. Every primitive cubic Dirichlet character of conductor +coprime to 3 is of the form χβ, where β = π1 . . . πk is a product of +distinct primary primes, (β, 3β) = 1, and where +(3.2) +χβ(α) = +�α +β +� +3 = +k +� +i=1 +� α +πi +� +3. +The cubic primitive characters of conductor divisible by 3 are of the +form χ3χβ where χ3 is one of two cubic characters of conductor 32, +and χβ is cubic of conductor coprime to 3. +Proof. The classification of such characters with conductor coprime to +3 is given by [BY10, Lemma 2.1], so it only remains to treat cubic +characters of conductor 3j. The primitive character of conductor 3 is +not cubic. Next, the group (Z/(9))× is cyclic of order 6, generated by +2. There are two cubic characters, determined by χ(2) = ω±1. Next +we argue that there is no primitive cubic character of conductor 3j +with j ≥ 3. For this, we first observe that χ(1 + 3j−1) = ω±1, since +primitivity implies χ(1 + 3j−1) ̸= 1, and χ(1 + 3j−1)3 = χ(1 + 3j) = 1. +Next we have χ(1 + 3j−2)3 = χ(1 + 3j−1) = ω±1, so χ(1 + 3j−2) is a +cube-root of ω±1. Therefore, χ cannot be cubic. +□ +3.3. Counting characters. To start, we count all the quartic and sex- +tic characters of conductor up to some bound and in each family. Such +counts were found for arbitrary order in [FMS10] by Finch, Martin and +Sebah, but since we are interested in only quartic and sextic charac- +ters, in which case the proofs simplify, we prove the results we need. +Moreover, we need other variants for which we cannot simply quote +[FMS10], so we will develop a bit of machinery that will be helpful for +these other questions as well. +We begin with a lemma based on the Perron formula. +Lemma 3.5. Suppose that a(n) is a multiplicative function such that +|a(n)| ≤ dk(n), the k-fold divisor function, for some k ≥ 0. Let Z(s) = +� +n≥1 a(n)n−s, for Re(s) > 1. Suppose that for some integer j ≥ 0, +(s−1)jZ(s) has a analytic continuation to a region of the form {σ+it : +σ > 1 − +c +log(2+|t|)}, for some c > 0. In addition, suppose that Z(s) is +bounded polynomially in log (2 + |t|) in this region. Then +(3.3) +� +n≤X +a(n) = XPj−1(log X) + O(X(log X)−100), +for Pj−1 some polynomial of degree ≤ j − 1 (interpreted as 0, if j = 0). + +VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS +13 +The basic idea is standard, yet we were unable to find a suitable refer- +ence. +Proof sketch. One begins by a use of the quantitative Perron formula, +for which a convenient reference is [MV07, Thm. 5.2]. This implies +(3.4) +� +n≤X +a(n) = +1 +2πi +� σ0+iT +σ0−iT +Z(s)Xsds +s + R, +where R is a remainder term, and we take σ0 = 1+ +c +log X . Using [MV07, +Cor. 5.3] and standard bounds on mean values of dk(n), one can show +R ≪ X +T Poly(log X). Next one shifts the contour of integration to the +line 1 − c/2 +log T . The pole (if it exists) of Z(s) leads to a main term of the +form XPj−1(log X), as desired. The new line of integration is bounded +by +(3.5) +Poly(log T)X1− c/2 +log T . +Choosing log T = (log X)1/2 gives an acceptable error term. +□ +3.3.1. Quartic characters. Let Ψtot,odd +4 +(X) ⊆ Ψtot +4 (X) denote the sub- +set of characters with odd conductor. +Proposition 3.6. For some constants Ktot +4 , Ktot,odd +4 +> 0, we have +(3.6) +|Ψtot +4 (X)| ∼ Ktot +4 X, +and +|Ψtot,odd +4 +(X)| ∼ Ktot,odd +4 +X. +Moreover, +(3.7) +|Ψ′ +4(X)| ∼ +X +log X . +Proof. By Lemma 3.2, +(3.8) +|Ψtot,odd +4 +(X)| = +� +0̸=(β)⊆Z[i] +(β,2β)=1 +β squarefree +N(β)≤X +1, +and +(3.9) +|Ψtot +4 (X)| = |Ψtot,odd +4 +(X)| + 4|Ψtot,odd +4 +(2−4X)|. + +14 +JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG +To show (3.6), it suffices to prove the asymptotic formula for |Ψtot,odd +4 +(X)|. +In view of Lemma 3.5, it will suffice to understand the Dirichlet series +(3.10) +Z4(s) = +� +0̸=(β)⊆Z[i] +(β,2β)=1 +β squarefree +1 +N(β)s = +� +π̸=π +(π,2)=1 +(1 + N(π)−s) = +� +p≡1 (mod 4) +(1 + p−s)2. +Let χ4 be the primitive character modulo 4, so that ζ(s)L(s, χ4) = +ζQ[i](s). Then +(3.11) Z4(s) = ζQ[i](s) +� +p +(1 − p−s)(1 − χ4(p)p−s) +� +p≡1 (mod 4) +(1 + p−s)2, +which can be simplified as +(3.12) +Z4(s) = ζQ[i](s)ζ−1(2s)(1 + 2−s)−1 +� +p≡1 (mod 4) +(1 − p−2s). +Therefore, Z4(s) has a simple pole at s = 1, and its residue is a positive +constant. Moreover, the standard analytic properties of ζQ[i](s) let us +apply Lemma 3.5, giving the result. +The asymptotic on Ψ′ +4(X) follows from the prime number theorem in +arithmetic progressions, since there are two quartic characters of prime +conductor p ≡ 1 (mod 4), and none with p ≡ 3 (mod 4). +□ +Lemma 3.7. We have +(3.13) +|Ψ4(X)| = K4X log X + O(X), +for some K4 > 0 +Proof. Every primitive quartic character factors uniquely as χ4χ2 with +χ4 totally quartic of conductor q4 > 1 and χ2 quadratic of conductor +q2, with (q4, q2) = 1. It is convenient to drop the condition q4 > 1, +thereby including the quadratic characters; this is allowable since the +number of quadratic characters is O(X), which is acceptable for the +claimed error term. +The Dirichlet series for |Ψ4(X)|, modified to include the quadratic char- +acters, is +(3.14) +Zall +4 (s) = +� +0̸=(β)⊆Z[i] +(β,2β)=1 +β squarefree +1 +N(β)s +� +q2∈Z≥1 +(q2,2N(β))=1 +1 +qs +2 +. + +VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS +15 +A calculation with Euler products shows Zall +4 (s) = ζQ[i](s)ζ(s)A(s), +where A(s) is given by an absolutely convergent Euler product for +Re(s) > 1/2. Since Zall +4 (s) has a double pole at s = 1, this shows the +claim, using Lemma 3.5. +□ +3.3.2. Sextic characters. Next we turn to the sextic case. The proof of +the following proposition is similar to the proof of Proposition 3.6 and +so we omit it here. +Proposition 3.8. For some Ktot +6 +> 0, we have +(3.15) +|Ψtot +6 (X)| ∼ Ktot +6 X, +and +|Ψ′ +6(X)| ∼ +X +log X . +A primitive totally sextic character factors uniquely as a primitive cubic +character (with odd conductor, since 2 ̸≡ 1 (mod 3)), times the Jacobi +symbol of the same modulus as the cubic character. +In general, a +primitive sextic character factors uniquely as χ6χ3χ2 of modulus q6q3q2, +pairwise coprime, with χ6 totally sextic of conductor q6, χ3 cubic of +conductor q3, and χ2 quadratic of conductor q2. +Lemma 3.9. We have |Ψ6(X)| = K6X(log X)2+O(X log X), for some +K6 > 0. +Proof. Write χ = χ6χ3χ2 as above. Note that membership in Ψ6(X) +requires q6 > 1, which is an unpleasant condition when working with +Euler products. However, the number of χ = χ3χ2, i.e., with χ6 = 1, +is O(X log X), so we may drop the condition q6 > 1 when estimating +|Ψ6(X)|. +For simplicity, we count the characters with q2 odd and (q6q3, 3) = +1; the general case follows similar lines. The Dirichlet series for this +counting function is +Zall +6 (s) = +� +0̸=(β6)⊆Z[ω] +(β6,3β6)=1 +β6 squarefree +1 +N(β6)s +� +0̸=(β3)⊆Z[ω] +(β3,3β3)=1 +β3 squarefree +(N(β3),N(β6)=1 +1 +N(β3)s +� +q2∈Z≥1 +(q2,2N(β3β6))=1 +1 +qs +2 +. +A calculation with Euler products shows Zall +6 (s) = ζQ[ω](s)2ζ(s)A(s), +where A(s) is given by an absolutely convergent Euler product for +Re(s) > 1/2. Since Zall +6 (s) has a triple pole at s = 1, this shows the +claim, using Lemma 3.5. +□ + +16 +JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG +3.4. Equidistribution of Gauss sums. We first focus on the quartic +case, and then turn to the sextic case. +3.4.1. Quartic characters. The following standard formula can be found +as [IK04, (3.16)]. +Lemma 3.10. Suppose that χ = χ1χ2 has conductor q = q1q2, with +(q1, q2) = 1, and χi of conductor qi. Then +(3.16) +τ(χ1χ2) = χ2(q1)χ1(q2)τ(χ1)τ(χ2). +Corollary 3.11. Let notation be as in Lemma 3.10. Suppose that χ is +totally quartic and q is odd. Then +(3.17) +τ(χ1χ2)2 = τ(χ1)2τ(χ2)2. +Proof. By Lemma 3.10, we will obtain the formula provided χ2 +2(q1)χ2 +1(q2) = +1. Note that χ2 +i is the Jacobi symbol, so χ2 +2(q1)χ2 +1(q2) = ( q1 +q2)( q2 +q1) = 1, +by quadratic reciprocity, using that q1 ≡ q2 ≡ 1 (mod 4). +□ +Lemma 3.12. Suppose π ∈ Z[i] is a primary prime, with N(π) = p ≡ +1 (mod 4). Let χπ(x) = [ x +π] be the quartic residue symbol. Then +(3.18) +τ(χπ)2 = −χπ(−1)√pπ. +More generally, if β is primary, squarefree, with (β, 2β) = 1, then +(3.19) +τ(χβ)2 = µ(β)χβ(−1) +� +N(β)β. +Proof. The formula for χπ follows from [IR90, Thm.1 (Chapter 8), +Prop. 9.9.4]. The formula for general β follows from Corollary 3.11 +and Lemma 3.2. +□ +Lemma 3.13. Suppose that χ = χ2χ4 is a primitive quartic character +with odd conductor q, with χ2 quadratic of conductor q2, χ4 totally +quartic of conductor q4, and with q2q4 = q. +Then +(3.20) +τ(χ)2 = +�−q4 +q2 +� +q2τ(χβ)2. +Proof. By Lemma 3.10, we have τ(χ)2 = χ2(q4)2χ4(q2)2τ(χ2)2τ(χ4)2. +To simplify this, note χ2(q4)2 = 1, χ2 +4(q2) = (q2 +q4) = (q4 +q2), and τ(χ2)2 = +ϵ2 +q2q2 = ( −1 +q2 )q2. +□ + +VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS +17 +Our next goal is to express τ(χβ)2 in terms of a Hecke Grossencharacter. +Define +(3.21) +λ∞(α) = α +|α|, +α ∈ Z[i], α ̸= 0. +Next define a particular character λ1+i : R× → S1, where R = Z[i]/(1+ +i)3, by +(3.22) +λ1+i(ik) = i−k, +k ∈ {0, 1, 2, 3}. +This indeed defines a character since R× ≃ Z/4Z, generated by i. For +α ∈ Z[i], (α, 1 + i) = 1, define +(3.23) +λ((α)) = λ1+i(α)λ∞(α). +For this to be well-defined, we need that the right hand side of (3.23) +is constant on units in Z[i]. This is easily seen, since λ∞(ik) = ik = +λ1+i(ik)−1. Therefore, λ defines a Hecke Grossencharacter, as in [IK04, +Section 3.8]. Moreover, we note that +(3.24) +τ(χβ)2 +N(β) = µ(β) +� +2 +N(β) +� +λ((β)) +since this agrees with (3.19) for β primary, and is constant on units. +According to [IK04, Theorem 3.8], the Dirichlet series +(3.25) +L(s, λk) = +� +0̸=(β)⊆Z[i] +λ((β))k +N(β)s , +(k ∈ Z), +defines an L-function having analytic continuation to s ∈ C with no +poles except for k = 0. The same statement holds when twisting λk by +a finite-order character. +For k ∈ Z, define the Dirichlet series +(3.26) +Z(k, s) = +� +0̸=(β)⊆Z[i] +(β,2β)=1 +β squarefree +(τ(χβ)2/N(β))k +N(β)s +, +Re(s) > 1. +Proposition 3.14. Let δk = −1 for k odd, and δk = +1 for k even. +We have +(3.27) Z(k, s) = A(k, s)L(s, (λ · χ2)k)δk, +where +χ2(β) = +� +2 +N(β) +� +, +and where A(k, s) is given by an Euler product absolutely convergent +for Re(s) > 1/2. + +18 +JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG +In particular, the zero free region (as in [IK04, Theorem 5.35]) implies +that Z(k, s) is analytic in a region of the type postulated in Lemma +3.5. Moreover, the proof of [MV07, Theorem 11.4] shows that Z(k, s) +is bounded polynomially in log(2 + |t|) in this region. +Proof. The formula (3.24) shows that Z(k, s) has an Euler product of +the form +(3.28) +Z(k, s) = +� +(π)̸=(π) +(1 + (−1)k χk +2(π)λk((π)) +N(π)s +). +This is an Euler product over the split primes in Z[i]. We extend this +to include the primes p ≡ 3 (mod 4) as well, with N(π) = p2. It is +convenient to define χ2(1 + i) = 0, so we can freely extend the product +to include the ramified prime 1 + i. In all, we get +(3.29) +Z(k, s) = +� � +p +(1 − χk +2(p)λk(p) +N(p)s +) +�−δk � +p +(1 + O(p−2s)). +Note the product over p is L(s, (λ · χ2)k)δk, as claimed. +□ +According to Weyl’s equidistribution criterion [IK04, Ch. 21.1], a se- +quence of real numbers θn, 1 ≤ n ≤ N is equidistributed modulo 1 if +and only if � +n≤N e(kθn) = o(N) for each integer k ̸= 0. We apply +this to e(θn) = (τ(χ)2/q), whence e(kθn) = (τ(χ)2/q)k. Due to the +twisted multiplicativity formula (3.16), the congruence class in which +2k lies modulo ℓ may have a simplifying effect on τ(χ)2k. For instance, +when ℓ = 4, then k even leads to a simpler formula than k odd. This +motivates treating these cases separately. As a minor simplification, +below we focus on the sub-family of characters of odd conductor. The +even conductor case is only a bit different. +Corollary 3.15. The Gauss sums τ(χ)2/q for χ totally quartic of odd +conductor q, equidistribute on the unit circle. +Proof. The complex numbers τ(χ)2/q lie on the unit circle. +Weyl’s +equidistribution criterion says that these normalized squared Gauss +sums equidistribute on the unit circle provided +(3.30) +� +0̸=(β)⊆Z[i] +(β,2β)=1 +β squarefree +N(β)≤X +(τ(χβ)2/N(β))k = o(X), + +VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS +19 +Figure 2. This histogram represents the distribution +of the argument of the τ(χ)2/cond(χ) for totally quartic +characters. Each histogram is made by calculating the +Gauss sums of characters of each order up to prime and +composite conductor 300000. +for each nonzero integer k. In turn, this bound is implied by Propo- +sition 3.14, using the zero-free region for the Hecke Grossencharacter +L-functions in [IK04, Theorem 5.35]. +□ +To contrast this, we will show that the normalized Gauss sums τ(χ)2/q, +with χ ranging over all quartic characters, equidistribute slowly. More +precisely, we have the following result. +Proposition 3.16. Let k ∈ 2Z, k ̸= 0. There exists ck ∈ C such that +(3.31) +� +q≤X +(q,2)=1 +� +χ:χ4=1 +cond(χ)=q +(τ(χ)2/q)k = ckX + o(X). +Remark 3.17. Recall from Lemma 3.7 that the total number of such +characters grows like X log X, so Proposition 3.16 shows that the rate +of equidistribution is only O((log X)−1) here. In contrast, in the family +of totally quartic characters, the GRH would imply a rate of equidis- +tribution of the form O(X−1/2+ε). This difference in rates of equidis- +tribution is supported by Figure 2 in which we see that the arguments + +5+00 +4000 +DODE +DOZ +1000 +E- +-1 +020 +JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG +of squares of the Gauss sums of totally quartic characters quickly con- +verge to being uniformly distributed, as compared to the Gauss sums +of all quartic characters. +In addition, one can derive a similar result when restricting to χ ∈ +Ψ4(X), simply by subtracting off the contribution from the quadratic +characters alone. +Proof. As in Lemma 3.13, write χ = χ2χ4, with χ2 quadratic and χ4 +totally quartic. Then τ(χ)4/(q1q2)2 = τ(χ4)4/q2 +2. The analog of Z(k, s), +using k even to simplify, is +(3.32) +Zall(k, s) = +� +0̸=(β)⊆Z[i] +(β,2β)=1 +β squarefree +τ(χβ)2k/N(β)k +N(β)s +� +q2∈Z≥1 +(q2,2N(β))=1 +1 +qs +2 +. +Referring to the calculation in Proposition 3.14, we obtain +(3.33) +Zall(k, s) = ζ(s)L(s, λk)A(s), +where A(s) is an Euler product absolutely convergent for Re(s) > 1/2. +Since this generating function has a simple pole at s = 1, we deduce +Proposition 3.16. +□ +As mentioned above, in order to deduce equidistribution, by Weyl’s +equidistribution criterion, we also need to consider odd values of k in +(3.31). This is more technical than the case for even k, so we content +ourselves with a conjecture. +Conjecture 3.18. For each odd k, there exists δ > 0 such that +(3.34) +� +q≤X +(q,2)=1 +� +χ:χ4=1 +cond(χ)=q +(τ(χ)2/q)k ≪k,δ X1−δ. +Remark 3.19. By Lemma 3.13 and (3.24), this problem reduces to +understanding sums of the rough shape +� +β,q2 +q2N(β)≤X +�−N(β) +q2 +� +µ(β) +� +2 +N(β) +� +λ((β))k, +where we have omitted many of the conditions on β and q2. In the +range where q2 is very small, the GRH gives cancellation in the sum +over β. Conversely, in the range where N(β) is very small, the GRH + +VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS +21 +gives cancellation in the sum over q2. This discussion indicates that +Conjecture 3.18 follows from GRH, with any δ < 1/4. +Unconditionally, one can deduce some cancellation using the zero-free +region for the β-sum (with q2 very small), and a subconvexity bound +for the q2-sum (with N(β) very small). In the range where both q2 +and N(β) have some size, then Heath-Brown’s quadratic large sieve +[HB95] gives some cancellation. +Since we logically do not need an +unconditional proof of equidistribution, we omit the details for brevity. +Remark 3.20. Conjecture 3.18 and Proposition 3.16 together imply +that the squares of the quartic Gauss sums do equidistribute in the full +family Ψ4(X). +3.4.2. Sextic characters. Now we turn to the sextic Gauss sums. +Lemma 3.21. Suppose that χ is totally sextic of conductor q, and say +χ = χ2χ3 with χ2 quadratic and χ3 cubic, each of conductor q. Suppose +χ3 = χβ, as in Lemma 3.4. Then +(3.35) +τ(χ) = µ(q)χ3(2)τ(χ2)τ(χ3)βq−1. +Proof. By [IK04, (3.18)], τ(χ2)τ(χ3) = J(χ2, χ3)τ(χ), where J(χ2, χ3) +is the Jacobi sum. +It is easy to show using the Chinese remainder +theorem that if χ2 = � +p χ(p) +2 +and χ3 = � +p χ(p) +3 , then +(3.36) +J(χ2, χ3) = +� +p +J(χ(p) +2 , χ(p) +3 ). +The Jacobi sum for characters of prime conductor can be evaluated +explicitly using the following facts. By [Lem00, Prop. 4.30], +(3.37) +J(χ(p) +2 , χ(p) +3 ) = χ(p) +3 (22)J(χ(p) +3 , χ(p) +3 ). +Suppose that χ(p) +3 += χπ, where ππ = p, and π is primary. Then [IR90, +Ch. 9, Lem. 1] implies J(χπ, χπ) = −π. (Warning: they state the +value π instead of −π, but recall their definition of primary is opposite +our convention. Also recall that χπ = χ−π.) Gathering the formulas, +we obtain +(3.38) +τ(χ2)τ(χ3) = τ(χ)χ3(2)2 � +πi|β +(−πi) = τ(χ)χ3(2)2µ(q)β. +Rearranging this and using ββ = q completes the proof. +□ + +22 +JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG +Corollary 3.22. Let conditions be as in Lemma 3.21. Then +(3.39) +τ(χ)2/q = χ3(4) +�−1 +q +� +τ(χβ)2β +2/q2. +Patterson [Pat78] showed that τ(χβ)/√q is uniformly distributed on +the unit circle, as χβ ranges over primitive cubic characters. The same +method gives equidistribution after multiplication by a Hecke Grossen- +character, and so similarly to the quartic case above, we deduce: +Corollary 3.23 (Patterson). The Gauss sums τ(χ)2/q, for χ totally +sextic of conductor q, equidistribute on the unit circle. +In light of Corollary 3.22, Proposition 3.16, and Conjecture 3.18, it +seems reasonable to conjecture that the points τ(χ)2/q are equidis- +tributed on the unit circle, as χ varies over all sextic characters. To +see a limitation in the rate of equidistribution, it is convenient to con- +sider τ(χ)6/q3, which is multiplicative for χ sextic. For q ≡ 1 (mod 4), +and χ = χ2 quadratic, we have τ(χ2)2/q = 1, so the quadratic part is +constant. For χ cubic and q ≡ 1 (mod 4), +(3.40) +τ(χβ)6/q3 = µ(β)τ(χβ)3β +3 = q−1β +2, +which is nearly a Hecke Grossencharacter. A similar formula holds for +χ totally sextic, namely +(3.41) +τ(χ)6/q3 = q−4β +8. +Therefore, carrying out the same steps as in Proposition 3.16 shows +that +(3.42) +� +q≤X +q≡1 (mod 4) +� +χ∈Ψ6 +cond(χ)=q +� +τ(χ)6/q3�k += CkX + o(X). +This is less of an obstruction than in the quartic case, since here the +rate of equidistribution is O((log X)−2) instead of O((log X)−1), due to +the fact that |Ψ6(X)| is approximately log X times as large as |Ψ4(X)|. +Similarly to the discussion of the quartic case in Remarks 3.19 and +3.20, we make the following conjecture without further explanation. +Conjecture 3.24. The Gauss sums τ(χ)2/q, for χ ranging in Ψ6(X), +equidistribute on the unit circle. + +VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS +23 +3.5. Estimates for quartic and sextic characters. In order to ap- +ply the random matrix theory conjectures, we need variants on Propo- +sition 3.6, Lemma 3.7, Proposition 3.8, and Lemma 3.9, as follows. +Lemma 3.25. For primitive Dirichlet characters χ of order ℓ we have +for ℓ = 4 and ℓ = 6 that +(3.43) +� +χ∈Ψℓ(X) +1 +� +cond(χ) +∼ 2Kℓ +√ +X(log X)d(ℓ)−2, +and +(3.44) +� +χ∈Ψtot +ℓ +(X) +1 +� +cond(χ) +∼ 2Ktot +ℓ +√ +X, +� +χ∈Ψ′ +ℓ(X) +1 +� +cond(χ) +∼ 2 +√ +X +log X . +Proof. These estimates follow from a straightforward application of +partial summation or from a minor modification of Lemma 3.5 since +the generating Dirichlet series for one of these sums has its pole at +s = 1/2 instead of at s = 1. +□ +4. Random matrix theory: Conjectural asymptotic +behavior +This section closely follows the exposition of §3 of [DFK04] and §4 of +[DFK07]. +Let U(N) be the set of unitary N×N matrices with complex coefficients +which forms a probability space with respect to the Haar measure. +For a family of L-functions with symmetry type U(N), Katz and Sar- +nak conjectured that the statistics of the low-lying zeros should agree +with those of the eigenangles of random matrices in U(N) [KS99]. Let +PA(λ) = det(A − λI) be the characteristic polynomial of A. Keating +and Snaith [KS00] suggest that the distribution of the values of the L- +functions at the critical point is related to the value distribution of the +characteristic polynomials |PA(1)| with respect to the Haar measure on +U(N). +For any s ∈ C we consider the moments +MU(s, N) := +� +U(N) +|PA(1)|s dHaar + +24 +JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG +for the distribution of |PA(1)| in U(N) with respect to the Haar mea- +sure. In [KS00], Keating and Snaith proved that +(4.1) +MU(s, N) = +N +� +j=1 +Γ(j)Γ(j + s) +Γ2(j + s/2) , +so that MU(s, N) is analytic for Re(s) > −1 and has meromorphic +continuation to the whole complex plane. The probability density of +|PA(1)| is given by the Mellin transform +pU(x, N) = +1 +2πi +� +Re(s)=c +MU(s, N)x−s−1 ds, +for some c > −1. +In the applications to the vanishing of twisted L-functions we consider +in this paper, we are only interested in small values of x where the value +of pU(x, N) is determined by the first pole of MU(s, N) at s = −1. More +precisely, for x ≤ N −1/2, one can show that +pU(x, N) ∼ G2(1/2)N 1/4 +as N → ∞, +where G(z) is the Barnes G-function with special value [Bar99] +G(1/2) = exp +�3 +2ζ′(−1) − 1 +4 log π + 1 +24 log 2 +� +. +We will now consider the moments for the special values of twists of +L-functions. +We then define, for any s ∈ C, the following sum of +evaluations at s = 1 of L-functions primitive order ℓ characters of +conductor less than X: +(4.2) +ME(s, X) = +1 +#FΨℓ,E(X) +� +L(E,s,χ)∈FΨℓ,E(X) +|L(E, 1, χ)|s. +Then, since the families of twists of order ℓ are expected to have unitary +symmetry, we have +Conjecture 4.1 (Keating and Snaith Conjecture for twists of order +ℓ). With the notation as above, +ME(s, X) ∼ aE(s/2)MU(s, N) +as N = 2 log X → ∞, +where aE(s/2) is an arithmetic factor depending only on the curve E. + +VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS +25 +From Conjecture 4.1, the probability density for the distribution of the +special values |L(E, 1, χ)| for characters of order ℓ is +pE(x, X) += +1 +2πi +� +Re(s)=c +ME(s, X)x−s−1 ds +(4.3) +∼ +1 +2πi +� +Re(s)=c +aE(s/2)MU(s, N)x−s−1 ds +(4.4) +as N = 2 log X → ∞. As above, when x ≤ N −1/2, the value of pE(x, X) +is determined by the residue of MU(s, N) at s = −1, thus it follows +from (4.4) that for x ≤ (2 log X)−1/2, +(4.5) +pE(x, X) ∼ 21/4aE(−1/2)G2(1/2) log1/4(X) +as X → ∞. +We now use the probability density of the random matrix model with +the properties of the integers nE(χ) to obtain conjectures for the van- +ishing of the L-values |L(E, 1, χ)|. When χ is either quartic or sextic, +the discretization nE(χ) is a rational integer since Z[ζℓ] ∩ R = Z when +ℓ = 4 or 6. +Lemma 4.2. Let χ be a primitive Dirichlet character of order ℓ = 4 +or 6. Then +|L(E, 1, χ)| = +cE,ℓ +� +cond(χ) +|nE(χ)|, +where cE,ℓ is a nonzero constant which depends only on the curve E +and ℓ. +Proof. By rearranging equation (2.2) we obtain +|L(E, 1, χ)| = +���� +Ωϵ(E) τ(χ) kE nE(χ) +cond(χ) +���� = |Ωϵ(E) kE nE(χ)| +� +cond(χ) += cE,ℓ|nE(χ)| +� +cond(χ) +, +where the nonzero constant kE is that of Proposition 2.2. +□ +We write +(4.6) +Prob{|L(E, 1, χ)| = 0} = Prob{|L(E, 1, χ)| < B(cond(χ))}, +for some function B(cond(χ)) of the character. By Lemma 4.2 we may +take B(cond(χ)) = +cE,ℓ +� +cond(χ) +. Note that since cE,ℓ ̸= 0, if +|nE(χ)|cE,ℓ +� +cond(χ) +< +cE,ℓ +� +cond(χ) +, +then |nE(χ)| < 1 and hence must vanish since |nE(χ)| ∈ Z≥0. + +26 +JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG +Using (4.5), we have +Prob{|L(E, 1, χ)| = 0} += +� B(cond(χ)) +0 +21/4aE(−1/2)G2(1/2) log1/4(X) dx += +21/4aE(−1/2)G2(1/2) log1/4(X)B(cond(χ)) +Summing the probabilities gives +|VΨℓ,E(X)| = 21/4cE,kaE(−1/2)G2(1/2) log1/4(X) +� +cond(χ)≤X +1 +� +cond(χ) +. +Thus, by the analysis in §3.3, we have +|VΨ4,E(X)| ∼ 25/4cE,4K4aE(−1/2)G2(1/2) log1/4(X) +√ +X log X +∼ bE,4X1/2 log5/4 X +and +|VΨ6,E(X)| ∼ 25/4cE,6K6aE(−1/2)G2(1/2) log1/4(X) +√ +X(log X)2 +∼ bE,6X1/2 log9/4 X +as X → ∞. +Moreover, if we restrict to those characters that are totally quartic or +sextic, we get the following estimates +|VΨtot +4 ,E(X)| ∼ 25/4cE,4Ktot +4 aE(−1/2)G2(1/2) log1/4(X) +√ +X +∼ btot +E,4X1/2 log1/4 X +and +|VΨtot +6 ,E(X)| ∼ 25/4cE,6Ktot +6 aE(−1/2)G2(1/2) +∼ btot +E,6X1/2 log1/4 X +as X → ∞. +Finally, if we restrict only to those twists by characters of prime con- +ductor, we conclude +|VΨ′ +4,E(X)| ∼ 25/4cE,4aE(−1/2)G2(1/2) log1/4(X) +√ +X +log X +∼ b′ +E,4X1/2 log−3/4 X + +VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS +27 +and +|VΨ′ +6,E(X)| ∼ 25/4cE,6aE(−1/2)G2(1/2) log1/4(X) +√ +X +log X +∼ b′ +E,6X1/2 log−3/4 X +as X → ∞. +4.1. Computations. Here we provide numerical evidence for Conjec- +ture 1.1. The computations of the Conrey labels for the characters were +done in SageMath [Sag21] and the computations of the L-functions were +done in PARI/GP [PAR22]. The L-function computations were done +in a distributed way on the Open Science Grid. For each curve, we +generated a PARI/GP script to calculate a twisted L-function for each +primitive character of order 4 and 6, and then combined the results into +one file at the end. The combined wall time of all the computations +was more than 50 years. The code and data are available at [BR23]. +In Figure 3 we plot the points +(X, X1/2 log5/4 X +|VΨ4,11.a.1(X)|), (X, X1/2 log−3/4 X +|VΨ′ +4,11.a.1(X)| ), (X, +X1/2 log1/4 X +|VΨtot +4 +,11.a.1(X)|) +that provides a comparison between the predicted vanishings of L(E, 1, χ) +for quartic characters and for the curve 11.a.1. In Figure 4 we plot the +analogous points for the same curve but for sextic twists. In Figure 5 +we plot the points +(X, X1/2 log−3/4 X +|VΨ′ +4,37.a.1(X)| ), (X, X1/2 log−3/4 X +|VΨ′ +6,37.a.1(X)| ) +Even though we are most interested in the families of all quartic and +sextic twists, we include the families of twists of prime conductor be- +cause there are far fewer such characters and so we can calculate the +number of vanishings up to a much larger X. We include the fami- +lies of twists by totally quartic and sextic characters to highlight the +transition between the family of prime conductors and the family of all +conductors. +References +[Bar99] +E.W. Barnes. The theory of the G-function. Quart. J. Math., 31:264–314, +1899. +[BCDT01] Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor. +On the modularity of elliptic curves over Q: wild 3-adic exercises. Journal +of the American Mathematical Society, pages 843–939, 2001. +[BE81] +Bruce C Berndt and Ronald J Evans. The determination of Gauss sums. +Bulletin of the American Mathematical Society, 5(2):107–129, 1981. + +28 +JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG +(a) +The +ratio +of +predicted +vanishings +to +empirical +van- +ishings +of +twists +of +the curve 11.a.1 by +quartic characters of +conductor ≤ 700000. +(b) The ratio of pre- +dicted +vanishings +to +empirical +vanishings +of twists of the curve +11.a.1 +by +quartic +characters +of +prime +conductor ≤ 2000000. +(c) The ratio of pre- +dicted +vanishings +to +empirical +vanishings +of twists of the curve +11.a.1 +by +totally +quartic characters of +conductor ≤ 700000. +Figure 3. Verification of Conjecture 1.1 for quartic +twists of 11.a.1. +(a) The ratio of pre- +dicted +vanishings +to +empirical vanishings of +twists +of +the +curve +11.a.1 by sextic char- +acters of conductor ≤ +300000. +(b) The ratio of pre- +dicted +vanishings +to +empirical vanishings of +twists +of +the +curve +11.a.1 by sextic char- +acters of prime con- +ductor ≤ 2000000. +(c) The ratio of pre- +dicted +vanishings +to +empirical vanishings of +twists +of +the +curve +11.a.1 by totally sex- +tic characters of con- +ductor ≤ 300000. +Figure 4. Verification of Conjecture 1.1 for sextic +twists of 11.a.1. +[BR23] +Jen Berg and Nathan C. Ryan. Code and data for quartic and sextic +twists of elliptic curve L-functions. http://eg.bucknell.edu/~ncr006/ +quartic-sextic-twists-website/, 2023. +[BY10] +Stephan Baier and Matthew P. Young. Mean values with cubic characters. +J. Number Theory, 130(4):879–903, 2010. +[CFK+05] J. Brian Conrey, David W Farmer, Jon P Keating, Michael O Rubin- +stein, and Nina C Snaith. Integral moments of L-functions. Proceedings +of the London Mathematical Society, 91(1):33–104, 2005. +[Cho87] +Sarvadaman Chowla. The Riemann hypothesis and Hilbert’s tenth prob- +lem, volume 4. CRC Press, 1987. + +0.38 +0.36 +tE'O +0.32 +0.30 +0.28 +0.26 +0.D0 +0.25 +0.50 +0.75 +LDo +125 +150 +175 +20o +1e614 +12 +LD +0.B +0.6 +0 + 11 +9 - +8 - +1 +61 +50000 +150000 +DO +250000 +3000000.45 +0.40 - +0.35 +0.30 +0.25 +0.D0 +0.25 +0.50 +0.75 +Lio +125 +150 +175 +200 +1e616 +15 +14 +13 +12 +11 +LD +0 +DODS +10dC0 +150000 +240000 +25000 +3+0dC04.5 +4.D +3.5- +3.0 +25 - +20 +15 +LD + VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS +29 +(a) The ratio of pre- +dicted +vanishings +to +empirical +vanishings +of twists of the curve +37.a.1 +by +quartic +characters +of +prime +conductor ≤ 2000000. +(b) The ratio of pre- +dicted +vanishings +to +empirical vanishings of +twists +of +the +curve +37.a.1 by sextic char- +acters of prime con- +ductor ≤ 2000000. +Figure 5. Verification of parts of Conjecture 1.1 for +twists of 37.a.1. +[CKRS00] JB Conrey, JP Keating, MO Rubinstein, and NC Snaith. On the fre- +quency of vanishing of quadratic twists of modular L-functions. In Pro- +ceedings of the Millennial Conference on Number Theory, Urbana, Illinois, +21-26 May, 2000. AK Peters, 2000. +[DFK04] Chantal David, Jack Fearnley, and Hershy Kisilevsky. On the vanishing of +twisted L-functions of elliptic curves. Experiment. Math., 13(2):185–198, +2004. +[DFK07] Chantal David, Jack Fearnley, and Hershy Kisilevsky. Vanishing of L- +functions of elliptic curves over number fields. In Ranks of elliptic curves +and random matrix theory, volume 341 of London Math. Soc. Lecture Note +Ser., pages 247–259. Cambridge Univ. Press, Cambridge, 2007. +[FMS10] Steven Finch, Greg Martin, and Pascal Sebah. Roots of unity and +nullity modulo n. Proceedings of the American Mathematical Society, +138(8):2729–2743, 2010. +[HB95] +D. R. Heath-Brown. A mean value estimate for real character sums. Acta +Arith., 72(3):235–275, 1995. +[HBP79] D. R. Heath-Brown and S. J. Patterson. The distribution of Kummer +sums at prime arguments. J. Reine Angew. Math., 310:111–130, 1979. +[IK04] +Henryk Iwaniec and Emmanuel Kowalski. Analytic number theory, vol- +ume 53 of American Mathematical Society Colloquium Publications. +American Mathematical Society, Providence, RI, 2004. +[IR90] +Kenneth Ireland and Michael Rosen. A classical introduction to modern +number theory, volume 84 of Graduate Texts in Mathematics. Springer- +Verlag, New York, second edition, 1990. +[KS99] +Nicholas Katz and Peter Sarnak. Zeroes of zeta functions and symmetry. +Bulletin of the American Mathematical Society, 36(1):1–26, 1999. +[KS00] +Jon P Keating and Nina C Snaith. Random matrix theory and L-functions +at s = 1/2. Communications in Mathematical Physics, 214(1):91–100, +2000. + +0.32 +0.28 +0.26 +0.24 +0.22 +0.0O +0.25 +0.50 +0.75 +125 +150 +175 +2b0 +1e6SLEO +0.350 +0.325 +0.300 - +0.275 +0.250 +0.225 +0.200 +0.175 +0.DO +0.25 +0.50 +0.75 +LDo +125 +150 +175 +2o +le630 +JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG +[Lem00] Franz Lemmermeyer. Reciprocity laws. Springer Monographs in Mathe- +matics. Springer-Verlag, Berlin, 2000. From Euler to Eisenstein. +[MR21] +Barry Mazur and Karl Rubin. Arithmetic conjectures suggested by the +statistical behavior of modular symbols. Experimental Mathematics, pages +1–16, 2021. +[MV07] +Hugh L. Montgomery and Robert C. Vaughan. Multiplicative number the- +ory. I. Classical theory, volume 97 of Cambridge Studies in Advanced +Mathematics. Cambridge University Press, Cambridge, 2007. +[PAR22] PARI Group, Univ. Bordeaux. PARI/GP version 2.13.4, 2022. available +from http://pari.math.u-bordeaux.fr/. +[Pat78] +S. J. Patterson. On the distribution of Kummer sums. J. Reine Angew. +Math., 303(304):126–143, 1978. +[Pat87] +Samuel J Patterson. The distribution of general Gauss sums and simi- +lar arithmetic functions at prime arguments. Proceedings of the London +Mathematical Society, 3(2):193–215, 1987. +[PHH81] SJ Patterson, H Halberstam, and C Hooley. The distribution of general +Gauss sums at prime arguments. Progress in Analytic Number Theory, +2:171–182, 1981. +[PPK+07] Ruth Pordes, Don Petravick, Bill Kramer, Doug Olson, Miron Livny, +Alain Roy, Paul Avery, Kent Blackburn, Torre Wenaus, Frank W¨urthwein, +Ian Foster, Rob Gardner, Mike Wilde, Alan Blatecky, John McGee, and +Rob Quick. The open science grid. In J. Phys. Conf. Ser., volume 78 of +78, page 012057, 2007. +[Sag21] +Sage Developers. SageMath, the Sage Mathematics Software System (Ver- +sion 9.4), 2021. https://www.sagemath.org. +[SBH+09] Igor Sfiligoi, Daniel C Bradley, Burt Holzman, Parag Mhashilkar, San- +jay Padhi, and Frank Wurthwein. The pilot way to grid resources using +glideinwms. In 2009 WRI World Congress on Computer Science and In- +formation Engineering, volume 2 of 2, pages 428–432, 2009. +[SY10] +K. Soundararajan and Matthew P. Young. The second moment of +quadratic twists of modular L-functions. J. Eur. Math. Soc. (JEMS), +12(5):1097–1116, 2010. +[TW95] +Richard Taylor and Andrew Wiles. Ring-theoretic properties of certain +Hecke algebras. Annals of Mathematics, 141(3):553–572, 1995. +[Wil95] +Andrew Wiles. Modular elliptic curves and Fermat’s last theorem. Annals +of mathematics, 141(3):443–551, 1995. +[WW20] Hanneke Wiersema and Christian Wuthrich. Integrality of twisted l-values +of elliptic curves, 2020. +Email address: jsb047@bucknell.edu +Email address: nathan.ryan@bucknell.edu +Department of Mathematics, Bucknell University, Lewisburg, PA 17837 +Email address: myoung@math.tamu.edu + +VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS +31 +Department of Mathematics, Texas A&M University, College Station, +TX 77843-3368 + diff --git a/DdE4T4oBgHgl3EQf6A7h/content/tmp_files/load_file.txt b/DdE4T4oBgHgl3EQf6A7h/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dddc8a6146475ca88730f1983aa2c93f4ca94a94 --- /dev/null +++ b/DdE4T4oBgHgl3EQf6A7h/content/tmp_files/load_file.txt @@ -0,0 +1,977 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf,len=976 +page_content='VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS JENNIFER BERG, NATHAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' RYAN, AND MATTHEW P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' YOUNG Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Let E be an elliptic curve over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' We conjecture asymptotic estimates for the number of vanishings of L(E, 1, χ) as χ varies over all primitive Dirichlet characters of orders 4 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Our conjectures about these families come from conjectures about random unitary matrices as predicted by the philosophy of Katz- Sarnak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' We support our conjectures with numerical evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Earlier work by David, Fearnley and Kisilevsky formulates anal- ogous conjectures for characters of any odd prime order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In the composite order case, however, we need to justify our use of random matrix theory heuristics by analyzing the equidistribution of the squares of normalized Gauss sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Along the way we introduce the notion of totally order ℓ characters to quantify how quickly quartic and sextic Gauss sums become equidistributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Surprisingly, the rate of equidistribution in the full family of quartic (sextic, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=') characters is much slower than in the sub-family of totally quar- tic (sextic, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=') characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' A conceptual explanation for this phenomenon is that the full family of order ℓ twisted elliptic curve L-functions, with ℓ even and composite, is a mixed family with both unitary and orthogonal aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Introduction Vanishings of elliptic curve L-functions at the value s = 1 (normalized so that the functional equation relates s and 2−s) is central to a great deal of modern number theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' For instance, if an L-function associated to an elliptic curve vanishes at s = 1, then the BSD conjecture predicts that the curve will have infinitely many rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Additionally, statistical questions about how often L-functions within a family vanish at the central value have also been of broad interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' For example, it is expected (as first conjectured by Chowla [Cho87]) that, for all primitive Dirichlet characters χ, we have L(χ, 1/2) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' A fruitful way of studying such questions has been to model L-functions using random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' For example, in [CKRS00] Conrey, Keat- ing, Rubinstein and Snaith consider the family of twisted L-functions 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='05329v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='NT] 12 Jan 2023 2 JENNIFER BERG, NATHAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' RYAN, AND MATTHEW P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' YOUNG L(f, s, χd) associated to a modular form f of weight k and quadratic characters χd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' They show that the random matrix theory model pre- dicts that infinitely many values L(f, s, χd) are zero when the weight of f is 2 or 4, but that only finitely many of the values are zero when the weight is at least 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Another example, due to David, Fearnley and Kisilevsky [DFK04, DFK07], instead uses the random matrix model to give conjectural asymptotics for the number of vanishings of elliptic curve L-functions twisted by families of Dirichlet characters of a fixed order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In particu- lar, they predict that for an elliptic curve E, the values L(E, 1, χ) are zero infinitely often if χ has order 3 or 5, but for characters χ with a fixed prime order ℓ ≥ 7, only finitely many values L(E, 1, χ) are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In recent work, inspired by the conjectures of [DFK04, DFK07], Mazur and Rubin [MR21] use statistical properties of modular symbols to heuristically estimate the probability that L(E, 1, χ) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Their Conjecture 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1 implies that for an elliptic curve E over Q, there should be only finitely many characters χ of a fixed order ℓ such that L(E, 1, χ) = 0 and ϕ(ℓ) > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' This further implies the following: Let E be an elliptic curve over Q and let be F/Q an infinite abelian exten- sion such that Gal(F/Q) has only finitely many characters of orders 2, 3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Then E(F) is finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Finally, for an elliptic curve E defined over Q, their Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='2 relates the (order of) vanishing of L(E, 1, χ) to the growth in rank of E over a finite abelian extension F/Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In particular, if BSD holds for E over both Q and F, then rank(E(F)) = rank(E(Q)) + � χ:Gal(F/Q)→C× ords=1L(E, s, χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Notation and statement of the Main Conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' We fix the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' See Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1 for the definition of totally order ℓ characters but, roughly speaking, these are order ℓ characters that, when factored, have all their factors also of order ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Set Ψℓ = {primitive Dirichlet characters χ of order ℓ} Ψtot ℓ = {χ ∈ Ψℓ that are totally order ℓ} Ψ′ ℓ = {χ ∈ Ψℓ with cond(χ) prime}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Note that Ψ′ ℓ ⊆ Ψtot ℓ ⊆ Ψℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS 3 Along the way we will need to estimate the number of characters in each family and so we define: Ψℓ(X) = {χ ∈ Ψℓ : cond(χ) ≤ X} Ψtot ℓ (X) = {χ ∈ Ψtot ℓ : cond(χ) ≤ X} Ψ′ ℓ(X) = {χ ∈ Ψ′ ℓ : cond(χ) ≤ X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' For an elliptic curve E over Q we also define: FΨℓ,E = {L(E, s, χ) : χ ∈ Ψℓ} FΨℓ,E(X) = {L(E, s, χ) ∈ FΨℓ,E : χ ∈ Ψℓ(X)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' We also define FΨtot ℓ ,E and FΨtot ℓ ,E(X) analogously for Ψtot ℓ in place of Ψℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' we do the same with Ψ′ ℓ, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Finally, let VΨℓ,E(X) = {L(E, s, χ) ∈ FΨℓ,E(X) : L(E, 1, χ) = 0} VΨtot ℓ ,E(X) = {L(E, s, χ) ∈ FΨtot ℓ ,E(X) : L(E, 1, χ) = 0} VΨ′ ℓ,E(X) = {L(E, s, χ) ∈ FΨ′ ℓ,E(X) : L(E, 1, χ) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' With this notation, we make the following conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Let E be an elliptic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Then, there exist con- stants bE,4 and bE,6 so that |VΨ4,E(X)| ∼ bE,4X1/2 log5/4 X and |VΨ6,E(X)| ∼ bE,6X1/2 log9/4 X as X → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Moreover, if we restrict only to those twists by totally quartic or totally sextic characters, then there exist constants btot E,4 and btot E,6 such that |VΨtot 4 ,E(X)| ∼ btot E,4X1/2 log1/4 X and |VΨtot 6 ,E(X)| ∼ btot E,6X1/2 log1/4 X as X → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Finally, if we restrict only to those twists by characters of prime con- ductor, then there exist constants b′ E,4 and b′ E,6 such that |VΨ′ 4,E(X)| ∼ b′ E,4X1/2 log−3/4 X and |VΨ′ 6,E(X)| ∼ b′ E,6X1/2 log−3/4 X as X → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' 4 JENNIFER BERG, NATHAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' RYAN, AND MATTHEW P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' YOUNG In particular, we conjecture that families of elliptic curve L-functions twisted by quartic and sextic characters vanish infinitely often at the central value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' To assist the reader in comparing the powers of log X in the above asymptotics, we point out here that for ℓ = 4, |Ψ4(X)| is roughly log X times as large as |Ψtot 4 (X)|, which in turn is roughly log X times as large as |Ψ′ 4(X)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' For ℓ = 6, then |Ψ6(X)|/|Ψtot 6 (X)| ≍ (log X)2, and |Ψtot 6 (X)|/|Ψ′ 6(X)| ≍ log X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Hence, in each of the three families with a given value of ℓ, the proportion of vanishing twists has the same order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' See Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='6, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='7, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='8, and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='9 below for asymptotics of the underlying families of characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Outline of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' There are two main ingredients needed to be able to apply random matrix theory predictions to our families of twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The first is a discretization for the central values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' As described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1 this can be done for curves E satisfying certain technical conditions as described in [WW20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' We need this discretization in order to approximate the probability that L(E, 1, χ) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The second ingredient is a proper identification of the symmetry type of the family, which is largely governed by the distribution of the sign of the functional equation within the family (see Section 4 of [CFK+05]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' This directly leads to an investigation around the equidistribution of squares of Gauss sums of quartic and sextic characters, which has con- nections to the theory of metaplectic automorphic forms [Pat87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' See Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1 for a thorough discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' It is a subtle feature that the families of twists of elliptic curve L- functions by the characters in Ψtot ℓ and Ψ′ ℓ have unitary symmetry type, but for composite even values of ℓ, the twists by Ψℓ should be viewed as a mixed family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' To elaborate on this point, consider the case that ℓ = 4, and first note that a character χ ∈ Ψ4 factors uniquely as a totally quartic character times a quadratic character of relatively prime conductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The totally quartic family has a unitary symmetry, but the family of twists of an elliptic curve by quadratic characters has orthogonal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' This tension between the totally quartic aspect and the quadratic aspect is what leads to the mixed symmetry type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The situation is analogous to the family L(E, 1 + it, χd);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' if t = 0 and d varies then one has an orthogonal family, while if d is fixed and t varies, then one has a unitary family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' See [SY10] for more discussion on this family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS 5 Another interesting feature of these families is that Ψℓ(X) is larger than Ψtot ℓ (X) by a logarithmic factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' For instance, when ℓ = 4, then Ψtot 4 (X) grows linearly in X (see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='6 below), and of course Ψ2(X) also grows linearly in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Similarly to how the average size of the divisor function is log X, this indicates that |Ψ4(X)| grows like X log X (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='7 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In the next section we give the necessary background and notation for L-functions and their central values and discuss the discretization we use in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In the subsequent section we estimate some sums involving quartic and sextic characters and discuss totally quartic and sextic characters in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In the final section, we motivate the asymptotics in Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1 and provide numerical evidence that supports them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' We thank David Farmer and Brian Conrey for helpful conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' This research was done using services provided by the OSG Consortium [PPK+07, SBH+09], which is supported by the National Science Foundation awards #2030508 and #1836650.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' This material is based upon work supported by the National Science Foun- dation under agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' DMS-2001306 (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Any opinions, find- ings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the Na- tional Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' L-functions and central values Let E be an elliptic curve defined over Q of conductor NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The L- function of E is given by the Euler product L(E, s) = � p∤NE � 1 − ap ps + 1 p2s−1 �−1 � p|NE � 1 − ap ps �−1 = � n≥1 an ns .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The modularity theorem [BCDT01, TW95, Wil95] implies that L(E, s) has an analytic continuation to all of C and satisfies the functional equation Λ(E, s) = � √NE 2π �s Γ(s)L(E, s) = wEΛ(E, 2 − s) where the sign of the functional equation is wE = ±1 and is the eigen- value of the Fricke involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Let χ be a primitive character and let cond(χ) be its conductor and suppose that cond(χ) is coprime to the 6 JENNIFER BERG, NATHAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' RYAN, AND MATTHEW P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' YOUNG conductor NE of the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The twisted L-function has Dirichlet series L(E, s, χ) = � n≥1 anχ(n) ns and the functional equation (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' [IK04, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='20]) Λ(E, s, χ) = � cond(χ)√NE 2π �s Γ(s)L(E, s, χ) = wEχ(NE)τ(χ)2 cond(χ) Λ(E, 2 − s, χ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1) where τ(χ) = � r∈Z/mZ χ(r)e2πir/m is the Gauss sum and m = cond(χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' To justify our Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1, we need a condi- tion that allows us to deduce that L(E, 1, χ) = 0, for a given E and χ of order ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In particular, we show that L(E, 1, χ) is discretized (see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='2) and so there exists a constant cE,ℓ such that |L(E, 1, χ)| < cE,ℓ/ � cond(χ) implies L(E, 1, χ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In this section we prove the results necessary for the discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Let E be an elliptic curve over Q with conductor NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Let χ be a nontrivial primitive Dirichlet character of conductor m and order ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Set ϵ = {±1} = χ(−1) depending on whether χ is an even or odd character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Let Ω+(E) and Ω−(E) denote the real and imaginary periods of E, respectively, with Ω+(E) > 0 and Ω−(E) ∈ iR>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The algebraic L-value is defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='2) Lalg(E, 1, χ) := L(E, 1, χ) · m τ(χ)Ωϵ(E) = ϵ · L(E, 1, χ)τ(χ) Ωϵ(E) While it has been known for some time that algebraic L-values are algebraic numbers, recent work of Weirsema and Wuthrich [WW20] characterizes conditions on E and χ which guarantee integrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In particular, under the assumption that the Manin constant c0(E) = 1, if the conductor m is not divisible by any prime of additive reduction for E, then Lalg(E, 1, χ) ∈ Z[ζℓ] is an algebraic integer [WW20, Theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' For a given curve E, we will avoid the finitely many characters χ for which Lalg(E, 1, χ) fails to be integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Let χ be a primitive Dirichlet character of odd order ℓ and conductor m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Then Lalg(E, 1, χ) = � χ(NE)(ℓ+1)/2 nE(χ), if wE = 1, (ζℓ − ζ−1 ℓ )−1 χ(NE)(ℓ+1)/2 nE(χ) if wE = −1, for some algebraic integer nE(χ) ∈ Z[ζℓ + ζ−1 ℓ ] = Z[ζℓ] ∩ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS 7 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Let χ be a primitive Dirichlet character of even order ℓ and conductor m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Then Lalg(E, 1, χ) = kE nE(χ) where nE(χ) is some algebraic integer in Z[ζℓ +ζ−1 ℓ ] = Z[ζℓ]∩R and kE is a constant depending only on the curve E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In particular, when wE = 1 we have kE = � � � � � (1 + χ(NE)) if χ(NE) ̸= −1 ζℓ/4 ℓ , if 4 | ℓ and χ(NE) = −1 (ζℓ − ζ−1 ℓ ) if 4 ∤ ℓ and χ(NE) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Proof of Prop 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1 and Prop 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Since E is defined over Q, we have L(E, 1, χ) = L(E, 1, χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Using the functional equation, we obtain Lalg(E, 1, χ) = ϵ · L(E, 1, χ)τ(χ) Ωϵ(E) = ϵ · wE χ(NE) τ(χ)τ(χ)2 m · Ωϵ(E) L(E, 1, χ) = wE χ(NE) τ(χ) Ωϵ(E) L(E, 1, χ) = wEχ(NE) ϵ · τ(χ)L(E, 1, χ) Ωϵ(E) = wEχ(NE) Lalg(E, 1, χ) Thus Lalg(E, 1, χ) is a solution to the equation z = wEχ(NE)z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Note that if z1, z2 ∈ Z[ζℓ] are two distinct solutions to this equation, then z1/z1 = z2/z2 so that z1/z2 = z1/z2 = (z1/z2), hence z1/z2 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Thus Lalg(E, 1, χ) = αz with α ∈ Z[ζℓ] ∩ R = Z[ζℓ + ζ−1 ℓ ] and z ∈ Z[ζℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Suppose that wE = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' When ℓ is odd, we can take z = χ(NE) ℓ+1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Now suppose that ℓ is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' If χ(NE) ̸= −1, since χ(NE) = ζr ℓ for some 1 ≤ r ≤ ℓ, we may take z = (1+χ(NE)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Indeed, we have wEχ(NE)z = ζr ℓ (1 + ζℓ−r ℓ ) = ζr ℓ + 1 = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' If 4 | ℓ and χ(NE) = −1 = ζℓ/2 ℓ , we take z = ζℓ/4 ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Finally, if 4 ∤ ℓ and χ(NE) = −1 take z = ζℓ−ζ−1 ℓ = 2i Im(ζℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' When wE = −1 and ℓ is odd, we may take z = (ζℓ − ζ−1 ℓ )−1χ(NE) ℓ+1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' When ℓ is even, if χ(NE) = −1 then we may take z = ζℓ + ζ−1 ℓ = 2 Re(ζℓ), and if χ(NE) ̸= −1 then we make take z = 1 − χ(NE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' We note that for ℓ even, |kE| ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' It is clear that |ζℓ/4 ℓ | = 1 and |2i Im(ζℓ)| ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Observe |(1 + χ(NE)| ≤ 2, by the triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' 8 JENNIFER BERG, NATHAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' RYAN, AND MATTHEW P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' YOUNG Note that since L(E, 1, χ) vanishes if and only if nE(χ) does, we may interpret the integers nE(χ) as a discretization of the special values L(E, 1, χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' This is similar to the case of cubic characters considered in [DFK04] since Q(ζ3)+ = Q, as opposed to characters of prime order ℓ ≥ 5 where further steps were needed to find an appropriate discretization [DFK07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Estimates for Dirichlet characters In this section we discuss various aspects of Dirichlet characters of order 4 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' A necessary condition for a family of L-functions to be modeled by the family of unitary matrices is that the signs must be uniformly distributed on the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1), L(E, s, χ) has sign wEχ(NE) τ(χ)2 cond(χ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' we will largely focus on the distribution of the square of the Gauss sums, viewing the extra factor χ(NE) as a minor perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' To obtain our estimates for the number of vanishings |VΨℓ,E(X)| (respectively, |VΨ′ ℓ,E(X)| and |VΨtot ℓ ,E(X)|) we must estimate the size of Ψℓ(X) (respectively, Ψ′ ℓ(X) and Ψtot ℓ (X)) as well as the size of an associated sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' We also discuss the family of totally quartic and sextic characters to explain some phenomena we observed in our computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Distributions of Gauss sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Patterson [Pat87], building on work of Heath-Brown and Patterson [HBP79] on the cubic case, showed that the normalized Gauss sum τ(χ)/ � cond(χ) is uniformly distributed on the circle for χ varying in each of Ψtot ℓ and Ψ′ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' This result was first announced in [PHH81];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' see [BE81] for an excellent summary of this and other work related to the distributions of Gauss sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Patter- son’s method moreover shows that the argument of τ(χ)χ(k) is equidis- tributed for any fixed nonzero integer k, and hence so is the argument of τ(χ)2χ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' For the case of quartic and sextic characters with arbitrary conductors, there do not appear to be any results in the literature that imply their Gauss sums are uniformly distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In Figure 1 we see the distri- butions of Gauss sums of characters of orders 3 through 9 of arbitrary conductor up to 200000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' We included characters of order 4 and 6 since those examples are the focus of the paper;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' we included characters of or- ders 3, 5, and 7 as consistency checks (in [DFK04, DFK07] the authors rely on them being uniformly distributed);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' and we included composite orders 8 and 9 to see if something similar happens in those cases as happens in the quartic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In all cases but the quartic case, we see that the distributions of the angles of the signs appear to be uniformly VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS 9 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Each histogram represents the distribution of the argument of the τ(χ)2/cond(χ) for characters of order 3 through 9, from top left to bottom right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Each histogram is made by calculating the Gauss sums of char- acters in Ψℓ of each conductor up to 200000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The quartic distribution has two obvious peaks that we discuss below, in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The images in Figure 1 suggest that the family of matrices that best models the vanishing of L(E, 1, χ) is unitary in every case except possi- bly the case of quartic characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Nevertheless, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='4 we show that the squares of the quartic Gauss sums are indeed equidistributed, despite what the data suggest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Indeed, we prove that the squares of the sextic and quartic Gauss sums are equidistributed, allowing us to apply the heuristics from random matrix theory as in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Totally quartic and sextic characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Much of the back- ground material in this section can be found with proofs in [IR90, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Let χ be a primitive Dirichlet character of conductor q and order ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' For prime p, let vp be the p-adic valuation, so that q = � p pvp(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' We correspondingly factor χ = � p χ(p), where χ(p) has conductor pvp(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' We say that χ is totally order ℓ if each χp is exact 5+00 5000 4000 DORE 2400 1400 E- 1 0ADODE 24000 E- 2 1 0 1DOS 4400 3000 DOZ 1400 2 1 0 112000 40000 E- 2 1 0 34400 DOSE DODE 2500 DOZ 1500 1400 500 0 E- 1 i 2 350000 40000 ADODE DO 4000 E- 2 1 0 i217500 15000 12500 14000 DOSr 5000 2500 E- i210 JENNIFER BERG, NATHAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' RYAN, AND MATTHEW P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' YOUNG order ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' By convention we also consider the trivial character to be totally order ℓ for every ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Quartic characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The construction of quartic characters uses the arithmetic in Z[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The ring Z[i] has class number 1, unit group {±1, ±i}, and discriminant −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' We say α ∈ Z[i] with (α, 2) = 1 is primary if α ≡ 1 (mod (1+i)3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Any odd element in Z[i] has a unique primary associate, which comes from the fact that the unit group in the ring Z[i]/(1+i)3 may be identified with {±1, ±i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' An odd prime p splits as p = ππ if and only if p ≡ 1 (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Given π with N(π) = p, define the quartic residue symbol [ α π] for α ∈ Z[i] with (α, π) = 1, by [ α π] ∈ {±1, ±i} and [ α π] ≡ α p−1 4 (mod π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The map χπ(α) = [ α π] from (Z[i]/(π))× to {±1, ±i} is a character of order 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' If α ∈ Z, then [ α π]2 ≡ α p−1 2 ≡ ( α p) (mod π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Therefore, χ2 π(α) = (α p), showing in particular that the restriction of the quartic residue symbol to Z defines a primitive quartic Dirichlet character of conductor p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Every primitive totally quartic character of odd conductor is of the form χβ, where β = π1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' πk is a product of distinct primary primes, (β, 2β) = 1, and where (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1) χβ(α) = �α β � = k � i=1 � α πi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The totally quartic primitive characters of even conductor are of the form χ2χβ where χ2 is one of four quartic characters of conductor 24, and χβ is totally quartic of odd conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' We begin by classifying the quartic characters of odd prime- power conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' If p ≡ 3 (mod 4), there is no quartic character of conductor pa, since φ(pa) = pa−1(p − 1) ̸≡ 0 (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Since φ(p) = p − 1, if p ≡ 1 (mod 4), there are two distinct quartic characters of conductor p, namely, χπ and χπ, where p = ππ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' There are no primitive quartic characters modulo pj for j ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' To see this, suppose χ is a character of conductor pj, and note that χ(1 + pj−1) ̸= 1, while χ(1 + pj−1)p = χ(1 + pj) = 1, so χ(1 + pj−1) is a nontrivial pth root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Since p is odd, χ(1+pj−1) is not a 4th root of unity, so χ cannot be quartic and primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' By the above classification, a primitive totally quartic character χ of odd conductor must factor over distinct primes pi ≡ 1 (mod 4), and the p-part of χ must be χπ or χπ, where ππ = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' We may assume that VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS 11 π and π are primary primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Hence χ factors as � i χπi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The property that β := π1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' πk is squarefree is equivalent to the condition that the πi are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Moreover, the property (β, β) = 1 is equivalent to that πiπi = pi ≡ 1 (mod 4), for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Hence, every quartic character of odd conductor arises uniquely in the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Next we treat p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' There are four primitive quartic characters of conductor 24, since (Z/(24))× ≃ Z/(2) × Z/(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' We claim there are no primitive quartic characters of conductor 2j, with j ̸= 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' For j ≤ 3 or j = 5 this is a simple finite computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' For j ≥ 6, one can show this as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' First, χ(1 + 2j−1) = −1, since χ2(1 + 2j−1) = χ(1 + 2j) = 1, and primitivity shows χ(1+2j−1) ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' By a similar idea, χ(1+2j−2)2 = χ(1 + 2j−1) = −1, so χ(1 + 2j−2) = ±i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' We finish the claim by noting χ2(1 + 2j−3) = χ(1 + 2j−2) = ±i, so χ(1 + 2j−3) is a square-root of ±i, and hence χ is not quartic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' With the claim established, we easily obtain the final sentence of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' □ Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The first totally quartic primitive character of compos- ite conductor has conductor 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' While there are 8 quartic primitive characters of conductor 65, the LMFDB labels of the totally quartic ones are 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='18, 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='47, 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='8, and 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Sextic characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The construction of sextic characters uses the arithmetic in the Eisenstein integers Z[ω], where ω = e2πi/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The ring Z[ω] has class number 1, unit group {±1, ±ω, ±ω2}, and discriminant −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' We say α ∈ Z[ω] with (α, 3) = 1 is primary1 if α ≡ 1 (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Warning: our usage of primary is consistent with [HBP79], but conflicts with the definition of [IR90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' However, it is easy to translate since α is primary in our sense if and only if −α is primary in the sense of [IR90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Any element in Z[ω] coprime to 3 has a unique primary associate, which comes from the fact that the unit group in the ring Z[ω]/(3) may be identified with {±1, ±ω, ±ω2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' An unramified prime p ∈ Z splits as p = ππ if and only if p ≡ 1 (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Given π with N(π) = p, define the cubic residue symbol ( α π)3 for α ∈ Z[ω] by ( α π)3 ∈ {1, ω, ω2} and ( α π)3 ≡ α p−1 3 (mod π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The map χπ(α) = ( α π)3 from (Z[ω]/(π))× to {1, ω, ω2} is a character of order 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The restriction of χπ to Z induces a primitive cubic Dirichlet character of conductor p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Note that χπ = χ−π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Motivated by the fact that a sextic character factors as a cubic times a quadratic, we next discuss the classification of cubic characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' 1We remark that the usage of primary is context-dependent, and that since we do not mix quartic and sextic characters, we hope there will not be any ambiguity 12 JENNIFER BERG, NATHAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' RYAN, AND MATTHEW P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' YOUNG Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Every primitive cubic Dirichlet character of conductor coprime to 3 is of the form χβ, where β = π1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' πk is a product of distinct primary primes, (β, 3β) = 1, and where (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='2) χβ(α) = �α β � 3 = k � i=1 � α πi � 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The cubic primitive characters of conductor divisible by 3 are of the form χ3χβ where χ3 is one of two cubic characters of conductor 32, and χβ is cubic of conductor coprime to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The classification of such characters with conductor coprime to 3 is given by [BY10, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1], so it only remains to treat cubic characters of conductor 3j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The primitive character of conductor 3 is not cubic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Next, the group (Z/(9))× is cyclic of order 6, generated by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' There are two cubic characters, determined by χ(2) = ω±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Next we argue that there is no primitive cubic character of conductor 3j with j ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' For this, we first observe that χ(1 + 3j−1) = ω±1, since primitivity implies χ(1 + 3j−1) ̸= 1, and χ(1 + 3j−1)3 = χ(1 + 3j) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Next we have χ(1 + 3j−2)3 = χ(1 + 3j−1) = ω±1, so χ(1 + 3j−2) is a cube-root of ω±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Therefore, χ cannot be cubic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Counting characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' To start, we count all the quartic and sex- tic characters of conductor up to some bound and in each family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Such counts were found for arbitrary order in [FMS10] by Finch, Martin and Sebah, but since we are interested in only quartic and sextic charac- ters, in which case the proofs simplify, we prove the results we need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Moreover, we need other variants for which we cannot simply quote [FMS10], so we will develop a bit of machinery that will be helpful for these other questions as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' We begin with a lemma based on the Perron formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Suppose that a(n) is a multiplicative function such that |a(n)| ≤ dk(n), the k-fold divisor function, for some k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Let Z(s) = � n≥1 a(n)n−s, for Re(s) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Suppose that for some integer j ≥ 0, (s−1)jZ(s) has a analytic continuation to a region of the form {σ+it : σ > 1 − c log(2+|t|)}, for some c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In addition, suppose that Z(s) is bounded polynomially in log (2 + |t|) in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='3) � n≤X a(n) = XPj−1(log X) + O(X(log X)−100), for Pj−1 some polynomial of degree ≤ j − 1 (interpreted as 0, if j = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS 13 The basic idea is standard, yet we were unable to find a suitable refer- ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Proof sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' One begins by a use of the quantitative Perron formula, for which a convenient reference is [MV07, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' This implies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='4) � n≤X a(n) = 1 2πi � σ0+iT σ0−iT Z(s)Xsds s + R, where R is a remainder term, and we take σ0 = 1+ c log X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Using [MV07, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='3] and standard bounds on mean values of dk(n), one can show R ≪ X T Poly(log X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Next one shifts the contour of integration to the line 1 − c/2 log T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The pole (if it exists) of Z(s) leads to a main term of the form XPj−1(log X), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The new line of integration is bounded by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='5) Poly(log T)X1− c/2 log T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Choosing log T = (log X)1/2 gives an acceptable error term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Quartic characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Let Ψtot,odd 4 (X) ⊆ Ψtot 4 (X) denote the sub- set of characters with odd conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' For some constants Ktot 4 , Ktot,odd 4 > 0, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='6) |Ψtot 4 (X)| ∼ Ktot 4 X, and |Ψtot,odd 4 (X)| ∼ Ktot,odd 4 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Moreover, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='7) |Ψ′ 4(X)| ∼ X log X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='8) |Ψtot,odd 4 (X)| = � 0̸=(β)⊆Z[i] (β,2β)=1 β squarefree N(β)≤X 1, and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='9) |Ψtot 4 (X)| = |Ψtot,odd 4 (X)| + 4|Ψtot,odd 4 (2−4X)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' 14 JENNIFER BERG, NATHAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' RYAN, AND MATTHEW P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' YOUNG To show (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='6), it suffices to prove the asymptotic formula for |Ψtot,odd 4 (X)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In view of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='5, it will suffice to understand the Dirichlet series (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='10) Z4(s) = � 0̸=(β)⊆Z[i] (β,2β)=1 β squarefree 1 N(β)s = � π̸=π (π,2)=1 (1 + N(π)−s) = � p≡1 (mod 4) (1 + p−s)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Let χ4 be the primitive character modulo 4, so that ζ(s)L(s, χ4) = ζQ[i](s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='11) Z4(s) = ζQ[i](s) � p (1 − p−s)(1 − χ4(p)p−s) � p≡1 (mod 4) (1 + p−s)2, which can be simplified as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='12) Z4(s) = ζQ[i](s)ζ−1(2s)(1 + 2−s)−1 � p≡1 (mod 4) (1 − p−2s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Therefore, Z4(s) has a simple pole at s = 1, and its residue is a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Moreover, the standard analytic properties of ζQ[i](s) let us apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='5, giving the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The asymptotic on Ψ′ 4(X) follows from the prime number theorem in arithmetic progressions, since there are two quartic characters of prime conductor p ≡ 1 (mod 4), and none with p ≡ 3 (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' We have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='13) |Ψ4(X)| = K4X log X + O(X), for some K4 > 0 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Every primitive quartic character factors uniquely as χ4χ2 with χ4 totally quartic of conductor q4 > 1 and χ2 quadratic of conductor q2, with (q4, q2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' It is convenient to drop the condition q4 > 1, thereby including the quadratic characters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' this is allowable since the number of quadratic characters is O(X), which is acceptable for the claimed error term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The Dirichlet series for |Ψ4(X)|, modified to include the quadratic char- acters, is (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='14) Zall 4 (s) = � 0̸=(β)⊆Z[i] (β,2β)=1 β squarefree 1 N(β)s � q2∈Z≥1 (q2,2N(β))=1 1 qs 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS 15 A calculation with Euler products shows Zall 4 (s) = ζQ[i](s)ζ(s)A(s), where A(s) is given by an absolutely convergent Euler product for Re(s) > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Since Zall 4 (s) has a double pole at s = 1, this shows the claim, using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Sextic characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Next we turn to the sextic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The proof of the following proposition is similar to the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='6 and so we omit it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' For some Ktot 6 > 0, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='15) |Ψtot 6 (X)| ∼ Ktot 6 X, and |Ψ′ 6(X)| ∼ X log X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' A primitive totally sextic character factors uniquely as a primitive cubic character (with odd conductor, since 2 ̸≡ 1 (mod 3)), times the Jacobi symbol of the same modulus as the cubic character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In general, a primitive sextic character factors uniquely as χ6χ3χ2 of modulus q6q3q2, pairwise coprime, with χ6 totally sextic of conductor q6, χ3 cubic of conductor q3, and χ2 quadratic of conductor q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' We have |Ψ6(X)| = K6X(log X)2+O(X log X), for some K6 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Write χ = χ6χ3χ2 as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Note that membership in Ψ6(X) requires q6 > 1, which is an unpleasant condition when working with Euler products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' However, the number of χ = χ3χ2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=', with χ6 = 1, is O(X log X), so we may drop the condition q6 > 1 when estimating |Ψ6(X)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' For simplicity, we count the characters with q2 odd and (q6q3, 3) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' the general case follows similar lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The Dirichlet series for this counting function is Zall 6 (s) = � 0̸=(β6)⊆Z[ω] (β6,3β6)=1 β6 squarefree 1 N(β6)s � 0̸=(β3)⊆Z[ω] (β3,3β3)=1 β3 squarefree (N(β3),N(β6)=1 1 N(β3)s � q2∈Z≥1 (q2,2N(β3β6))=1 1 qs 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' A calculation with Euler products shows Zall 6 (s) = ζQ[ω](s)2ζ(s)A(s), where A(s) is given by an absolutely convergent Euler product for Re(s) > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Since Zall 6 (s) has a triple pole at s = 1, this shows the claim, using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' □ 16 JENNIFER BERG, NATHAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' RYAN, AND MATTHEW P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' YOUNG 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Equidistribution of Gauss sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' We first focus on the quartic case, and then turn to the sextic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Quartic characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The following standard formula can be found as [IK04, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='16)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Suppose that χ = χ1χ2 has conductor q = q1q2, with (q1, q2) = 1, and χi of conductor qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='16) τ(χ1χ2) = χ2(q1)χ1(q2)τ(χ1)τ(χ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Let notation be as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Suppose that χ is totally quartic and q is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='17) τ(χ1χ2)2 = τ(χ1)2τ(χ2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='10, we will obtain the formula provided χ2 2(q1)χ2 1(q2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Note that χ2 i is the Jacobi symbol, so χ2 2(q1)χ2 1(q2) = ( q1 q2)( q2 q1) = 1, by quadratic reciprocity, using that q1 ≡ q2 ≡ 1 (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Suppose π ∈ Z[i] is a primary prime, with N(π) = p ≡ 1 (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Let χπ(x) = [ x π] be the quartic residue symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='18) τ(χπ)2 = −χπ(−1)√pπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' More generally, if β is primary, squarefree, with (β, 2β) = 1, then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='19) τ(χβ)2 = µ(β)χβ(−1) � N(β)β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The formula for χπ follows from [IR90, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1 (Chapter 8), Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The formula for general β follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='11 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Suppose that χ = χ2χ4 is a primitive quartic character with odd conductor q, with χ2 quadratic of conductor q2, χ4 totally quartic of conductor q4, and with q2q4 = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='20) τ(χ)2 = �−q4 q2 � q2τ(χβ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='10, we have τ(χ)2 = χ2(q4)2χ4(q2)2τ(χ2)2τ(χ4)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' To simplify this, note χ2(q4)2 = 1, χ2 4(q2) = (q2 q4) = (q4 q2), and τ(χ2)2 = ϵ2 q2q2 = ( −1 q2 )q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' □ VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS 17 Our next goal is to express τ(χβ)2 in terms of a Hecke Grossencharacter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Define (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='21) λ∞(α) = α |α|, α ∈ Z[i], α ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Next define a particular character λ1+i : R× → S1, where R = Z[i]/(1+ i)3, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='22) λ1+i(ik) = i−k, k ∈ {0, 1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' This indeed defines a character since R× ≃ Z/4Z, generated by i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' For α ∈ Z[i], (α, 1 + i) = 1, define (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='23) λ((α)) = λ1+i(α)λ∞(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' For this to be well-defined, we need that the right hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='23) is constant on units in Z[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' This is easily seen, since λ∞(ik) = ik = λ1+i(ik)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Therefore, λ defines a Hecke Grossencharacter, as in [IK04, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Moreover, we note that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='24) τ(χβ)2 N(β) = µ(β) � 2 N(β) � λ((β)) since this agrees with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='19) for β primary, and is constant on units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' According to [IK04, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='8], the Dirichlet series (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='25) L(s, λk) = � 0̸=(β)⊆Z[i] λ((β))k N(β)s , (k ∈ Z), defines an L-function having analytic continuation to s ∈ C with no poles except for k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The same statement holds when twisting λk by a finite-order character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' For k ∈ Z, define the Dirichlet series (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='26) Z(k, s) = � 0̸=(β)⊆Z[i] (β,2β)=1 β squarefree (τ(χβ)2/N(β))k N(β)s , Re(s) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Let δk = −1 for k odd, and δk = +1 for k even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' We have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='27) Z(k, s) = A(k, s)L(s, (λ · χ2)k)δk, where χ2(β) = � 2 N(β) � , and where A(k, s) is given by an Euler product absolutely convergent for Re(s) > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' 18 JENNIFER BERG, NATHAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' RYAN, AND MATTHEW P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' YOUNG In particular, the zero free region (as in [IK04, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='35]) implies that Z(k, s) is analytic in a region of the type postulated in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Moreover, the proof of [MV07, Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='4] shows that Z(k, s) is bounded polynomially in log(2 + |t|) in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='24) shows that Z(k, s) has an Euler product of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='28) Z(k, s) = � (π)̸=(π) (1 + (−1)k χk 2(π)λk((π)) N(π)s ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' This is an Euler product over the split primes in Z[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' We extend this to include the primes p ≡ 3 (mod 4) as well, with N(π) = p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' It is convenient to define χ2(1 + i) = 0, so we can freely extend the product to include the ramified prime 1 + i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In all, we get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='29) Z(k, s) = � � p (1 − χk 2(p)λk(p) N(p)s ) �−δk � p (1 + O(p−2s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Note the product over p is L(s, (λ · χ2)k)δk, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' □ According to Weyl’s equidistribution criterion [IK04, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1], a se- quence of real numbers θn, 1 ≤ n ≤ N is equidistributed modulo 1 if and only if � n≤N e(kθn) = o(N) for each integer k ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' We apply this to e(θn) = (τ(χ)2/q), whence e(kθn) = (τ(χ)2/q)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Due to the twisted multiplicativity formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='16), the congruence class in which 2k lies modulo ℓ may have a simplifying effect on τ(χ)2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' For instance, when ℓ = 4, then k even leads to a simpler formula than k odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' This motivates treating these cases separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' As a minor simplification, below we focus on the sub-family of characters of odd conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The even conductor case is only a bit different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The Gauss sums τ(χ)2/q for χ totally quartic of odd conductor q, equidistribute on the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The complex numbers τ(χ)2/q lie on the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Weyl’s equidistribution criterion says that these normalized squared Gauss sums equidistribute on the unit circle provided (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='30) � 0̸=(β)⊆Z[i] (β,2β)=1 β squarefree N(β)≤X (τ(χβ)2/N(β))k = o(X), VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS 19 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' This histogram represents the distribution of the argument of the τ(χ)2/cond(χ) for totally quartic characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Each histogram is made by calculating the Gauss sums of characters of each order up to prime and composite conductor 300000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' for each nonzero integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In turn, this bound is implied by Propo- sition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='14, using the zero-free region for the Hecke Grossencharacter L-functions in [IK04, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' □ To contrast this, we will show that the normalized Gauss sums τ(χ)2/q, with χ ranging over all quartic characters, equidistribute slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' More precisely, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Let k ∈ 2Z, k ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' There exists ck ∈ C such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='31) � q≤X (q,2)=1 � χ:χ4=1 cond(χ)=q (τ(χ)2/q)k = ckX + o(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Recall from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='7 that the total number of such characters grows like X log X, so Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='16 shows that the rate of equidistribution is only O((log X)−1) here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In contrast, in the family of totally quartic characters, the GRH would imply a rate of equidis- tribution of the form O(X−1/2+ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' This difference in rates of equidis- tribution is supported by Figure 2 in which we see that the arguments 5+00 4000 DODE DOZ 1000 E- 1 020 JENNIFER BERG, NATHAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' RYAN, AND MATTHEW P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' YOUNG of squares of the Gauss sums of totally quartic characters quickly con- verge to being uniformly distributed, as compared to the Gauss sums of all quartic characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In addition, one can derive a similar result when restricting to χ ∈ Ψ4(X), simply by subtracting off the contribution from the quadratic characters alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' As in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='13, write χ = χ2χ4, with χ2 quadratic and χ4 totally quartic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Then τ(χ)4/(q1q2)2 = τ(χ4)4/q2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The analog of Z(k, s), using k even to simplify, is (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='32) Zall(k, s) = � 0̸=(β)⊆Z[i] (β,2β)=1 β squarefree τ(χβ)2k/N(β)k N(β)s � q2∈Z≥1 (q2,2N(β))=1 1 qs 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Referring to the calculation in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='14, we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='33) Zall(k, s) = ζ(s)L(s, λk)A(s), where A(s) is an Euler product absolutely convergent for Re(s) > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Since this generating function has a simple pole at s = 1, we deduce Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' □ As mentioned above, in order to deduce equidistribution, by Weyl’s equidistribution criterion, we also need to consider odd values of k in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' This is more technical than the case for even k, so we content ourselves with a conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' For each odd k, there exists δ > 0 such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='34) � q≤X (q,2)=1 � χ:χ4=1 cond(χ)=q (τ(χ)2/q)k ≪k,δ X1−δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='13 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='24), this problem reduces to understanding sums of the rough shape � β,q2 q2N(β)≤X �−N(β) q2 � µ(β) � 2 N(β) � λ((β))k, where we have omitted many of the conditions on β and q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In the range where q2 is very small, the GRH gives cancellation in the sum over β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Conversely, in the range where N(β) is very small, the GRH VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS 21 gives cancellation in the sum over q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' This discussion indicates that Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='18 follows from GRH, with any δ < 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Unconditionally, one can deduce some cancellation using the zero-free region for the β-sum (with q2 very small), and a subconvexity bound for the q2-sum (with N(β) very small).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In the range where both q2 and N(β) have some size, then Heath-Brown’s quadratic large sieve [HB95] gives some cancellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Since we logically do not need an unconditional proof of equidistribution, we omit the details for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='18 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='16 together imply that the squares of the quartic Gauss sums do equidistribute in the full family Ψ4(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Sextic characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Now we turn to the sextic Gauss sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Suppose that χ is totally sextic of conductor q, and say χ = χ2χ3 with χ2 quadratic and χ3 cubic, each of conductor q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Suppose χ3 = χβ, as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='35) τ(χ) = µ(q)χ3(2)τ(χ2)τ(χ3)βq−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' By [IK04, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='18)], τ(χ2)τ(χ3) = J(χ2, χ3)τ(χ), where J(χ2, χ3) is the Jacobi sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' It is easy to show using the Chinese remainder theorem that if χ2 = � p χ(p) 2 and χ3 = � p χ(p) 3 , then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='36) J(χ2, χ3) = � p J(χ(p) 2 , χ(p) 3 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The Jacobi sum for characters of prime conductor can be evaluated explicitly using the following facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' By [Lem00, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='30], (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='37) J(χ(p) 2 , χ(p) 3 ) = χ(p) 3 (22)J(χ(p) 3 , χ(p) 3 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Suppose that χ(p) 3 = χπ, where ππ = p, and π is primary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Then [IR90, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' 9, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' 1] implies J(χπ, χπ) = −π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' (Warning: they state the value π instead of −π, but recall their definition of primary is opposite our convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Also recall that χπ = χ−π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=') Gathering the formulas, we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='38) τ(χ2)τ(χ3) = τ(χ)χ3(2)2 � πi|β (−πi) = τ(χ)χ3(2)2µ(q)β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Rearranging this and using ββ = q completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' □ 22 JENNIFER BERG, NATHAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' RYAN, AND MATTHEW P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' YOUNG Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Let conditions be as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='39) τ(χ)2/q = χ3(4) �−1 q � τ(χβ)2β 2/q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Patterson [Pat78] showed that τ(χβ)/√q is uniformly distributed on the unit circle, as χβ ranges over primitive cubic characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The same method gives equidistribution after multiplication by a Hecke Grossen- character, and so similarly to the quartic case above, we deduce: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='23 (Patterson).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The Gauss sums τ(χ)2/q, for χ totally sextic of conductor q, equidistribute on the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In light of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='22, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='16, and Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='18, it seems reasonable to conjecture that the points τ(χ)2/q are equidis- tributed on the unit circle, as χ varies over all sextic characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' To see a limitation in the rate of equidistribution, it is convenient to con- sider τ(χ)6/q3, which is multiplicative for χ sextic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' For q ≡ 1 (mod 4), and χ = χ2 quadratic, we have τ(χ2)2/q = 1, so the quadratic part is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' For χ cubic and q ≡ 1 (mod 4), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='40) τ(χβ)6/q3 = µ(β)τ(χβ)3β 3 = q−1β 2, which is nearly a Hecke Grossencharacter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' A similar formula holds for χ totally sextic, namely (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='41) τ(χ)6/q3 = q−4β 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Therefore, carrying out the same steps as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='16 shows that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='42) � q≤X q≡1 (mod 4) � χ∈Ψ6 cond(χ)=q � τ(χ)6/q3�k = CkX + o(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' This is less of an obstruction than in the quartic case, since here the rate of equidistribution is O((log X)−2) instead of O((log X)−1), due to the fact that |Ψ6(X)| is approximately log X times as large as |Ψ4(X)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Similarly to the discussion of the quartic case in Remarks 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='19 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='20, we make the following conjecture without further explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The Gauss sums τ(χ)2/q, for χ ranging in Ψ6(X), equidistribute on the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS 23 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Estimates for quartic and sextic characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In order to ap- ply the random matrix theory conjectures, we need variants on Propo- sition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='6, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='7, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='8, and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='9, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' For primitive Dirichlet characters χ of order ℓ we have for ℓ = 4 and ℓ = 6 that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='43) � χ∈Ψℓ(X) 1 � cond(χ) ∼ 2Kℓ √ X(log X)d(ℓ)−2, and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='44) � χ∈Ψtot ℓ (X) 1 � cond(χ) ∼ 2Ktot ℓ √ X, � χ∈Ψ′ ℓ(X) 1 � cond(χ) ∼ 2 √ X log X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' These estimates follow from a straightforward application of partial summation or from a minor modification of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='5 since the generating Dirichlet series for one of these sums has its pole at s = 1/2 instead of at s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Random matrix theory: Conjectural asymptotic behavior This section closely follows the exposition of §3 of [DFK04] and §4 of [DFK07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Let U(N) be the set of unitary N×N matrices with complex coefficients which forms a probability space with respect to the Haar measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' For a family of L-functions with symmetry type U(N), Katz and Sar- nak conjectured that the statistics of the low-lying zeros should agree with those of the eigenangles of random matrices in U(N) [KS99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Let PA(λ) = det(A − λI) be the characteristic polynomial of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Keating and Snaith [KS00] suggest that the distribution of the values of the L- functions at the critical point is related to the value distribution of the characteristic polynomials |PA(1)| with respect to the Haar measure on U(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' For any s ∈ C we consider the moments MU(s, N) := � U(N) |PA(1)|s dHaar 24 JENNIFER BERG, NATHAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' RYAN, AND MATTHEW P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' YOUNG for the distribution of |PA(1)| in U(N) with respect to the Haar mea- sure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In [KS00], Keating and Snaith proved that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1) MU(s, N) = N � j=1 Γ(j)Γ(j + s) Γ2(j + s/2) , so that MU(s, N) is analytic for Re(s) > −1 and has meromorphic continuation to the whole complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The probability density of |PA(1)| is given by the Mellin transform pU(x, N) = 1 2πi � Re(s)=c MU(s, N)x−s−1 ds, for some c > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In the applications to the vanishing of twisted L-functions we consider in this paper, we are only interested in small values of x where the value of pU(x, N) is determined by the first pole of MU(s, N) at s = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' More precisely, for x ≤ N −1/2, one can show that pU(x, N) ∼ G2(1/2)N 1/4 as N → ∞, where G(z) is the Barnes G-function with special value [Bar99] G(1/2) = exp �3 2ζ′(−1) − 1 4 log π + 1 24 log 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' We will now consider the moments for the special values of twists of L-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' We then define, for any s ∈ C, the following sum of evaluations at s = 1 of L-functions primitive order ℓ characters of conductor less than X: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='2) ME(s, X) = 1 #FΨℓ,E(X) � L(E,s,χ)∈FΨℓ,E(X) |L(E, 1, χ)|s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Then, since the families of twists of order ℓ are expected to have unitary symmetry, we have Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1 (Keating and Snaith Conjecture for twists of order ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' With the notation as above, ME(s, X) ∼ aE(s/2)MU(s, N) as N = 2 log X → ∞, where aE(s/2) is an arithmetic factor depending only on the curve E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS 25 From Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1, the probability density for the distribution of the special values |L(E, 1, χ)| for characters of order ℓ is pE(x, X) = 1 2πi � Re(s)=c ME(s, X)x−s−1 ds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='3) ∼ 1 2πi � Re(s)=c aE(s/2)MU(s, N)x−s−1 ds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='4) as N = 2 log X → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' As above, when x ≤ N −1/2, the value of pE(x, X) is determined by the residue of MU(s, N) at s = −1, thus it follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='4) that for x ≤ (2 log X)−1/2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='5) pE(x, X) ∼ 21/4aE(−1/2)G2(1/2) log1/4(X) as X → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' We now use the probability density of the random matrix model with the properties of the integers nE(χ) to obtain conjectures for the van- ishing of the L-values |L(E, 1, χ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' When χ is either quartic or sextic, the discretization nE(χ) is a rational integer since Z[ζℓ] ∩ R = Z when ℓ = 4 or 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Let χ be a primitive Dirichlet character of order ℓ = 4 or 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Then |L(E, 1, χ)| = cE,ℓ � cond(χ) |nE(χ)|, where cE,ℓ is a nonzero constant which depends only on the curve E and ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' By rearranging equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='2) we obtain |L(E, 1, χ)| = ���� Ωϵ(E) τ(χ) kE nE(χ) cond(χ) ���� = |Ωϵ(E) kE nE(χ)| � cond(χ) = cE,ℓ|nE(χ)| � cond(χ) , where the nonzero constant kE is that of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' □ We write (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='6) Prob{|L(E, 1, χ)| = 0} = Prob{|L(E, 1, χ)| < B(cond(χ))}, for some function B(cond(χ)) of the character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='2 we may take B(cond(χ)) = cE,ℓ � cond(χ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Note that since cE,ℓ ̸= 0, if |nE(χ)|cE,ℓ � cond(χ) < cE,ℓ � cond(χ) , then |nE(χ)| < 1 and hence must vanish since |nE(χ)| ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' 26 JENNIFER BERG, NATHAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' RYAN, AND MATTHEW P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' YOUNG Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='5), we have Prob{|L(E, 1, χ)| = 0} = � B(cond(χ)) 0 21/4aE(−1/2)G2(1/2) log1/4(X) dx = 21/4aE(−1/2)G2(1/2) log1/4(X)B(cond(χ)) Summing the probabilities gives |VΨℓ,E(X)| = 21/4cE,kaE(−1/2)G2(1/2) log1/4(X) � cond(χ)≤X 1 � cond(χ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Thus, by the analysis in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='3, we have |VΨ4,E(X)| ∼ 25/4cE,4K4aE(−1/2)G2(1/2) log1/4(X) √ X log X ∼ bE,4X1/2 log5/4 X and |VΨ6,E(X)| ∼ 25/4cE,6K6aE(−1/2)G2(1/2) log1/4(X) √ X(log X)2 ∼ bE,6X1/2 log9/4 X as X → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Moreover, if we restrict to those characters that are totally quartic or sextic, we get the following estimates |VΨtot 4 ,E(X)| ∼ 25/4cE,4Ktot 4 aE(−1/2)G2(1/2) log1/4(X) √ X ∼ btot E,4X1/2 log1/4 X and |VΨtot 6 ,E(X)| ∼ 25/4cE,6Ktot 6 aE(−1/2)G2(1/2) ∼ btot E,6X1/2 log1/4 X as X → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Finally, if we restrict only to those twists by characters of prime con- ductor, we conclude |VΨ′ 4,E(X)| ∼ 25/4cE,4aE(−1/2)G2(1/2) log1/4(X) √ X log X ∼ b′ E,4X1/2 log−3/4 X VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS 27 and |VΨ′ 6,E(X)| ∼ 25/4cE,6aE(−1/2)G2(1/2) log1/4(X) √ X log X ∼ b′ E,6X1/2 log−3/4 X as X → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Here we provide numerical evidence for Conjec- ture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The computations of the Conrey labels for the characters were done in SageMath [Sag21] and the computations of the L-functions were done in PARI/GP [PAR22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The L-function computations were done in a distributed way on the Open Science Grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' For each curve, we generated a PARI/GP script to calculate a twisted L-function for each primitive character of order 4 and 6, and then combined the results into one file at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The combined wall time of all the computations was more than 50 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The code and data are available at [BR23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In Figure 3 we plot the points (X, X1/2 log5/4 X |VΨ4,11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1(X)|), (X, X1/2 log−3/4 X |VΨ′ 4,11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1(X)| ), (X, X1/2 log1/4 X |VΨtot 4 ,11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1(X)|) that provides a comparison between the predicted vanishings of L(E, 1, χ) for quartic characters and for the curve 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In Figure 4 we plot the analogous points for the same curve but for sextic twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In Figure 5 we plot the points (X, X1/2 log−3/4 X |VΨ′ 4,37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1(X)| ), (X, X1/2 log−3/4 X |VΨ′ 6,37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1(X)| ) Even though we are most interested in the families of all quartic and sextic twists, we include the families of twists of prime conductor be- cause there are far fewer such characters and so we can calculate the number of vanishings up to a much larger X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' We include the fami- lies of twists by totally quartic and sextic characters to highlight the transition between the family of prime conductors and the family of all conductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' References [Bar99] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Barnes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The theory of the G-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Quart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=', 31:264–314, 1899.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' [BCDT01] Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' On the modularity of elliptic curves over Q: wild 3-adic exercises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Journal of the American Mathematical Society, pages 843–939, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' [BE81] Bruce C Berndt and Ronald J Evans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The determination of Gauss sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Bulletin of the American Mathematical Society, 5(2):107–129, 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' 28 JENNIFER BERG, NATHAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' RYAN, AND MATTHEW P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' YOUNG (a) The ratio of predicted vanishings to empirical van- ishings of twists of the curve 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1 by quartic characters of conductor ≤ 700000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' (b) The ratio of pre- dicted vanishings to empirical vanishings of twists of the curve 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1 by quartic characters of prime conductor ≤ 2000000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' (c) The ratio of pre- dicted vanishings to empirical vanishings of twists of the curve 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1 by totally quartic characters of conductor ≤ 700000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Verification of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1 for quartic twists of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' (a) The ratio of pre- dicted vanishings to empirical vanishings of twists of the curve 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1 by sextic char- acters of conductor ≤ 300000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' (b) The ratio of pre- dicted vanishings to empirical vanishings of twists of the curve 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1 by sextic char- acters of prime con- ductor ≤ 2000000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' (c) The ratio of pre- dicted vanishings to empirical vanishings of twists of the curve 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1 by totally sex- tic characters of con- ductor ≤ 300000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Verification of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1 for sextic twists of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' [BR23] Jen Berg and Nathan C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Ryan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Code and data for quartic and sextic twists of elliptic curve L-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' http://eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='bucknell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='edu/~ncr006/ quartic-sextic-twists-website/, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' [BY10] Stephan Baier and Matthew P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Young.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Mean values with cubic characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Number Theory, 130(4):879–903, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' [CFK+05] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Brian Conrey, David W Farmer, Jon P Keating, Michael O Rubin- stein, and Nina C Snaith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Integral moments of L-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Proceedings of the London Mathematical Society, 91(1):33–104, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' [Cho87] Sarvadaman Chowla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The Riemann hypothesis and Hilbert’s tenth prob- lem, volume 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' CRC Press, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content="36 tE'O 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='D0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='75 LDo 125 150 175 20o 1e614 12 LD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='6 0 11 9 - 8 - 1 61 50000 150000 DO 250000 3000000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='40 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='D0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='75 Lio 125 150 175 200 1e616 15 14 13 12 11 LD 0 DODS 10dC0 150000 240000 25000 3+0dC04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='D 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='5- 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='0 25 - 20 15 LD VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS 29 (a) The ratio of pre- dicted vanishings to empirical vanishings of twists of the curve 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1 by quartic characters of prime conductor ≤ 2000000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' (b) The ratio of pre- dicted vanishings to empirical vanishings of twists of the curve 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1 by sextic char- acters of prime con- ductor ≤ 2000000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Verification of parts of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1 for twists of 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' [CKRS00] JB Conrey, JP Keating, MO Rubinstein, and NC Snaith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' On the fre- quency of vanishing of quadratic twists of modular L-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In Pro- ceedings of the Millennial Conference on Number Theory, Urbana, Illinois, 21-26 May, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' AK Peters, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' [DFK04] Chantal David, Jack Fearnley, and Hershy Kisilevsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' On the vanishing of twisted L-functions of elliptic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=', 13(2):185–198, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' [DFK07] Chantal David, Jack Fearnley, and Hershy Kisilevsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Vanishing of L- functions of elliptic curves over number fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' In Ranks of elliptic curves and random matrix theory, volume 341 of London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Lecture Note Ser.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' The second moment of quadratic twists of modular L-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' (JEMS), 12(5):1097–1116, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' [TW95] Richard Taylor and Andrew Wiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Ring-theoretic properties of certain Hecke algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Annals of Mathematics, 141(3):553–572, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' [Wil95] Andrew Wiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Modular elliptic curves and Fermat’s last theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Annals of mathematics, 141(3):443–551, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' [WW20] Hanneke Wiersema and Christian Wuthrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Integrality of twisted l-values of elliptic curves, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content=' Email address: jsb047@bucknell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='edu Email address: nathan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='ryan@bucknell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='edu Department of Mathematics, Bucknell University, Lewisburg, PA 17837 Email address: myoung@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='tamu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} +page_content='edu VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS 31 Department of Mathematics, Texas A&M University, College Station, TX 77843-3368' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'} diff --git a/EdE2T4oBgHgl3EQfSgfT/content/2301.03794v1.pdf b/EdE2T4oBgHgl3EQfSgfT/content/2301.03794v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..3d7085ba7d33d801e0c39913149160eb976bf77b --- /dev/null +++ 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a/HNE4T4oBgHgl3EQfHwxv/content/tmp_files/2301.04906v1.pdf.txt b/HNE4T4oBgHgl3EQfHwxv/content/tmp_files/2301.04906v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5184042dc45ec8c34e3f6fcc1603ebdf79173ab6 --- /dev/null +++ b/HNE4T4oBgHgl3EQfHwxv/content/tmp_files/2301.04906v1.pdf.txt @@ -0,0 +1,1679 @@ +Practical challenges in data-driven +interpolation: dealing with noise, enforcing +stability, and computing realizations +Quirin Aumann∗ +Ion Victor Gosea† +∗Max Planck Instiute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg, +Germany. +Email: aumann@mpi-magdeburg.mpg.de, ORCID: 0000-0001-7942-5703 +†Max Planck Instiute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg, +Germany. +Email: gosea@mpi-magdeburg.mpg.de, ORCID: 0000-0003-3580-4116 +Abstract: In this contribution, we propose a detailed study of interpolation-based data- +driven methods that are of relevance in the model reduction and also in the systems and +control communities. The data are given by samples of the transfer function of the under- +lying (unknown) model, i.e., we analyze frequency-response data. We also propose novel +approaches that combine some of the main attributes of the established methods, for ad- +dressing particular issues. +This includes placing poles and hence, enforcing stability of +reduced-order models, robustness to noisy or perturbed data, and switching from different +rational function representations. We mention here the classical state-space format and +also various barycentric representations of the fitted rational interpolants. We show that +the newly-developed approaches yield, in some cases, superior numerical results, when com- +paring to the established methods. The numerical results include a thorough analysis of +various aspects related to approximation errors, choice of interpolation points, or placing +dominant poles, which are tested on some benchmark models and data-sets. +Keywords: Data-driven methods, rational approximation, interpolatory methods, least +squares fit, Loewner framework, frequency response data, pole placement, noisy measure- +ments, Loewner and Cauchy matrices. +Novelty statement: This note shows that by combining the features of established data- +driven rational approximation methods based on interpolation (and/or least squares fit), +one can devise methods that offer additional important advantages. These include stabil- +ity enforcement by placing poles in an elegant and numerically stable manner, together +with robustness to noisy data. +1. Introduction +Approximation of large-scale dynamical systems is pivotal for serving the scopes of efficient simulation +and designing control laws in real-time. The technique for reducing the complexity of a system is known +as model order reduction (MOR) [1, 5, 13, 14]. There exist a number of methodologies for reducing +large-scale models, and each method is tailored to some specific applications (mostly, but not restricted +Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg). +2023-01-13 +arXiv:2301.04906v1 [math.NA] 12 Jan 2023 + +Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions +2 +to mechanical and electrical engineering) and to achieving certain goals (stability, passivity or structure +preservation), on top of the complexity reduction part. Data-driven MOR approaches are of particular +importance when access to high-fidelity models is not explicitly granted. This means that a state- +space formulation with access to internal variables is not available, yet input/output data are. Such +methods circumvent the need to access an exact description of the original model and are applicable +whenever classical projection-based MOR is not. Here, we mention system and control methodologies +that are based on interpolation or least-square fit of data (i.e., frequency response measurements), +such as vector fitting [31], the Loewner framework [41], or the AAA algorithm [42]. Methods that use +time-domain data are also of interest, including the ones that require input-output data together with +those which use snapshot data (access to the state evolution), such as the classical ones in [36,58,59], +followed by [8], [47] or [54]. +We focus on interpolation-based or so-called moment matching (MM) methods which have emerged, +were developed, and improved continuously in the last decades. The backbone of such methods rep- +resent rational Krylov-type approaches together with the Sylvester matrix equation interpretation +[1, 11, 20]. Apart from being computationally efficient and easy to implement, MM approaches have +another advantage: they do not (necessarily) require access to a full-state realization of the original +dynamical system. Hence, they can be viewed as data-driven methods. Here data are given by the +moments of the system, i.e., samples of the underlying transfer function of the system (and of its +derivatives) evaluated in a particular frequency range; for more details, we refer the readers to [5,8,41] +and to Chapter 3 in [13]. The notion of a moment with respect to systems and control theory is related +to the unique solution of a Sylvester matrix equation [20]. +The purpose of this note is twofold; first, we intend to review and to connect three important system +theoretical model reduction methods based on interpolation that were introduced in the last 15 years: +• The Loewner framework (LF) by Mayo and Antoulas from 2007 in [41]; +• The Astolfi framework (AF) from 2010 in [8]; +• The Adaptive Antoulas Anderson (AAA) algorithm by Nakatsukasa, Set´e and Trefethen from +2018 in [42]. +Together, these three approaches were cited multiple times in various research publications, being +arguably quite popular methods. However, until now, not too many connections between them were +provided, neither in the automatic control, nor in the model reduction, or numerical analysis commu- +nities. Together with the vector fitting algorithm (VF) in [31] (which is not based on interpolation, +and is hence a purely optimization approach based on least-squares fitting), these methods repre- +sent arguably the most prolific rational approximation schemes developed in the system and control +community. However, VF is not the object of this study since it is not based on interpolation. +The other scope of this note is to propose a new method that is based on the three methods +enumerated above, and that addresses some of the shortcomings and challenges associated with them. +Basically, the idea is to combine the attributes of each method, by following the steps below. +• We make use of the order-revealing property of the LF (encoded by the rank of augmented +Loewner matrices); additionally, the selection of interpolation points is done via a Loewner-CUR +technique proposed in [38]. +• We utilize the elegant state-space parameterization of the LTI system proposed by the AF (after +imposing k interpolation conditions); this is the backbone of the methods (we also show the +connection between state-space forms and barycentric forms). +• We use either the fitting step from AAA (that chooses free parameters to fit the un-interpolated +data in a least square sense) or we impose pole placing (dominant poles are selected from those +of the Loewner model); in both cases, a linear system of equations needs to be solved. +Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg). +2023-01-13 + +Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions +3 +In what follows, we consider a multiple-input multiple-output (MIMO) linear time-invariant (LTI) +system ΣL of dimension n described by the following system of differential equations: +ΣL : +� +˙x(t) = Ax(t) + Bu(t), +y(t) = Cx(t), +(1) +with x(t) ∈ Rn as the state variable, u(t) ∈ Rm as the control inputs, and y(t) ∈ Rp as the observed +outputs. Here, we have that A ∈ Cn×n, B ∈ Cn×m and C ∈ Cp×n. The transfer (matrix) function of +the LTI system is given by H(s) ∈ Cp×n, with s ∈ C, as +H(s) = C(sIn − A)−1B. +(2) +It is to be noted that, for m = p = 1, the system becomes single-input single-output (SISO). We will +sometime switch between MIMO and SISO formats while presenting the methods covered in this note, +since the latter allows a more easy exposition for some of the results shown here. +Let si ∈ C \ σ(A), where σ(A) denotes the spectrum of matrix A ∈ Cn×n, i.e., the set of its +eigenvalues. +The j-moment of system ΣL at si is given by ηj(si) = +(−1)j +j! +� +dj +dsj H(s) +� +s=si, for any +integer j ⩾ 1. The 0-moment is obtained by sampling the transfer function H(s) in (2) at si, i.e., +η0 = H(si). In this contribution, we restrict the analysis to matching 0-moments, i.e., samples of +the transfer function H(s), and not of its derivatives. However, all methodologies shown here can be +expected to cope with this as well. Moreover, in practice, inferring 0-moments from time-domain data +is usually a more straight-forward task; this is performed by exciting the system with harmonic inputs, +and by applying spectral transformations to the outputs. Additionally, the inference of derivative +values (of the transfer function) is typically susceptible to perturbations and it is more challenging to +attain, from a numerical point of view. +The paper is structured in the following way; after the introduction session sets up the stage, we +propose a survey of three established interpolation-based methods in Section 2. Then, the proposed +methodologies are developed in Section 3, with emphasis on the one-step approach that combines +optimal selection of interpolation points (chosen using CUR-DEIM) with LS fit on the rest of the +data, and also the pole placement method in barycentric form that enforces dominant poles from the +Loewner data-driven model. Then, Section 4 illustrates the numerical aspects of applying the methods +discussed/proposed in the previous two sections to a variety of test cases (various models and data +sets). Finally, Section 5 presents the conclusions and the outlook into future research. +2. A survey of established methods +In this section we discuss three established data-driven methods for rational approximation (AF, LF, +and AAA, as mentioned in the previous section). +The data are samples of the transfer function +corresponding to the underlying dynamical system, measured on a particular frequency grid. In what +follows, we mention some state-of-the-art methodologies used to measure such data, i.e., frequency +response data. Typically, such measurements can be produced in practice from experiments conducted +in scientific laboratories using carefully calibrated machines, called spectrum analyzers (SAs). In this +category we mention swept-tuned spectrum analyzers, scalar network analyzers (SNAs), and vector +network analyzers (VNAs). +The SNA is an instrument that measures microwave signals by converting them to a DC voltage +using a diode detector. In a VNA, information regarding both the magnitude and the phase of a +microwave signal is extracted. +While there are different ways to perform such measurements, the +method employed by commercial products (such as the Anritsu series described in [18]) of VNAs is +to down-convert the signal to a lower intermediate frequency in a process called harmonic sampling. +This signal can then be measured directly by a tuned receiver. Compared to the SNA, the VNA is +a more powerful analyzer tool. The major difference is that the VNA can also measure the phase, +and not only the amplitude. With this property enforced, then so-called scattering parameters (or +Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg). +2023-01-13 + +Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions +4 +S-parameters) can be processed. These can be used for identifying forward and reverse transmission +and reflection characteristics. More details can be found in [18]. +The harmonic balance method (or HBM) [43], is an established methodology in the field of elec- +tromagnetics. The HBM is used in many (if not most) commercial radio-frequency (RF) simulation +tools. This is due to the fact that it has certain advantages over other common methods, namely +modified nodal analysis (MNA), which makes it more appropriate to use for stiff problems and circuits +containing transmission lines, nonlinearities and dispersive effects. More details can be found in the +survey paper [49]. +2.1. The one-sided moment-matching approach in [8] +The framework introduced by Astolfi in [8] (referred to as AF throughout the paper) deals with the +problem of model reduction by moment matching. Although classically interpreted as a problem of +interpolation of points in the complex plane, it has instead been recast as a problem of interpolation +of steady-state responses. In the following we briefly review its application to linear systems. It is to +be noted that the AF was steadily extended and applied to different scenarios (including nonlinear +dynamical systems, pole-zero placement, and least-squares fit) [33–35,45,53,55]. +The moments of a linear system can be characterized in terms of the solution of Sylvester equations. +By using this observation, it has been shown that the moments are in one-to-one relation with the +steady-state output response of the interconnection between a signal generator and the original linear +system. +In what follows, for simplicity of exposition, it is assumed that ΣL is a minimal system (both fully +controllable and fully observable). For exact definitions on minimality, controllability, or observability +of LTI systems, we refer the reader to [1]. +Let k ⩽ n and S ∈ Ck×k a non-derogatory matrix (for which the characteristic and minimal +polynomials coincide) with σ(S) ∩ σ(A) = ∅ and R ∈ C1×k so that (S, R) is observable. Consider the +signal generator system Σsg described by the equations +Σsg : +� +˙ω(t) = Sω(t), +u(t) = Rω(t). +(3) +Then, the explicit solution of (3) can be written as ω(t) = eStω(0). Hence, the control input is written +as u(t) = ReStω(0). In addition, the eigenvalues of S are called interpolation points. +For a linear system ΣL, and interpolation points si ∈ C \ σ(A), for i = 1, . . . , k, consider a non- +derogatory matrix S ∈ Rk×k. It follows that there exists a one-to-one relation between the moments +of the system ΣL and +1. the matrix CΠ, where Π is the (unique) solution of the Sylvester equation AΠ + BR = ΠS, +for any row vector R ∈ R1×k so that (R, S) is observable; +2. the steady-state response of the output y of the interconnection of system ΣL and the system +Σsg, for any R and ω(0) such that the triplet (R, S, ω(0)) is minimal. +More precisely, let ∆ ∈ Rk be a column vector containing k free parameters (denoted here by +δ1, δ2, . . . , δk, with δi ̸= 0, 1 ≤ i ≤ k). Then, as stated in [8], the family of linear time-invariant +systems that interpolates the moments of system ΣL at the eigenvalues of matrix S, is given by +ˆΣ∆ : +� ˙ˆx(t) = (S − ∆R) +� +�� +� += ˆA +ˆx(t) + ∆ +���� += ˆB +u(t), +ˆy(t) = CΠ +���� += ˆC +ˆx(t), +(4) +where the matrices S and R are as before and Π is the unique solution of the Sylvester equation +AΠ + BR = ΠS. Additionally, the condition σ(S) ∩ σ(S − ∆R) = ∅ needs to be enforced. It is +to be noted that the free parameters explicitly enter the vector ˆB = ∆, but also the matrix ˆA, as +ˆA = S − ∆R. Finally, ˆC = CΠ has fixed entries. +Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg). +2023-01-13 + +Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions +5 +The Sylvester matrix equation for the reduced-order system is written as ˆA ˆΠ+ ˆBR = ˆΠS. This can +be explained by the fact that the reduced-order system matches the prescribed moments of the original +system, hence the same format of the two equations. Without loss of generality, one can consider +that the matrix ˆΠ is the identity matrix, i.e. +ˆΠ = Ik (this can be achieved by applying similarity +transformations). By replacing this value into the reduced-dimension Sylvester matrix equation above, +the formula ˆA = S − ∆R directly follows. +Afterwards, the free parameters collected in the vector ∆ can be chosen in order to enforce or impose +additional conditions as mentioned in [8], such as: matching with imposing additional k interpolation +conditions, matching with prescribed eigenvalues, matching with prescribed relative degree, matching +with prescribed zeros, matching with a passivity constraint, matching with L2-gain, or matching with +a compartmental constraint. +An important aspect of the AF is the characterization of all, i.e., infinitely many families of reduced- +order models that satisfy k prescribed interpolation conditions. This is done by explicitly computing +such parameterized models, for which the free parameters are the variables entering in the vector ∆. +The main parameterization developed here will be used as a “backbone” of the methods developed in +Section 3. +As stated in the original paper, the main advantage of the AF (characterization of moments in +terms of steady-state responses) is that it allows the definition of moments for systems which do not +admit a clear/immediate representation in terms of transfer function(s). Hence, the author provides +as examples the case of linear time-varying systems, and the case of nonlinear systems. Moreover, +it is stated in [8] that one disadvantage of the framework is that it requires the existence of steady- +state responses. Consequently, the original system has to be (exponentially) stable. However, in most +practical applications, this is a realistic requirement. +2.2. The Loewner framework in [41] +In this section we present a short summary of the Loewner framework (LF), as introduced in [41]. It +is to mentioned that LF has its roots in the earlier work of [4], and that LF can be considered to be a +double-sided moment-matching approach (as opposed to AF, which is one-sided). +For a tutorial paper on LF for LTI systems, we refer the reader to [7], and for a recent extension +that uses time-domain data, we refer the reader to [48]. The Loewner framework has been recently +extended to certain classes of nonlinear systems, such as bilinear systems in [6], and quadratic-bilinear +(QB) systems in [24], but also to linear parameter-varying systems in [28]. Additionally, issues such +as stability preservation or enforcement, or passivity preservation, were tackled in the LF in [23, 29], +for the former, and in [2,12], for the latter. +The LF is based on processing frequency-domain measurements D = {(ωℓ, H(ωℓ)) , ℓ = 1, . . . , N} +(with ωℓ ∈ R for 1 ≤ ℓ ≤ N) corresponding to evaluations of the transfer function of the underlying +(unknown/hidden) dynamical system. +The interpolation problem is formulated as shown below (for convenience of exposition, we show +here only the SISO formulation). We are given data nodes and data points in the set D, partitioned +into two disjoint subsets DL and DR, with DL ∪ DR = D and k + q = N, as +right data : DL = {(λj, H(λj)) , j = 1, . . . , k}, and, +left data : DR = {(µi, H(µi)) , i = 1, . . . , q}, +(5) +and we seek to find a rational function ˆH(s), such that the following interpolation conditions hold: +ˆH(µi) = H(µi) := vi, +ˆH(λj) = H(λj) := wj. +(6) +The Loewner matrix L ∈ Cq×k and the shifted Loewner matrix Ls ∈ Cq×k play an important role in +the LF, and are given by +L(i,j) = vi − wj +µi − λj +, +Ls(i,j) = µivi − λjwj +µi − λj +, +(7) +Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg). +2023-01-13 + +Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions +6 +while the data vectors V ∈ Cq, WT ∈ Ck are given by +V(i) = vi, +W(j) = wj, for i = 1, . . . , q, j = 1, . . . , k. +(8) +Moreover, the following Sylvester matrix equations ([1, Ch. 6]) are satisfied by the Loewner and shifted +Loewner matrices (here, 1q = +� +1 +· · · +1 +�T ∈ Cq) +� +ML − LΛ = V1T +k − 1qW, +MLs − LsΛ = MV1T +k − 1qWΛ, +(9) +where M = diag(µ1, . . . , µq) and Λ = diag(λ1, . . . , λk) are diagonal matrices. The following relation +holds true +Ls = LΛ + V1T +k = ML + 1qW. +(10) +The unprocessed Loewner surrogate model, provided that k = q, is composed of the matrices +ˆE = −L, +ˆA = −Ls, +ˆB = V, +ˆC = W, +(11) +and if the pencil (L, Ls) is regular, then the function ˆH(s) satisfying the interpolation conditions in +(6) can be explicitly computed in terms of the matrices in (11), as ˆH(s) = ˆC(sˆE − ˆA)−1 ˆB. +In practical applications (when processing a fairly large number of measurements), the pencil (Ls, L) +is often singular. Hence, a post-processing step is required for the Loewner model in (11). In such +cases, one needs to perform a singular value decomposition (SVD) of augmented Loewner matrices, to +extract the dominant features and remove inherent redundancies in the data. By doing so, projection +matrices X, Y ∈ Ck×r are obtained, as left, and respectively, right truncated singular vector matrices: +[L Ls] = YS(1) +r +˜X +H � L +Ls +� += ˜YS(2) +r XH, +(12) +where S(1) +r , S(2) +r +∈ Rr×r, +Y ∈ Ck×r, X ∈ Cq×r, ˜Y ∈ C2q×r, ˜X ∈ Cr×2k. The truncation index r +can be chosen as the numerical rank (based on a tolerance value τ > 0) or as the exact rank of the +Loewner pencil (in exact arithmetic), depending on the application and data size. More details can be +found in [7]. +The system matrices corresponding to a projected Loewner model of dimension r can be computed +as follows: +˜E = −XHLY, +˜A = −XHLsY, +˜B = XHV, +˜C = WY. +We note that MIMO extensions of the LF were already proposed in the original contribution [41]. +There, a tangential interpolation framework is considered. Instead of imposing interpolation of full +p × m blocks, the authors prefer to interpolate the original transfer matrix function samples along +certain vectors (or tangential directions). We also note that a first attempt of re-interpreting the LF +in [41] as a one-sided method was made in [25]. In the latter, the main difference to the classical work +in [4] was that a compression of the left (un-interpolated) data set was enforced. However, in [25], it +was still unclear how to split the data, i.e., what the right data set should be (where interpolation is +enforced). Finally, it is to be noted that the choice of interpolation points is crucial in the LF. An +exhaustive study of different choices was proposed in [37], while a greedy strategy was proposed in +[17], for scenarios in which limited experimental data are available. +2.3. The AAA algorithm in [42] +The AAA algorithm introduced in [42] is an adaptive and iterative extension of the interpolation +method based on Loewner matrices, originally proposed in [4]. The main steps are as follows +1. Express the fitted rational approximants in a barycentric representation, which represents a +numerically stable way of expressing rational functions [15]. +Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg). +2023-01-13 + +Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions +7 +Algorithm 1 The AAA algorithm. +Require: A (discrete) set of sample points Γ ⊂ C with N points, function f (or the evaluations of f +on the set Γ, i.e., the sample values), and an error tolerance ϵ > 0. +Ensure: A rational approximant rn(s) of order (n, n) displayed in a barycentric form. +1: Initialize j = 0, Γ(0) ← Γ, and r−1 ← N −1 �N +i=1 f(γi). +2: while |f(s) − rj−1(s)| > ϵ do +3: +Select a point zj ∈ Γ(j) for which |f(s) − rj−1(s)| attains a maximal value, where for j ≥ 1, it +follows: +rj−1(s) := +�j−1 +� +k=0 +ω(j−1) +k +s − zk +�−1 �j−1 +� +k=0 +ω(j−1) +k +fk +s − zk +� +. +(13) +4: +if |f(zj) − rj−1(zj)| ≤ ε then +5: +Return rj−1. +6: +else +7: +fj ← f(zj) and Γ(j+1) ← Γ(j) \ {zj}. +8: +end if +9: +Find the weights ω(j) = [ω(j) +0 , . . . , ω(j) +j ] by solving a least squares problem over z ∈ Γ(j+1) +j +� +k=0 +ω(j) +k +s − zk +f(s) ≈ +j +� +k=0 +ω(j) +k fk +s − zk +⇔ +� +j +� +k=0 +f(s) − fk +s − zk +� +ω(j) +k +≈ 0 ⇔ L(j)ω(j) = 0. +(14) +The solution of (14) is given by the (j + 1)th right singular vector of the Loewner matrix +L(j) ∈ C(N−j−1)×(j+1). +10: +j ← j + 1. +11: end while +2. Select the next interpolation (support) points via a greedy scheme; basically, interpolation is +enforced at the point where the (absolute or relative) error at the previous step is maximal. +3. Compute the other variables (the so-called barycentric weights) in order to enforce least squares +approximation on the non-interpolated data. +In recent years, the AAA algorithm has proven to be an accurate, fast, and reliable rational ap- +proximation tool with a fairly large range of applications. Here, we will mention only a few: nonlinear +eigenvalue problems [39], MOR of parameterized linear dynamical systems [16], MOR of linear sys- +tems with quadratic outputs [26], rational approximation of periodic functions [10], representation of +conformal maps [22], rational approximation of matrix-valued functions [27], or signal processing with +trigonometric rational functions [60]. The procedure is sketched in Algorithm 1. +It is to be mentioned that a modified version of AAA that enforces real-valued and strictly- +proper rational appoximants was recently proposed in [30]. There, the format of the function in (13) +was modified by inserting a 1 into the denominator, as follows +˜rj(s) := +� +1 + +j−1 +� +k=0 +ω(j−1) +k +s − zk +�−1 �j−1 +� +k=0 +ω(j−1) +k +fk +s − zk +� +. +(15) +Consequently, the equation in (14) becomes L(j)ω(j−1) = −f(j−1), where the vector f(j−1) ∈ Cj is +given by f(j−1) = +�f0 +f2 +· · · +fj−1 +�T. It is to be noted that ˜rj(s) in (15) is theoretically a rational +approximant of order (j − 1, j), if we do not take into account pole/zero cancellations or any other +zero cancellations of coefficients in the numerator or denominator. +Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg). +2023-01-13 + +Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions +8 +3. The proposed methodologies +3.1. Skeleton of the main methods +Similar to the methods reviewed in Section 2 we want to find an LTI system with a transfer function of +the structure (1) that interpolates data provided as measurements H (si) , i = 1, . . . , k of the transfer +function of the original system. We can directly put together an LTI parametrized model of dimension +r = km, having km2 degrees of freedom with transfer function +ˆH(s) = ˆC(sIr − ˆA)−1 ˆB, +(16) +with the underlying data concatenated to +ˆC = +�H(λ1) +· · · +H(λk)� +∈ Cp×r, +(17) +a matrix of weights ˆ +Wi +ˆB = +� +ˆ +W +H +1 +· · · +ˆ +W +H +k +�H +∈ Cr×m, +(18) +and ˆA ∈ Cr×r formed from a diagonal matrix populated with the interpolation points λi disturbed by +ˆB, such that +ˆA = Λ − ˆBR = diag (λ1, . . . , λk) ⊗ Im − ˆB +� +1T +r ⊗ Im +� +. +(19) +Making use of the Woodbury matrix identity and denoting Λs = sIkm − Λ, the transfer function (16) +can be rewritten as +ˆH(s) = ˆCΛ−1 +s +ˆB +� +Im + RΛ−1 +s +ˆB +�−1 +. +(20) +A complete derivation of (20) is given in Appendix A.1. +In the single-input single-output case (m = p = 1, hence r = k), the barycentric weights reduce to +scalars and the matrices for a ROM of structure (16) are given by +ˆA = Λ − ˆBR ∈ Ck×k, +ˆB = +� ˆw1 +· · · +ˆwk +�T ∈ Ck×1, +ˆC = +�H (λ1) +· · · +H (λk)� +∈ C1×k. +(21) +By inserting the formulae in (21) into (20), and using the notation hi := H (λi), leads to +ˆCΛ−1 +s +ˆB = +k +� +i=1 +ˆwihi +s − λi +, +� +Im + RΛ−1 +s +ˆB +�−1 += +1 +1 + �k +i=1 +ˆwi +s − λi +. +(22) +Hence, the transfer function of the model in (21) is given in barycentric representation by +ˆH(s) = +�k +i=1 +ˆwihi +s − λi +1 + �k +i=1 +ˆwi +s − λi +. +(23) +This can be performed analogously for a multi-input multi-output case (m = p, r = km). The first +part of (20) becomes +ˆCΛ−1 +s +ˆB = +�H(λ1)Im(s − λ1)−1 +· · · +H(λk)Im(s − λk)−1� +� +�� +ˆ +W1 +... +ˆ +Wk +� +�� = +k +� +i=1 +H(λi) ˆ +Wi +s − λi +, +(24) +Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg). +2023-01-13 + +Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions +9 +the second part +� +Im + RΛ−1 +s +ˆB +�−1 += +� +� +� +�Im + +�Im +· · · +Im +� +� +�� +Im(s − λ1)−1 +... +Im(s − λk)−1 +� +�� +−1 � +�� +ˆ +W1 +... +ˆ +Wk +� +�� +� +� +� +� +−1 += +� +Im + +k +� +i=1 +ˆ +Wi +s − λi +�−1 +. +(25) +Consequently, the transfer function in (20) has also a barycentric form given by +ˆH(s) = +� k +� +i=1 +H(λi) ˆ +Wi +s − λi +� � +Im + +k +� +i=1 +ˆ +Wi +s − λi +�−1 +. +(26) +The transfer function is defined by the choice of the interpolation points and of the weights. The +interpolation points can be chosen as dominant parts of the available data or based on their location in +the frequency spectrum. The weights can be computed such that the data which are not interpolated, +are approximated in an optimal way. Alternatively, the weights can be chosen to enforce poles at +specific locations. In the following, we show different strategies for both choices. +3.2. Automatic choice of interpolation points +The approximation quality of a surrogate model of the form (16) is greatly influenced by the choice of +the interpolation points λ. This choice is not always obvious, so automatic strategies are frequently +employed. The Loewner framework uses the SVD to identify dominant subsets of the available data +to enforce interpolation on. Alternatively, the AAA algorithm uses a greedy scheme to minimize the +error between surrogate and original data. Another approach, originally introduced by [37], makes use +of the CUR decomposition to extract interpolation points from a relevant subset of the available data. +The CUR decomposition approximates a matrix A by a product of three low-rank matrices ˇA = +ˇC ˇU ˇR, where ˇC and ˇR represent subsets of the columns respectively rows of A [40,56]. In our case the +three matrices are only a byproduct, we are more interested in the interpolation points λ and µ that +are associated to the columns and rows extracted as ˇC and ˇR. In combination with the skeleton for a +realization described in Section 3.1, Algorithm 2 computes a surrogate model approximating a set of +given transfer function data. We use the algorithm from [56] to compute the CUR decomposition and +thus identify dominant parts of the original data set and their corresponding left and right interpolation +points. Contrary to [37], we decompose the original Loewner matrix L rather than the augmented +Loewner matrices +�L +Ls +� +and +� +LH +LH +s +�H. Using all interpolation points obtained from the CUR +decomposition would introduce redundant data into the surrogate. Therefore we choose only a subset +of the interpolation points: either only the left points, only the right points, or every other entry from +a concatenated and sorted vector of left and right points. Together with the data associated to the +chosen interpolation points they are used to populate a rectangular Loewner matrix. We now need +to compute weights for barycentric interpolation as described in the following section. After having +obtained the weights, a surrogate model (16) can be computed from (17)–(19). +Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg). +2023-01-13 + +Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions +10 +Algorithm 2 LS-Loewner with CUR. +Require: Transfer function samples {H (si)}N +i=1, corresponding sampling points Ξ = {si}N +i=1. +Ensure: Surrogate model ˆH(s) = ˆC(sIr − ˆA)−1 ˆB. +1: Partition data and compute Loewner matrix L as in (7). +2: Compute CUR decomposition, such that L = ˇC ˇU ˇR with ˇC ∈ CN×k, ˇR ∈ Ck×N. +3: Obtain interpolation points {λi}k +i=1 , {µi}k +i=1 corresponding to the columns and rows in ˇC, ˇR. +4: Postprocess interpolation points to obtain ν = {ν}k +i=1 and χ = Ξ \ ν. +5: Populate a rectangular Loewner matrix L(i,j) = H(χi)−H(νj) +χi−νj +. +6: Compute the weights Ω = −L† � +H (ν1)H +· · · +H (νk)H�H +, where L† is the pseudo-inverse of L and +Ω = +� +ˆ +W +H +1 +· · · +ˆ +W +H +k +�H +. +7: Compute ˆA, ˆB, ˆC with (17)–(19). +3.3. Computing the barycentric weights +3.3.1. Least-squares approach +The matrix-valued weights ˆ +Wi can be computed similarly to AAA [27] by solving the minimization +problem +min +ˆ +Wi +h +� +j=1 +� +� +� k +� +i=1 +H(λi) ˆ +Wi +sj − λi +� � +Im + +k +� +i=1 +ˆ +Wi +sj − λi +�−1 +− H(sj) +� +� +2 +. +(27) +This solution can, for example, be obtained from an optimization in least-squares sense. The weights +for the SISO case are computed analogously. Here, the matrix-values weights and transfer function +values reduce to scalars. +3.3.2. Pole placement +The next step would be to take advantage of the degrees of freedom in the vector ˆB from (21), so +that the ROM thus constructed has particular (stable) poles [21, 35, 46]. These will be denoted with +ζ1, ζ2, . . . , ζk. The following derivations assume a SISO model. To enforce that this happens, we need +to make sure that the matrix ζjIk − ˆA loses rank for all 1 ≤ j ≤ k. In what follows, we show how to +enforce this property in an elegant, straightforward way. Remember that the transfer function of the +parameterized AF model is given by: +ˆH(s) = +�k +i=1 +ˆwihi +s − λi +1 + �k +i=1 +ˆwi +s − λi += N(s) +D(s). +(28) +Now, let’s say we would like this transfer function to have k poles at the selected values ζj’s. Clearly, +the condition is D(ζj) = 0 and hence we need to enforce: +1 + +k +� +i=1 +ˆwi +ζj − λi += 0, ∀1 ≤ j ≤ k ⇔ Cζ,λ ˆB = −1k ⇔ ˆB = −C−1 +ζ,λ1k, +(29) +where Cζ,λ is a Cauchy matrix defined by: (Cζ,λ)i,j = +1 +ζi−λj . Details on how to obtain the above +expression by following the procedure in [3] are given in Appendix A.2. We note that placing poles +is a difficult numerical problem which requires the inversion of a Cauchy matrix, which is highly +ill-conditioned, by nature. +Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg). +2023-01-13 + +Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions +11 +Algorithm 3 Loewner framework with pole placement (LFPP). +Require: Transfer function samples {H (si)}N +i=1, corresponding sampling points Ξ = {si}N +i=1, loca- +tions for poles ζ = {ζi}k +i=1, interpolation points λ = {λi}k +i=1. +Ensure: Surrogate model ˆH(s) = ˆC(sIr − ˆA)−1 ˆB. +1: Compute ΣD from {H (si)}N +i=1 and {si}N +i=1 using the Loewner framework (cf. Section 2.2). +2: ˆC ← +�HD (λ1) +· · · +HD (λk)� +. +3: ˆB ← −C−1 +ζ,λ1r. +4: ˆA ← diag (λ1, . . . , λk) − ˆB1T +k. +Instead of doing this, we could solve Cζ,λ ˆB = −1r, without inverting the Cauchy matrix explicitly, +i.e., by solving a linear systems of equations. Algorithm 3 summarizes this procedure in a data-driven +context. The required underlying model is obtained from a set of transfer function evaluations by +applying the Loewner framework. The method is illustrated for SISO systems, but can readily be +extended to the MIMO case. +3.4. Automatic choice of poles and interpolation points +A reasonable choice of poles and interpolation points for Algorithm 3 is not always readily available, +but the approximation of the surrogate is heavily influenced by this choice. In the following, we show +an extension to Algorithm 3 which computes a surrogate model (16) without requiring sets of poles +and interpolation points as input parameters. Algorithm 4 sketches the skeleton of such automatic +algorithm. Similar to Algorithm 3 it employs the Loewner framework to obtain a realization of a +surrogate interpolating the provided data. +Subsequently, a generalized eigendecomposition of the +Loewner realization of the original data is computed to find suitable locations for poles. From this +is is possible to compute the dominance of all eigenvalues; for details, see, e.g. [51]. The algorithm +now chooses the k most dominant eigenvalues as poles to enforce in the surrogate. It should be noted +that only eigenvalues with negative real parts should be considered, if the stability of the surrogate is +important. The required interpolation points can now be chosen similar to Algorithm 2 by computing +a CUR decomposition and using the interpolation points associated to the rows or columns of the +decomposition as interpolation points for the new surrogate. +The approximation of dominant poles of the underlying model from data is less robust if the transfer +function samples are disturbed by noise. This leads to a reduced approximation quality. For a better +performance if applied to noisy data, Algorithm 4 can be modified as follows: To obtain poles which +should be enforced, first choose manually the most prominent features in the transfer function, e.g. +peaks, which should be approximated by the surrogate model. +Now choose the eigenvalues which +imaginary parts are closest to the frequencies, where the chosen features of the transfer function are +located. The CUR decomposition also fails at extracting the most dominant rows and columns of +the Loewner matrix if noisy data is assessed. Therefore another heuristic is employed to choose the +interpolation points: Use the value si which corresponds to the lowest amplitude of the transfer function +between the locations of two enforced poles. This leads to reasonable approximations, especially for +lightly damped systems. Other approaches include choosing simply the middle between the location +of two poles or specifying an offset between pole and interpolation point location. +4. Numerical results +In the following, we demonstrate the methods discussed in Section 3 by applying them on three +benchmark examples available from the MOR-Wiki1: +1http://modelreduction.org +Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg). +2023-01-13 + +Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions +12 +Algorithm 4 Loewner framework with automatic pole placement (LFaPP). +Require: Transfer function samples {H (si)}N +i=1, corresponding sampling points Ξ = {si}N +i=1. +Ensure: Surrogate model ˆH(s) = ˆC(sIr − ˆA)−1 ˆB. +1: Compute ΣD from {H (si)}N +i=1 and {si}N +i=1 using the Loewner framework (cf. Section 2.2). +2: Compute the generalized eigenvalue decompositions AX = EXα and YHA = αYHE for the +matrices of right and left eigenvectors X, Y and the matrix of eigenvalues α = diag (α1, . . . , αn). +3: Compute eigenvalue dominance di = |CY(:,i)αiX(:,i)HB| +|ℜ(αi)| +, i = 1, . . . , n and sort α accordingly +4: Set ζ to the k most dominant eigenvalues. +5: Compute CUR decomposition of L. +6: Set λ to the k right or left interpolation points corresponding to the CUR decomposition. +7: Compute surrogate as in Algorithm 3. +ISS This system models the structural response of the Russian Service Module of the International +Space Station (ISS) [52]. The model has n = 270 states, m = 3 inputs, and p = 3 outputs. The +dataset used for the computations contains transfer function measurements at 400 logarithmically +distributed points in the range +� +10−1, 102� +· ı. The model is also part of the SLICOT benchmark +collection [44]. +Flexible aircraft This system models lift and drag along the flexible wing of an aircraft. The system +matrices are not available, we only have access to a dataset of 420 transfer functions samples +at linearly distributed frequencies between 0.1 and 42.0 Hz. The original dataset has one input +(the gust disturbance) and 92 outputs. For the following experiments, we choose the 91st output +which corresponds to the first flexible mode [50]. The dataset is available from [57]. +Sound transmission This system models the sound transmission through a system of two brass plates +with an air enclosure between them. The transfer function measures the sound pressure in an +adjacent acoustic cavity. The geometry is based on [32]; the data—transfer function evaluations +at 1000 linearly-distributed frequency values between 1 and 1000 Hz—is available from [9]. +We note that no tangential interpolation (as described in [41]) is applied for the MIMO model. +Instead, the Loewner matrices are constructed in a block-wise manner. The case of tangential inter- +polation, within the proposed approaches in this note, will be investigated in future works. +We enforce realness of all surrogate models (all matrices contain only real entries) by applying the +transformation described in [7]. For this, all data must be available in complex conjugate pairs. The +required transformation matrix is given by +J = Iℓ ⊗ +� 1 +√ +2 +� Im +Im +−ıIm +ıIm +�� +, +(30) +with ℓ = k +2 and the real-valued quantities are obtained from ˆA +(ℜ) = J ˆAJH, ˆB +(ℜ) = J ˆB, and ˆC +(ℜ) = +ˆCJH. +For some of the experiments we add artificial noise to the measurements, in order to obtain perturbed +data. The modified measurements are given by +ˇH (si) = H (si) (1 + Zi) , i = 1, . . . , n, +(31) +where Zi ∈ C is the ith sample drawn from a set of random numbers Z ∼ CN +� +µ, σ2� +following a +complex normal distribution with mean µ and standard deviation σ2. Here, the real and imaginary +parts of Z are independent normally distributed variables [19]. +We assess the approximation error of the surrogate models with an approximated L∞ norm, because +many surrogates have unstable poles and hence, the H∞ can not be computed. For a given reduced +Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg). +2023-01-13 + +Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions +13 +order r, the L∞ error in the considered frequency range ω ∈ [ωmin, ωmax] is approximated by +ε(r) = +max +ω∈[ωmin,ωmax] +���H(ωı) − ˆHr(ωı) +��� +2 +max +ω∈[ωmin,ωmax] ∥H(ωı)∥2 +≈ +���H − ˆHr +��� +L∞ +∥H∥L∞ +. +(32) +Note that strategies to post-process surrogates to obtain stable models have been studied in [29]. +The numerical experiments have been conducted on a laptop equipped with an AMD Ryzen™ +7 PRO 5850U and 12 GB RAM running Linux Mint 21 as operating system. All algorithms have been +implemented and run with MATLAB R2021b Update 2 (9.11.0.1837725). +Code and data availability +The data that support the findings of this study are openly available in Zenodo at +doi:10.5281/zenodo.7490158 +under the BSD-2-Clause license, authored by Quirin Aumann and Ion Victor Gosea. +4.1. Case of exact measurement data +In the following, we compare the performance of the new approach LS-Loewner to the following estab- +lished strategies: +• Loewner-SVD: Truncate Loewner matrices populated with the complete dataset to order r using +an SVD [7]. +• Loewner-CUR: Construct a purely interpolatory model of order r using all data points chosen by +the CUR decomposition, similar to [37]. +• Modified AAA: Apply the strictly-proper variant of AAA [29] to the complete dataset to compute +a reduced-order model of size r. +We first consider the original MIMO ISS example and a SISO variant where we select the first input +and output, respectively, from the MIMO system. To evaluate the overall performance of the different +methods related to the size of a surrogate model, we compute the approximated L∞ errors for models +with orders 6 ≤ r ≤ 60. The approximation error versus the dimension of the respective surrogate +model is depicted in Figure 1 for all four methods. +Since tangential interpolation was not employed here, the order of the MIMO surrogates rises by m +for each additional interpolation point, i.e., r = km. This explains the lower accuracy of the MIMO +surrogate. For the maximum reduced order r = 60, k = 20 interpolation points are considered. The +errors of the SISO surrogates for r = 20, i.e., k = 20, is in a similar range as in the MIMO case. The +SISO surrogates reach similar levels of approximation for all employed methods. In the MIMO case, +Loewner-SVD performs best. This can be explained by the following observation: the other methods +always consider the complete transfer function measurement H(λi) ∈ Cp×m per interpolation point, +while Loewner-SVD extracts only the r most dominant singular vectors for projection, regardless of +to which interpolation point they belong to. In turn, the other methods also consider probably less +important parts of the data as long as one input/output combination of the respective sample is +relevant for approximation. It can also be noted that LS-Loewner and Loewner-CUR perform very +similar. This was expected, as both methods rely on the same interpolation points. +All four methods are now employed to compute a surrogate model of size r = 108 to approximate +the transfer function of the flexible aircraft model. The size of the surrogate model is determined by +truncating all singular values τ < 1·10−6 of an underlying Loewner matrix. +The transfer functions of all resulting models and their respective relative errors are given in Figure 2. +Again, all methods succeed in computing a sufficiently accurate surrogate. However, the approximation +Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg). +2023-01-13 + +Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions +14 +10 +20 +30 +40 +50 +60 +10−6 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +Reduced order r +L∞ error +SISO +10 +20 +30 +40 +50 +60 +10−6 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +Reduced order r +L∞ error +MIMO +LS-Loewner +Loewner-SVD +Loewner-CUR +Modified AAA +Figure 1: The approximated L∞ errors of reduced-order models of order r computed from the ISS +data. Left: SISO with first input and output, respectively. Right: MIMO with three inputs +and three outputs m = p = 3 (the number of interpolation points is k = r +m). +quality of Loewner-CUR is noticeably worse than that of the other three methods. Given that both +Loewner-CUR and LS-Loewner use the same interpolation points, the weights computed from the least +squares problem show a better performance compared to the partitioning approach used in Loewner- +CUR. +4.2. Perturbed measurement data +Analyzing measurement data perturbed by noise is a challenging task for interpolatory methods, such +as the Loewner framework and the AAA algorithm (as pointed out in, e.g., [27]). In this experiment +we investigate the effect of noise to the performance of the four methods described above and show, +how enforcing poles and/or interpolation points can increase the approximation quality. In the first +experiment we consider transfer function data from the ISS model perturbed by noise with mean µ = 0 +and standard deviation σ2 = 0.15. We employ LFaPP and enforce poles at ı[.77, 2, 4, 5.6, 9.33, 37.9] +near peaks of the transfer function. The resulting real-valued surrogate model has order r = 12. The +transfer functions of the surrogate model with enforced poles and reduced models computed from the +same noisy data with LS-Loewner, Loewner-SVD, Loewner-CUR, and Modified AAA are given in Figure 3. +Enforcing the poles near peaks in the transfer function of the underlying data allows the surrogate +to capture the behavior of the original data in a wider frequency range than applying LS-Loewner, +Loewner-SVD, and Loewner-CUR. The choice of the locations, in which vicinity the poles should be +chosen is, however, not automatized. Figure 3 also shows the relative errors of all surrogate models +referenced to the original data without noise. While the enforced poles all have a negative real part, +the models computed from the variants of the LF and AAA exhibit unstable eigenvalues. Thus, pole +placement can be seen also as a means to enforce stability of the surrogate models. Alternatively, +a post-processing step can be added to enforce stable models (for both LF and AAA methods), as +performed in [29]. +We now evaluate the performance of the algorithms by applying them to heavily distorted trans- +fer function measurements of the sound transmission problem. Noise with a standard deviation of +σ2 = 0.25 is considered and three algorithms are employed to compute surrogates: Loewner-SVD, +LFPP (Algorithm 3), and LFaPP (Algorithm 4). We also test the modifications to LFaPP described in +Section 3.4. These results are denoted by “LFaPP mod.”. For LFPP we enforce poles at the eigenval- +ues of the underlying Loewner model which imaginary parts are near 2πı [72, 189, 392, 401, 706, 856]. +These locations correspond to characteristic peaks in the transfer function. Further, we choose the in- +Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg). +2023-01-13 + +Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions +15 +5 +10 +15 +20 +25 +30 +35 +40 +10−4 +10−2 +100 +Magnitude +Original data +LS-Loewner +Loewner-SVD +Loewner-CUR +Modified AAA +5 +10 +15 +20 +25 +30 +35 +40 +10−12 +10−9 +10−6 +10−3 +100 +Frequency [Hz] +Relative error +LS-Loewner +Loewner-SVD +Loewner-CUR +Modified AAA +Figure 2: Transfer function (top) and relative pointwise errors (bottom) for reduced-order models of +size r = 108 for the aircraft model. The error is plotted only at frequencies which do not +coincide to interpolation points of the respective method. +terpolation points at 2πı [138, 339, 369, 569, 712, 954], which lie at the dips between the enforced poles. +Loewner-SVD and LFaPP do not require input parameters in addition to the measured data. Figure 4 +shows the transfer function of the resulting surrogate models in comparison to the original and noisy +underlying data. It can be observed, that the automatic approaches Loewner-SVD and LFaPP (mod.) +cannot approximate the transfer function well after the first two peaks, i.e., for frequencies higher than +200 Hz, while LFPP approximates the original data over the complete frequency range with decent +accuracy. The importance of reasonable interpolation points can be seen in the difference of LFPP and +LFaPP mod., which have the same poles. It should be noted that the surrogate model computed by +Loewner-SVD has two unstable poles while the other three surrogate models are stable. It is, however, +not always clear a priori how to choose the poles and interpolation points for LFPP in order to achieve +the best approximation quality possible. In this example, the noise level is too high for one of the +automatic approaches to yield reasonable dominant interpolation points or poles. +5. Conclusion and outlook +In this contribution, we have proposed an extensive study of interpolation-based data-driven ap- +proaches for approximating the response of linear dynamical systems. +All methods require input +and output data, i.e., transfer function measurements, while direct access to the system operators or +the states is not required. We showed different approaches how to achieve compact surrogate models +approximating the input/output behavior of the original system and how to ensure various properties +of the surrogate models, such as stability. Strategies how to work with noisy measurement data have +also been addressed. +A natural extension of the framework described here is to apply the ideas of tangential interpolation +as a means of modeling a MIMO system from data. Here, the tangential directions need to be incorpo- +rated in the parameterized one-sided realization. Further topics include enforcing different structures +of the original model in the surrogate model, e.g., second-order or delay structures. It would also be +interesting to study the possibility of placing certain stable poles while achieving interpolation in a +Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg). +2023-01-13 + +Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions +16 +10−1 +100 +101 +102 +10−5 +10−3 +10−1 +Magnitude +Noisy data +Original data +LFaPP +LS-Loewner +Loewner-SVD +Loewner-CUR +Modified AAA +10−1 +100 +101 +102 +10−4 +10−3 +10−2 +10−1 +100 +101 +102 +Frequency +Relative error +Noise +LFaPP +LS-Loewner +Loewner-SVD +Loewner-CUR +Modified AAA +Figure 3: Transfer function of a surrogate with enforced poles compared to the noisy and original +transfer function values. The transfer function of a model r = 12 computed from Loewner- +SVD is given for reference. +least-squares sense. Application cases for the proposed methodology could include damping optimiza- +tion. Here, a family of parameterized interpolants could be used to find optimal positions for viscous +dampers in a structural system. +A. Appendix +A.1. The Woodbury matrix identity +We can expand the right part of (19), such that: +ˆA = Λ − ˆBR ⇒ sIkm − ˆA = sIkm − Λ +� +�� +� +ˆ +M ++ +� +�� +ˆ +W1 +... +ˆ +Wk +� +�� +� �� � +ˆU +�Im +� +���� +ˆT +�Im +· · · +Im +� +� +�� +� +ˆV +. +(33) +The Woodbury matrix identity is as follows: +� +ˆM + ˆU ˆT ˆV +�−1 += ˆM +−1 − ˆM +−1 ˆU +� +ˆT +−1 + ˆV ˆM +−1 ˆU +�−1 ˆV ˆM +−1, +where ˆM, ˆU, ˆT and ˆV are conformable matrices: ˆM is n × n, ˆT is k × k, ˆU is n × k, and ˆV is k × n. +This can be derived using blockwise matrix inversion. By denoting with Λs = sIkm − Λ, then the first +Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg). +2023-01-13 + +Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions +17 +100 +200 +300 +400 +500 +600 +700 +800 +900 +1,000 +10−6 +10−2 +102 +Magnitude +Noisy data +Original data +Loewner-SVD +LFPP +LFaPP +LFaPP mod. +100 +200 +300 +400 +500 +600 +700 +800 +900 +1,000 +10−3 +10−2 +10−1 +100 +101 +102 +103 +Frequency [Hz] +Relative error +Noise +Loewner-SVD +LFPP +LFaPP +LFaPP mod. +Figure 4: Transfer function (top) and relative pointwise errors (bottom) as well as the added noise for +reduced-order models of size r = 12 for the aircraft model. +transfer function of the fitted model is written: +ˆH(s) = ˆC +� +sI3m − ˆA +�−1 ˆB = ˆC +� +Λs + ˆU ˆV +�−1 ˆB += ˆCΛ−1 +s +ˆB − ˆCΛ−1 +s +ˆU +� +Im + ˆVΛ−1 +s +ˆU +�−1 ˆVΛ−1 +s +ˆB += ˆCΛ−1 +s +ˆB − ˆCΛ−1 +s +ˆB +� +Im + RΛ−1 +s +ˆB +�−1 +RΛ−1 +s +ˆB += ˆCΛ−1 +s +ˆB +� +Im − +� +Im + ˆX +�−1 ˆX +� += ˆCΛ−1 +s +ˆB +� +Im + ˆX +�−1 +, +(34) +where ˆX = RΛ−1 +s +ˆB. Hence, we arrive at (20) and the transfer function ˆH(s) can be written as follows: +ˆH(s) = ˆCΛ−1 +s +ˆB +� +Im + RΛ−1 +s +ˆB +�−1 +. +(20) +A.2. Pole placement as in [3] +In order to enforce both prescribed poles and certain interpolation conditions in the ROM, we follow +the derivations from [3]. +It is to be noted that this approach is intrusive, i.e., requires access to +the system’s matrices. Hence, a descriptor model characterized in (generalized) state-space by the +following equations +ΣDes : +� +E ˙x(t) = Ax(t) + Bu(t), +y(t) = Cx(t), +(35) +with corresponding transfer function HDes(s) = C(sE − A)−1B is considered to be given. For the +(right) interpolation points λi, i = 1, . . . , k (where interpolation is imposed), and the desired poles to +be placed, denoted with ζj’s, the author in [3] starts by finding a row vector Cζ ∈ C1×n so that: +Cζ +� +(λ1E − A)−1B · · · (λkE − A)−1B +� += 01×k. +(36) +Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg). +2023-01-13 + +Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions +18 +Then, the next step is to choose projection matrices W, V ∈ Cn×k as +WH = +� +�� +Cζ(ζ1E − A)−1 +... +˜C(ζkE − A)−1 +� +�� , +V = +�(λ1E − A)−1B · · · (λkE − A)−1B� +. +(37) +As explained in [3], the choice of WH above is explained by imposing the required poles for the +reduced model, while V is chosen to match the interpolation conditions at the λi’s. Moreover, using +these notations, it follows that ˜CV = 0. Next, put together the following matrices ˜E = WHEV, +˜A = +WHAV. Then, it follows that (s˜E − ˜A) loses rank when s ∈ {ζ1, . . . , ζr}. To show this, we simply +write +eT +j (ζj ˜E − ˜A) = eT +j WH(ζjE − A)V = Cζ(ζjE − A)−1(ζjE − A)V = CζV = 0. +(38) +Let Hζ(s) = Cζ(sE − A)−1B be a rational function in s and we note that ˆE and ˆA are a special type +of diagonally scaled Cauchy matrices, with the following exact definition: +˜Ei,j = −Cζ(ζiE − A)−1B − Cζ(λjE − A)−1B +ζi − λj += − Hζ(ζi) +ζi − λj +˜Ai,j = −ζiCζ(ζiE − A)−1B − λjCζ(λjE − A)−1B +ζi − λj += −ζiHζ(ζi) +ζi − λj +(39) +From the definition in (39), it follows that ˜E = −D ˜BCζ,λ, where D ˜B = diag( ˜B) is a diagonal matrix. +Similarly, it follows that ˜A = −ZD ˜BCζ,λ, where Z = diag(ζ1, . . . , ζk). +Next, we write the other projected quantities as +˜B = WHB = +�Hζ(ζ1) +· · · +Hζ(ζk)�T , +˜C = CV = +�H(λ1) +· · · +H(λk)� +(40) +Hence, the reduced-order linear descriptor system Σpp : (˜E, ˜A, ˜B, ˜C) matches k interpolation conditions +and has the required poles. +Next, we show that this model can be written equivalently in the AF format. 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Comput., 44(3):C185–C209, 2022. doi:10.1137/ +21M1420277. +Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg). +2023-01-13 + diff --git a/HNE4T4oBgHgl3EQfHwxv/content/tmp_files/load_file.txt b/HNE4T4oBgHgl3EQfHwxv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7d8b2babeb5d20799a72f8e9be586923eb7854ac --- /dev/null +++ b/HNE4T4oBgHgl3EQfHwxv/content/tmp_files/load_file.txt @@ -0,0 +1,1325 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf,len=1324 +page_content='Practical challenges in data-driven interpolation: dealing with noise, enforcing stability, and computing realizations Quirin Aumann∗ Ion Victor Gosea† ∗Max Planck Instiute for Dynamics of Complex Technical Systems, Sandtorstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 1, 39106 Magdeburg, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Email: aumann@mpi-magdeburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='de, ORCID: 0000-0001-7942-5703 †Max Planck Instiute for Dynamics of Complex Technical Systems, Sandtorstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 1, 39106 Magdeburg, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Email: gosea@mpi-magdeburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='de, ORCID: 0000-0003-3580-4116 Abstract: In this contribution, we propose a detailed study of interpolation-based data- driven methods that are of relevance in the model reduction and also in the systems and control communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The data are given by samples of the transfer function of the under- lying (unknown) model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=', we analyze frequency-response data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' We also propose novel approaches that combine some of the main attributes of the established methods, for ad- dressing particular issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' This includes placing poles and hence, enforcing stability of reduced-order models, robustness to noisy or perturbed data, and switching from different rational function representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' We mention here the classical state-space format and also various barycentric representations of the fitted rational interpolants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' We show that the newly-developed approaches yield, in some cases, superior numerical results, when com- paring to the established methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The numerical results include a thorough analysis of various aspects related to approximation errors, choice of interpolation points, or placing dominant poles, which are tested on some benchmark models and data-sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Keywords: Data-driven methods, rational approximation, interpolatory methods, least squares fit, Loewner framework, frequency response data, pole placement, noisy measure- ments, Loewner and Cauchy matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Novelty statement: This note shows that by combining the features of established data- driven rational approximation methods based on interpolation (and/or least squares fit), one can devise methods that offer additional important advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' These include stabil- ity enforcement by placing poles in an elegant and numerically stable manner, together with robustness to noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Introduction Approximation of large-scale dynamical systems is pivotal for serving the scopes of efficient simulation and designing control laws in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The technique for reducing the complexity of a system is known as model order reduction (MOR) [1, 5, 13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' There exist a number of methodologies for reducing large-scale models, and each method is tailored to some specific applications (mostly, but not restricted Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 2023-01-13 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='04906v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='NA] 12 Jan 2023 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Gosea: Data-driven interpolation: challenges and solutions 2 to mechanical and electrical engineering) and to achieving certain goals (stability, passivity or structure preservation), on top of the complexity reduction part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Data-driven MOR approaches are of particular importance when access to high-fidelity models is not explicitly granted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' This means that a state- space formulation with access to internal variables is not available, yet input/output data are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Such methods circumvent the need to access an exact description of the original model and are applicable whenever classical projection-based MOR is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Here, we mention system and control methodologies that are based on interpolation or least-square fit of data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=', frequency response measurements), such as vector fitting [31], the Loewner framework [41], or the AAA algorithm [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Methods that use time-domain data are also of interest, including the ones that require input-output data together with those which use snapshot data (access to the state evolution), such as the classical ones in [36,58,59], followed by [8], [47] or [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' We focus on interpolation-based or so-called moment matching (MM) methods which have emerged, were developed, and improved continuously in the last decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The backbone of such methods rep- resent rational Krylov-type approaches together with the Sylvester matrix equation interpretation [1, 11, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Apart from being computationally efficient and easy to implement, MM approaches have another advantage: they do not (necessarily) require access to a full-state realization of the original dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Hence, they can be viewed as data-driven methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Here data are given by the moments of the system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=', samples of the underlying transfer function of the system (and of its derivatives) evaluated in a particular frequency range;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' for more details, we refer the readers to [5,8,41] and to Chapter 3 in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The notion of a moment with respect to systems and control theory is related to the unique solution of a Sylvester matrix equation [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The purpose of this note is twofold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' first, we intend to review and to connect three important system theoretical model reduction methods based on interpolation that were introduced in the last 15 years: The Loewner framework (LF) by Mayo and Antoulas from 2007 in [41];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The Astolfi framework (AF) from 2010 in [8];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The Adaptive Antoulas Anderson (AAA) algorithm by Nakatsukasa, Set´e and Trefethen from 2018 in [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Together, these three approaches were cited multiple times in various research publications, being arguably quite popular methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' However, until now, not too many connections between them were provided, neither in the automatic control, nor in the model reduction, or numerical analysis commu- nities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Together with the vector fitting algorithm (VF) in [31] (which is not based on interpolation, and is hence a purely optimization approach based on least-squares fitting), these methods repre- sent arguably the most prolific rational approximation schemes developed in the system and control community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' However, VF is not the object of this study since it is not based on interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The other scope of this note is to propose a new method that is based on the three methods enumerated above, and that addresses some of the shortcomings and challenges associated with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Basically, the idea is to combine the attributes of each method, by following the steps below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' We make use of the order-revealing property of the LF (encoded by the rank of augmented Loewner matrices);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' additionally, the selection of interpolation points is done via a Loewner-CUR technique proposed in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' We utilize the elegant state-space parameterization of the LTI system proposed by the AF (after imposing k interpolation conditions);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' this is the backbone of the methods (we also show the connection between state-space forms and barycentric forms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' We use either the fitting step from AAA (that chooses free parameters to fit the un-interpolated data in a least square sense) or we impose pole placing (dominant poles are selected from those of the Loewner model);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' in both cases, a linear system of equations needs to be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 2023-01-13 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Gosea: Data-driven interpolation: challenges and solutions 3 In what follows, we consider a multiple-input multiple-output (MIMO) linear time-invariant (LTI) system ΣL of dimension n described by the following system of differential equations: ΣL : � ˙x(t) = Ax(t) + Bu(t), y(t) = Cx(t), (1) with x(t) ∈ Rn as the state variable, u(t) ∈ Rm as the control inputs, and y(t) ∈ Rp as the observed outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Here, we have that A ∈ Cn×n, B ∈ Cn×m and C ∈ Cp×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The transfer (matrix) function of the LTI system is given by H(s) ∈ Cp×n, with s ∈ C, as H(s) = C(sIn − A)−1B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' (2) It is to be noted that, for m = p = 1, the system becomes single-input single-output (SISO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' We will sometime switch between MIMO and SISO formats while presenting the methods covered in this note, since the latter allows a more easy exposition for some of the results shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Let si ∈ C \\ σ(A), where σ(A) denotes the spectrum of matrix A ∈ Cn×n, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=', the set of its eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The j-moment of system ΣL at si is given by ηj(si) = (−1)j j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' � dj dsj H(s) � s=si, for any integer j ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The 0-moment is obtained by sampling the transfer function H(s) in (2) at si, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=', η0 = H(si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' In this contribution, we restrict the analysis to matching 0-moments, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=', samples of the transfer function H(s), and not of its derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' However, all methodologies shown here can be expected to cope with this as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Moreover, in practice, inferring 0-moments from time-domain data is usually a more straight-forward task;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' this is performed by exciting the system with harmonic inputs, and by applying spectral transformations to the outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Additionally, the inference of derivative values (of the transfer function) is typically susceptible to perturbations and it is more challenging to attain, from a numerical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The paper is structured in the following way;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' after the introduction session sets up the stage, we propose a survey of three established interpolation-based methods in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Then, the proposed methodologies are developed in Section 3, with emphasis on the one-step approach that combines optimal selection of interpolation points (chosen using CUR-DEIM) with LS fit on the rest of the data, and also the pole placement method in barycentric form that enforces dominant poles from the Loewner data-driven model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Then, Section 4 illustrates the numerical aspects of applying the methods discussed/proposed in the previous two sections to a variety of test cases (various models and data sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Finally, Section 5 presents the conclusions and the outlook into future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' A survey of established methods In this section we discuss three established data-driven methods for rational approximation (AF, LF, and AAA, as mentioned in the previous section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The data are samples of the transfer function corresponding to the underlying dynamical system, measured on a particular frequency grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' In what follows, we mention some state-of-the-art methodologies used to measure such data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=', frequency response data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Typically, such measurements can be produced in practice from experiments conducted in scientific laboratories using carefully calibrated machines, called spectrum analyzers (SAs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' In this category we mention swept-tuned spectrum analyzers, scalar network analyzers (SNAs), and vector network analyzers (VNAs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The SNA is an instrument that measures microwave signals by converting them to a DC voltage using a diode detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' In a VNA, information regarding both the magnitude and the phase of a microwave signal is extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' While there are different ways to perform such measurements, the method employed by commercial products (such as the Anritsu series described in [18]) of VNAs is to down-convert the signal to a lower intermediate frequency in a process called harmonic sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' This signal can then be measured directly by a tuned receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Compared to the SNA, the VNA is a more powerful analyzer tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The major difference is that the VNA can also measure the phase, and not only the amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' With this property enforced, then so-called scattering parameters (or Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 2023-01-13 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Gosea: Data-driven interpolation: challenges and solutions 4 S-parameters) can be processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' These can be used for identifying forward and reverse transmission and reflection characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' More details can be found in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The harmonic balance method (or HBM) [43], is an established methodology in the field of elec- tromagnetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The HBM is used in many (if not most) commercial radio-frequency (RF) simulation tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' This is due to the fact that it has certain advantages over other common methods, namely modified nodal analysis (MNA), which makes it more appropriate to use for stiff problems and circuits containing transmission lines, nonlinearities and dispersive effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' More details can be found in the survey paper [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The one-sided moment-matching approach in [8] The framework introduced by Astolfi in [8] (referred to as AF throughout the paper) deals with the problem of model reduction by moment matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Although classically interpreted as a problem of interpolation of points in the complex plane, it has instead been recast as a problem of interpolation of steady-state responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' In the following we briefly review its application to linear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' It is to be noted that the AF was steadily extended and applied to different scenarios (including nonlinear dynamical systems, pole-zero placement, and least-squares fit) [33–35,45,53,55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The moments of a linear system can be characterized in terms of the solution of Sylvester equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' By using this observation, it has been shown that the moments are in one-to-one relation with the steady-state output response of the interconnection between a signal generator and the original linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' In what follows, for simplicity of exposition, it is assumed that ΣL is a minimal system (both fully controllable and fully observable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' For exact definitions on minimality, controllability, or observability of LTI systems, we refer the reader to [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Let k ⩽ n and S ∈ Ck×k a non-derogatory matrix (for which the characteristic and minimal polynomials coincide) with σ(S) ∩ σ(A) = ∅ and R ∈ C1×k so that (S, R) is observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Consider the signal generator system Σsg described by the equations Σsg : � ˙ω(t) = Sω(t), u(t) = Rω(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' (3) Then, the explicit solution of (3) can be written as ω(t) = eStω(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Hence, the control input is written as u(t) = ReStω(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' In addition, the eigenvalues of S are called interpolation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' For a linear system ΣL, and interpolation points si ∈ C \\ σ(A), for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' , k, consider a non- derogatory matrix S ∈ Rk×k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' It follows that there exists a one-to-one relation between the moments of the system ΣL and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' the matrix CΠ, where Π is the (unique) solution of the Sylvester equation AΠ + BR = ΠS, for any row vector R ∈ R1×k so that (R, S) is observable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' the steady-state response of the output y of the interconnection of system ΣL and the system Σsg, for any R and ω(0) such that the triplet (R, S, ω(0)) is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' More precisely, let ∆ ∈ Rk be a column vector containing k free parameters (denoted here by δ1, δ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' , δk, with δi ̸= 0, 1 ≤ i ≤ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Then, as stated in [8], the family of linear time-invariant systems that interpolates the moments of system ΣL at the eigenvalues of matrix S, is given by ˆΣ∆ : � ˙ˆx(t) = (S − ∆R) � �� � = ˆA ˆx(t) + ∆ ���� = ˆB u(t), ˆy(t) = CΠ ���� = ˆC ˆx(t), (4) where the matrices S and R are as before and Π is the unique solution of the Sylvester equation AΠ + BR = ΠS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Additionally, the condition σ(S) ∩ σ(S − ∆R) = ∅ needs to be enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' It is to be noted that the free parameters explicitly enter the vector ˆB = ∆, but also the matrix ˆA, as ˆA = S − ∆R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Finally, ˆC = CΠ has fixed entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 2023-01-13 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Gosea: Data-driven interpolation: challenges and solutions 5 The Sylvester matrix equation for the reduced-order system is written as ˆA ˆΠ+ ˆBR = ˆΠS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' This can be explained by the fact that the reduced-order system matches the prescribed moments of the original system, hence the same format of the two equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Without loss of generality, one can consider that the matrix ˆΠ is the identity matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' ˆΠ = Ik (this can be achieved by applying similarity transformations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' By replacing this value into the reduced-dimension Sylvester matrix equation above, the formula ˆA = S − ∆R directly follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Afterwards, the free parameters collected in the vector ∆ can be chosen in order to enforce or impose additional conditions as mentioned in [8], such as: matching with imposing additional k interpolation conditions, matching with prescribed eigenvalues, matching with prescribed relative degree, matching with prescribed zeros, matching with a passivity constraint, matching with L2-gain, or matching with a compartmental constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' An important aspect of the AF is the characterization of all, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=', infinitely many families of reduced- order models that satisfy k prescribed interpolation conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' This is done by explicitly computing such parameterized models, for which the free parameters are the variables entering in the vector ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The main parameterization developed here will be used as a “backbone” of the methods developed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' As stated in the original paper, the main advantage of the AF (characterization of moments in terms of steady-state responses) is that it allows the definition of moments for systems which do not admit a clear/immediate representation in terms of transfer function(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Hence, the author provides as examples the case of linear time-varying systems, and the case of nonlinear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Moreover, it is stated in [8] that one disadvantage of the framework is that it requires the existence of steady- state responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Consequently, the original system has to be (exponentially) stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' However, in most practical applications, this is a realistic requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The Loewner framework in [41] In this section we present a short summary of the Loewner framework (LF), as introduced in [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' It is to mentioned that LF has its roots in the earlier work of [4], and that LF can be considered to be a double-sided moment-matching approach (as opposed to AF, which is one-sided).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' For a tutorial paper on LF for LTI systems, we refer the reader to [7], and for a recent extension that uses time-domain data, we refer the reader to [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The Loewner framework has been recently extended to certain classes of nonlinear systems, such as bilinear systems in [6], and quadratic-bilinear (QB) systems in [24], but also to linear parameter-varying systems in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Additionally, issues such as stability preservation or enforcement, or passivity preservation, were tackled in the LF in [23, 29], for the former, and in [2,12], for the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The LF is based on processing frequency-domain measurements D = {(ωℓ, H(ωℓ)) , ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' , N} (with ωℓ ∈ R for 1 ≤ ℓ ≤ N) corresponding to evaluations of the transfer function of the underlying (unknown/hidden) dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The interpolation problem is formulated as shown below (for convenience of exposition, we show here only the SISO formulation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' We are given data nodes and data points in the set D, partitioned into two disjoint subsets DL and DR, with DL ∪ DR = D and k + q = N, as right data : DL = {(λj, H(λj)) , j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' , k}, and, left data : DR = {(µi, H(µi)) , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' , q}, (5) and we seek to find a rational function ˆH(s), such that the following interpolation conditions hold: ˆH(µi) = H(µi) := vi, ˆH(λj) = H(λj) := wj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' (6) The Loewner matrix L ∈ Cq×k and the shifted Loewner matrix Ls ∈ Cq×k play an important role in the LF, and are given by L(i,j) = vi − wj µi − λj , Ls(i,j) = µivi − λjwj µi − λj , (7) Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 2023-01-13 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Gosea: Data-driven interpolation: challenges and solutions 6 while the data vectors V ∈ Cq, WT ∈ Ck are given by V(i) = vi, W(j) = wj, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' , q, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' (8) Moreover, the following Sylvester matrix equations ([1, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 6]) are satisfied by the Loewner and shifted Loewner matrices (here, 1q = � 1 · · 1 �T ∈ Cq) � ML − LΛ = V1T k − 1qW, MLs − LsΛ = MV1T k − 1qWΛ, (9) where M = diag(µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' , µq) and Λ = diag(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' , λk) are diagonal matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The following relation holds true Ls = LΛ + V1T k = ML + 1qW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' (10) The unprocessed Loewner surrogate model, provided that k = q, is composed of the matrices ˆE = −L, ˆA = −Ls, ˆB = V, ˆC = W, (11) and if the pencil (L, Ls) is regular, then the function ˆH(s) satisfying the interpolation conditions in (6) can be explicitly computed in terms of the matrices in (11), as ˆH(s) = ˆC(sˆE − ˆA)−1 ˆB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' In practical applications (when processing a fairly large number of measurements), the pencil (Ls, L) is often singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Hence, a post-processing step is required for the Loewner model in (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' In such cases, one needs to perform a singular value decomposition (SVD) of augmented Loewner matrices, to extract the dominant features and remove inherent redundancies in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' By doing so, projection matrices X, Y ∈ Ck×r are obtained, as left, and respectively, right truncated singular vector matrices: [L Ls] = YS(1) r ˜X H � L Ls � = ˜YS(2) r XH, (12) where S(1) r , S(2) r ∈ Rr×r, Y ∈ Ck×r, X ∈ Cq×r, ˜Y ∈ C2q×r, ˜X ∈ Cr×2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The truncation index r can be chosen as the numerical rank (based on a tolerance value τ > 0) or as the exact rank of the Loewner pencil (in exact arithmetic), depending on the application and data size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' More details can be found in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The system matrices corresponding to a projected Loewner model of dimension r can be computed as follows: ˜E = −XHLY, ˜A = −XHLsY, ˜B = XHV, ˜C = WY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' We note that MIMO extensions of the LF were already proposed in the original contribution [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' There, a tangential interpolation framework is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Instead of imposing interpolation of full p × m blocks, the authors prefer to interpolate the original transfer matrix function samples along certain vectors (or tangential directions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' We also note that a first attempt of re-interpreting the LF in [41] as a one-sided method was made in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' In the latter, the main difference to the classical work in [4] was that a compression of the left (un-interpolated) data set was enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' However, in [25], it was still unclear how to split the data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=', what the right data set should be (where interpolation is enforced).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Finally, it is to be noted that the choice of interpolation points is crucial in the LF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' An exhaustive study of different choices was proposed in [37], while a greedy strategy was proposed in [17], for scenarios in which limited experimental data are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The AAA algorithm in [42] The AAA algorithm introduced in [42] is an adaptive and iterative extension of the interpolation method based on Loewner matrices, originally proposed in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The main steps are as follows 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Express the fitted rational approximants in a barycentric representation, which represents a numerically stable way of expressing rational functions [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 2023-01-13 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Gosea: Data-driven interpolation: challenges and solutions 7 Algorithm 1 The AAA algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Require: A (discrete) set of sample points Γ ⊂ C with N points, function f (or the evaluations of f on the set Γ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=', the sample values), and an error tolerance ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Ensure: A rational approximant rn(s) of order (n, n) displayed in a barycentric form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 1: Initialize j = 0, Γ(0) ← Γ, and r−1 ← N −1 �N i=1 f(γi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 2: while |f(s) − rj−1(s)| > ϵ do 3: Select a point zj ∈ Γ(j) for which |f(s) − rj−1(s)| attains a maximal value, where for j ≥ 1, it follows: rj−1(s) := �j−1 � k=0 ω(j−1) k s − zk �−1 �j−1 � k=0 ω(j−1) k fk s − zk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' (13) 4: if |f(zj) − rj−1(zj)| ≤ ε then 5: Return rj−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 6: else 7: fj ← f(zj) and Γ(j+1) ← Γ(j) \\ {zj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 8: end if 9: Find the weights ω(j) = [ω(j) 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' , ω(j) j ] by solving a least squares problem over z ∈ Γ(j+1) j � k=0 ω(j) k s − zk f(s) ≈ j � k=0 ω(j) k fk s − zk ⇔ � j � k=0 f(s) − fk s − zk � ω(j) k ≈ 0 ⇔ L(j)ω(j) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' (14) The solution of (14) is given by the (j + 1)th right singular vector of the Loewner matrix L(j) ∈ C(N−j−1)×(j+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 10: j ← j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 11: end while 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Select the next interpolation (support) points via a greedy scheme;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' basically, interpolation is enforced at the point where the (absolute or relative) error at the previous step is maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Compute the other variables (the so-called barycentric weights) in order to enforce least squares approximation on the non-interpolated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' In recent years, the AAA algorithm has proven to be an accurate, fast, and reliable rational ap- proximation tool with a fairly large range of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Here, we will mention only a few: nonlinear eigenvalue problems [39], MOR of parameterized linear dynamical systems [16], MOR of linear sys- tems with quadratic outputs [26], rational approximation of periodic functions [10], representation of conformal maps [22], rational approximation of matrix-valued functions [27], or signal processing with trigonometric rational functions [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The procedure is sketched in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' It is to be mentioned that a modified version of AAA that enforces real-valued and strictly- proper rational appoximants was recently proposed in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' There, the format of the function in (13) was modified by inserting a 1 into the denominator, as follows ˜rj(s) := � 1 + j−1 � k=0 ω(j−1) k s − zk �−1 �j−1 � k=0 ω(j−1) k fk s − zk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' (15) Consequently, the equation in (14) becomes L(j)ω(j−1) = −f(j−1), where the vector f(j−1) ∈ Cj is given by f(j−1) = �f0 f2 · · fj−1 �T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' It is to be noted that ˜rj(s) in (15) is theoretically a rational approximant of order (j − 1, j), if we do not take into account pole/zero cancellations or any other zero cancellations of coefficients in the numerator or denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 2023-01-13 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Gosea: Data-driven interpolation: challenges and solutions 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The proposed methodologies 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Skeleton of the main methods Similar to the methods reviewed in Section 2 we want to find an LTI system with a transfer function of the structure (1) that interpolates data provided as measurements H (si) , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' , k of the transfer function of the original system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' We can directly put together an LTI parametrized model of dimension r = km, having km2 degrees of freedom with transfer function ˆH(s) = ˆC(sIr − ˆA)−1 ˆB, (16) with the underlying data concatenated to ˆC = �H(λ1) · · H(λk)� ∈ Cp×r, (17) a matrix of weights ˆ Wi ˆB = � ˆ W H 1 · · ˆ W H k �H ∈ Cr×m, (18) and ˆA ∈ Cr×r formed from a diagonal matrix populated with the interpolation points λi disturbed by ˆB, such that ˆA = Λ − ˆBR = diag (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' , λk) ⊗ Im − ˆB � 1T r ⊗ Im � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' (19) Making use of the Woodbury matrix identity and denoting Λs = sIkm − Λ, the transfer function (16) can be rewritten as ˆH(s) = ˆCΛ−1 s ˆB � Im + RΛ−1 s ˆB �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' (20) A complete derivation of (20) is given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' In the single-input single-output case (m = p = 1, hence r = k), the barycentric weights reduce to scalars and the matrices for a ROM of structure (16) are given by ˆA = Λ − ˆBR ∈ Ck×k, ˆB = � ˆw1 · · ˆwk �T ∈ Ck×1, ˆC = �H (λ1) · · H (λk)� ∈ C1×k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' (21) By inserting the formulae in (21) into (20), and using the notation hi := H (λi), leads to ˆCΛ−1 s ˆB = k � i=1 ˆwihi s − λi , � Im + RΛ−1 s ˆB �−1 = 1 1 + �k i=1 ˆwi s − λi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' (22) Hence, the transfer function of the model in (21) is given in barycentric representation by ˆH(s) = �k i=1 ˆwihi s − λi 1 + �k i=1 ˆwi s − λi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' (23) This can be performed analogously for a multi-input multi-output case (m = p, r = km).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The first part of (20) becomes ˆCΛ−1 s ˆB = �H(λ1)Im(s − λ1)−1 · · H(λk)Im(s − λk)−1� � �� ˆ W1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' ˆ Wk � �� = k � i=1 H(λi) ˆ Wi s − λi , (24) Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 2023-01-13 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Gosea: Data-driven interpolation: challenges and solutions 9 the second part � Im + RΛ−1 s ˆB �−1 = � � � �Im + �Im · · Im � � �� Im(s − λ1)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Im(s − λk)−1 � �� −1 � �� ˆ W1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' ˆ Wk � �� � � � � −1 = � Im + k � i=1 ˆ Wi s − λi �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' (25) Consequently, the transfer function in (20) has also a barycentric form given by ˆH(s) = � k � i=1 H(λi) ˆ Wi s − λi � � Im + k � i=1 ˆ Wi s − λi �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' (26) The transfer function is defined by the choice of the interpolation points and of the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The interpolation points can be chosen as dominant parts of the available data or based on their location in the frequency spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The weights can be computed such that the data which are not interpolated, are approximated in an optimal way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Alternatively, the weights can be chosen to enforce poles at specific locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' In the following, we show different strategies for both choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Automatic choice of interpolation points The approximation quality of a surrogate model of the form (16) is greatly influenced by the choice of the interpolation points λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' This choice is not always obvious, so automatic strategies are frequently employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The Loewner framework uses the SVD to identify dominant subsets of the available data to enforce interpolation on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Alternatively, the AAA algorithm uses a greedy scheme to minimize the error between surrogate and original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Another approach, originally introduced by [37], makes use of the CUR decomposition to extract interpolation points from a relevant subset of the available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The CUR decomposition approximates a matrix A by a product of three low-rank matrices ˇA = ˇC ˇU ˇR, where ˇC and ˇR represent subsets of the columns respectively rows of A [40,56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' In our case the three matrices are only a byproduct, we are more interested in the interpolation points λ and µ that are associated to the columns and rows extracted as ˇC and ˇR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' In combination with the skeleton for a realization described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='1, Algorithm 2 computes a surrogate model approximating a set of given transfer function data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' We use the algorithm from [56] to compute the CUR decomposition and thus identify dominant parts of the original data set and their corresponding left and right interpolation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Contrary to [37], we decompose the original Loewner matrix L rather than the augmented Loewner matrices �L Ls � and � LH LH s �H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Using all interpolation points obtained from the CUR decomposition would introduce redundant data into the surrogate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Therefore we choose only a subset of the interpolation points: either only the left points, only the right points, or every other entry from a concatenated and sorted vector of left and right points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Together with the data associated to the chosen interpolation points they are used to populate a rectangular Loewner matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' We now need to compute weights for barycentric interpolation as described in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' After having obtained the weights, a surrogate model (16) can be computed from (17)–(19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 2023-01-13 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Gosea: Data-driven interpolation: challenges and solutions 10 Algorithm 2 LS-Loewner with CUR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Require: Transfer function samples {H (si)}N i=1, corresponding sampling points Ξ = {si}N i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Ensure: Surrogate model ˆH(s) = ˆC(sIr − ˆA)−1 ˆB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 1: Partition data and compute Loewner matrix L as in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 2: Compute CUR decomposition, such that L = ˇC ˇU ˇR with ˇC ∈ CN×k, ˇR ∈ Ck×N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 3: Obtain interpolation points {λi}k i=1 , {µi}k i=1 corresponding to the columns and rows in ˇC, ˇR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 4: Postprocess interpolation points to obtain ν = {ν}k i=1 and χ = Ξ \\ ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 5: Populate a rectangular Loewner matrix L(i,j) = H(χi)−H(νj) χi−νj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 6: Compute the weights Ω = −L† � H (ν1)H · · H (νk)H�H , where L† is the pseudo-inverse of L and Ω = � ˆ W H 1 · · ˆ W H k �H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 7: Compute ˆA, ˆB, ˆC with (17)–(19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Computing the barycentric weights 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Least-squares approach The matrix-valued weights ˆ Wi can be computed similarly to AAA [27] by solving the minimization problem min ˆ Wi h � j=1 � � � k � i=1 H(λi) ˆ Wi sj − λi � � Im + k � i=1 ˆ Wi sj − λi �−1 − H(sj) � � 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' (27) This solution can, for example, be obtained from an optimization in least-squares sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The weights for the SISO case are computed analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Here, the matrix-values weights and transfer function values reduce to scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Pole placement The next step would be to take advantage of the degrees of freedom in the vector ˆB from (21), so that the ROM thus constructed has particular (stable) poles [21, 35, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' These will be denoted with ζ1, ζ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' , ζk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The following derivations assume a SISO model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' To enforce that this happens, we need to make sure that the matrix ζjIk − ˆA loses rank for all 1 ≤ j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' In what follows, we show how to enforce this property in an elegant, straightforward way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Remember that the transfer function of the parameterized AF model is given by: ˆH(s) = �k i=1 ˆwihi s − λi 1 + �k i=1 ˆwi s − λi = N(s) D(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' (28) Now, let’s say we would like this transfer function to have k poles at the selected values ζj’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Clearly, the condition is D(ζj) = 0 and hence we need to enforce: 1 + k � i=1 ˆwi ζj − λi = 0, ∀1 ≤ j ≤ k ⇔ Cζ,λ ˆB = −1k ⇔ ˆB = −C−1 ζ,λ1k, (29) where Cζ,λ is a Cauchy matrix defined by: (Cζ,λ)i,j = 1 ζi−λj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Details on how to obtain the above expression by following the procedure in [3] are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' We note that placing poles is a difficult numerical problem which requires the inversion of a Cauchy matrix, which is highly ill-conditioned, by nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 2023-01-13 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Gosea: Data-driven interpolation: challenges and solutions 11 Algorithm 3 Loewner framework with pole placement (LFPP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Require: Transfer function samples {H (si)}N i=1, corresponding sampling points Ξ = {si}N i=1, loca- tions for poles ζ = {ζi}k i=1, interpolation points λ = {λi}k i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Ensure: Surrogate model ˆH(s) = ˆC(sIr − ˆA)−1 ˆB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 1: Compute ΣD from {H (si)}N i=1 and {si}N i=1 using the Loewner framework (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 2: ˆC ← �HD (λ1) · · HD (λk)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 3: ˆB ← −C−1 ζ,λ1r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 4: ˆA ← diag (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' , λk) − ˆB1T k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Instead of doing this, we could solve Cζ,λ ˆB = −1r, without inverting the Cauchy matrix explicitly, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=', by solving a linear systems of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Algorithm 3 summarizes this procedure in a data-driven context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The required underlying model is obtained from a set of transfer function evaluations by applying the Loewner framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The method is illustrated for SISO systems, but can readily be extended to the MIMO case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Automatic choice of poles and interpolation points A reasonable choice of poles and interpolation points for Algorithm 3 is not always readily available, but the approximation of the surrogate is heavily influenced by this choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' In the following, we show an extension to Algorithm 3 which computes a surrogate model (16) without requiring sets of poles and interpolation points as input parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Algorithm 4 sketches the skeleton of such automatic algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Similar to Algorithm 3 it employs the Loewner framework to obtain a realization of a surrogate interpolating the provided data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Subsequently, a generalized eigendecomposition of the Loewner realization of the original data is computed to find suitable locations for poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' From this is is possible to compute the dominance of all eigenvalues;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' for details, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The algorithm now chooses the k most dominant eigenvalues as poles to enforce in the surrogate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' It should be noted that only eigenvalues with negative real parts should be considered, if the stability of the surrogate is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The required interpolation points can now be chosen similar to Algorithm 2 by computing a CUR decomposition and using the interpolation points associated to the rows or columns of the decomposition as interpolation points for the new surrogate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The approximation of dominant poles of the underlying model from data is less robust if the transfer function samples are disturbed by noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' This leads to a reduced approximation quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' For a better performance if applied to noisy data, Algorithm 4 can be modified as follows: To obtain poles which should be enforced, first choose manually the most prominent features in the transfer function, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' peaks, which should be approximated by the surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Now choose the eigenvalues which imaginary parts are closest to the frequencies, where the chosen features of the transfer function are located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The CUR decomposition also fails at extracting the most dominant rows and columns of the Loewner matrix if noisy data is assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Therefore another heuristic is employed to choose the interpolation points: Use the value si which corresponds to the lowest amplitude of the transfer function between the locations of two enforced poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' This leads to reasonable approximations, especially for lightly damped systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Other approaches include choosing simply the middle between the location of two poles or specifying an offset between pole and interpolation point location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Numerical results In the following, we demonstrate the methods discussed in Section 3 by applying them on three benchmark examples available from the MOR-Wiki1: 1http://modelreduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='org Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 2023-01-13 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Gosea: Data-driven interpolation: challenges and solutions 12 Algorithm 4 Loewner framework with automatic pole placement (LFaPP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Require: Transfer function samples {H (si)}N i=1, corresponding sampling points Ξ = {si}N i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Ensure: Surrogate model ˆH(s) = ˆC(sIr − ˆA)−1 ˆB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 1: Compute ΣD from {H (si)}N i=1 and {si}N i=1 using the Loewner framework (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 2: Compute the generalized eigenvalue decompositions AX = EXα and YHA = αYHE for the matrices of right and left eigenvectors X, Y and the matrix of eigenvalues α = diag (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' , αn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 3: Compute eigenvalue dominance di = |CY(:,i)αiX(:,i)HB| |ℜ(αi)| , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' , n and sort α accordingly 4: Set ζ to the k most dominant eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 5: Compute CUR decomposition of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 6: Set λ to the k right or left interpolation points corresponding to the CUR decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 7: Compute surrogate as in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' ISS This system models the structural response of the Russian Service Module of the International Space Station (ISS) [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The model has n = 270 states, m = 3 inputs, and p = 3 outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The dataset used for the computations contains transfer function measurements at 400 logarithmically distributed points in the range � 10−1, 102� ı.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The model is also part of the SLICOT benchmark collection [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Flexible aircraft This system models lift and drag along the flexible wing of an aircraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The system matrices are not available, we only have access to a dataset of 420 transfer functions samples at linearly distributed frequencies between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='1 and 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='0 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The original dataset has one input (the gust disturbance) and 92 outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' For the following experiments, we choose the 91st output which corresponds to the first flexible mode [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The dataset is available from [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Sound transmission This system models the sound transmission through a system of two brass plates with an air enclosure between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The transfer function measures the sound pressure in an adjacent acoustic cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The geometry is based on [32];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' the data—transfer function evaluations at 1000 linearly-distributed frequency values between 1 and 1000 Hz—is available from [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' We note that no tangential interpolation (as described in [41]) is applied for the MIMO model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Instead, the Loewner matrices are constructed in a block-wise manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The case of tangential inter- polation, within the proposed approaches in this note, will be investigated in future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' We enforce realness of all surrogate models (all matrices contain only real entries) by applying the transformation described in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' For this, all data must be available in complex conjugate pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The required transformation matrix is given by J = Iℓ ⊗ � 1 √ 2 � Im Im −ıIm ıIm �� , (30) with ℓ = k 2 and the real-valued quantities are obtained from ˆA (ℜ) = J ˆAJH, ˆB (ℜ) = J ˆB, and ˆC (ℜ) = ˆCJH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' For some of the experiments we add artificial noise to the measurements, in order to obtain perturbed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The modified measurements are given by ˇH (si) = H (si) (1 + Zi) , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' , n, (31) where Zi ∈ C is the ith sample drawn from a set of random numbers Z ∼ CN � µ, σ2� following a complex normal distribution with mean µ and standard deviation σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Here, the real and imaginary parts of Z are independent normally distributed variables [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' We assess the approximation error of the surrogate models with an approximated L∞ norm, because many surrogates have unstable poles and hence, the H∞ can not be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' For a given reduced Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 2023-01-13 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Gosea: Data-driven interpolation: challenges and solutions 13 order r, the L∞ error in the considered frequency range ω ∈ [ωmin, ωmax] is approximated by ε(r) = max ω∈[ωmin,ωmax] ���H(ωı) − ˆHr(ωı) ��� 2 max ω∈[ωmin,ωmax] ∥H(ωı)∥2 ≈ ���H − ˆHr ��� L∞ ∥H∥L∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' (32) Note that strategies to post-process surrogates to obtain stable models have been studied in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The numerical experiments have been conducted on a laptop equipped with an AMD Ryzen™ 7 PRO 5850U and 12 GB RAM running Linux Mint 21 as operating system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' All algorithms have been implemented and run with MATLAB R2021b Update 2 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='1837725).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Code and data availability The data that support the findings of this study are openly available in Zenodo at doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='7490158 under the BSD-2-Clause license, authored by Quirin Aumann and Ion Victor Gosea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Case of exact measurement data In the following, we compare the performance of the new approach LS-Loewner to the following estab- lished strategies: Loewner-SVD: Truncate Loewner matrices populated with the complete dataset to order r using an SVD [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Loewner-CUR: Construct a purely interpolatory model of order r using all data points chosen by the CUR decomposition, similar to [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Modified AAA: Apply the strictly-proper variant of AAA [29] to the complete dataset to compute a reduced-order model of size r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' We first consider the original MIMO ISS example and a SISO variant where we select the first input and output, respectively, from the MIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' To evaluate the overall performance of the different methods related to the size of a surrogate model, we compute the approximated L∞ errors for models with orders 6 ≤ r ≤ 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The approximation error versus the dimension of the respective surrogate model is depicted in Figure 1 for all four methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Since tangential interpolation was not employed here, the order of the MIMO surrogates rises by m for each additional interpolation point, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=', r = km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' This explains the lower accuracy of the MIMO surrogate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' For the maximum reduced order r = 60, k = 20 interpolation points are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The errors of the SISO surrogates for r = 20, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=', k = 20, is in a similar range as in the MIMO case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The SISO surrogates reach similar levels of approximation for all employed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' In the MIMO case, Loewner-SVD performs best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' This can be explained by the following observation: the other methods always consider the complete transfer function measurement H(λi) ∈ Cp×m per interpolation point, while Loewner-SVD extracts only the r most dominant singular vectors for projection, regardless of to which interpolation point they belong to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' In turn, the other methods also consider probably less important parts of the data as long as one input/output combination of the respective sample is relevant for approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' It can also be noted that LS-Loewner and Loewner-CUR perform very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' This was expected, as both methods rely on the same interpolation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' All four methods are now employed to compute a surrogate model of size r = 108 to approximate the transfer function of the flexible aircraft model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The size of the surrogate model is determined by truncating all singular values τ < 1·10−6 of an underlying Loewner matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The transfer functions of all resulting models and their respective relative errors are given in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Again, all methods succeed in computing a sufficiently accurate surrogate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' However, the approximation Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 2023-01-13 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Gosea: Data-driven interpolation: challenges and solutions 14 10 20 30 40 50 60 10−6 10−5 10−4 10−3 10−2 10−1 100 Reduced order r L∞ error SISO 10 20 30 40 50 60 10−6 10−5 10−4 10−3 10−2 10−1 100 Reduced order r L∞ error MIMO LS-Loewner Loewner-SVD Loewner-CUR Modified AAA Figure 1: The approximated L∞ errors of reduced-order models of order r computed from the ISS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Left: SISO with first input and output, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Right: MIMO with three inputs and three outputs m = p = 3 (the number of interpolation points is k = r m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' quality of Loewner-CUR is noticeably worse than that of the other three methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Given that both Loewner-CUR and LS-Loewner use the same interpolation points, the weights computed from the least squares problem show a better performance compared to the partitioning approach used in Loewner- CUR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Perturbed measurement data Analyzing measurement data perturbed by noise is a challenging task for interpolatory methods, such as the Loewner framework and the AAA algorithm (as pointed out in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=', [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' In this experiment we investigate the effect of noise to the performance of the four methods described above and show, how enforcing poles and/or interpolation points can increase the approximation quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' In the first experiment we consider transfer function data from the ISS model perturbed by noise with mean µ = 0 and standard deviation σ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' We employ LFaPP and enforce poles at ı[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='77, 2, 4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='6, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='33, 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='9] near peaks of the transfer function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The resulting real-valued surrogate model has order r = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The transfer functions of the surrogate model with enforced poles and reduced models computed from the same noisy data with LS-Loewner, Loewner-SVD, Loewner-CUR, and Modified AAA are given in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Enforcing the poles near peaks in the transfer function of the underlying data allows the surrogate to capture the behavior of the original data in a wider frequency range than applying LS-Loewner, Loewner-SVD, and Loewner-CUR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The choice of the locations, in which vicinity the poles should be chosen is, however, not automatized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Figure 3 also shows the relative errors of all surrogate models referenced to the original data without noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' While the enforced poles all have a negative real part, the models computed from the variants of the LF and AAA exhibit unstable eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Thus, pole placement can be seen also as a means to enforce stability of the surrogate models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Alternatively, a post-processing step can be added to enforce stable models (for both LF and AAA methods), as performed in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' We now evaluate the performance of the algorithms by applying them to heavily distorted trans- fer function measurements of the sound transmission problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Noise with a standard deviation of σ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='25 is considered and three algorithms are employed to compute surrogates: Loewner-SVD, LFPP (Algorithm 3), and LFaPP (Algorithm 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' We also test the modifications to LFaPP described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' These results are denoted by “LFaPP mod.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' For LFPP we enforce poles at the eigenval- ues of the underlying Loewner model which imaginary parts are near 2πı [72, 189, 392, 401, 706, 856].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' These locations correspond to characteristic peaks in the transfer function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Further, we choose the in- Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 2023-01-13 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Gosea: Data-driven interpolation: challenges and solutions 15 5 10 15 20 25 30 35 40 10−4 10−2 100 Magnitude Original data LS-Loewner Loewner-SVD Loewner-CUR Modified AAA 5 10 15 20 25 30 35 40 10−12 10−9 10−6 10−3 100 Frequency [Hz] Relative error LS-Loewner Loewner-SVD Loewner-CUR Modified AAA Figure 2: Transfer function (top) and relative pointwise errors (bottom) for reduced-order models of size r = 108 for the aircraft model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The error is plotted only at frequencies which do not coincide to interpolation points of the respective method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' terpolation points at 2πı [138, 339, 369, 569, 712, 954], which lie at the dips between the enforced poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Loewner-SVD and LFaPP do not require input parameters in addition to the measured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Figure 4 shows the transfer function of the resulting surrogate models in comparison to the original and noisy underlying data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' It can be observed, that the automatic approaches Loewner-SVD and LFaPP (mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=') cannot approximate the transfer function well after the first two peaks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=', for frequencies higher than 200 Hz, while LFPP approximates the original data over the complete frequency range with decent accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The importance of reasonable interpolation points can be seen in the difference of LFPP and LFaPP mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=', which have the same poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' It should be noted that the surrogate model computed by Loewner-SVD has two unstable poles while the other three surrogate models are stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' It is, however, not always clear a priori how to choose the poles and interpolation points for LFPP in order to achieve the best approximation quality possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' In this example, the noise level is too high for one of the automatic approaches to yield reasonable dominant interpolation points or poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Conclusion and outlook In this contribution, we have proposed an extensive study of interpolation-based data-driven ap- proaches for approximating the response of linear dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' All methods require input and output data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=', transfer function measurements, while direct access to the system operators or the states is not required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' We showed different approaches how to achieve compact surrogate models approximating the input/output behavior of the original system and how to ensure various properties of the surrogate models, such as stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Strategies how to work with noisy measurement data have also been addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' A natural extension of the framework described here is to apply the ideas of tangential interpolation as a means of modeling a MIMO system from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Here, the tangential directions need to be incorpo- rated in the parameterized one-sided realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Further topics include enforcing different structures of the original model in the surrogate model, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=', second-order or delay structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' It would also be interesting to study the possibility of placing certain stable poles while achieving interpolation in a Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 2023-01-13 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Gosea: Data-driven interpolation: challenges and solutions 16 10−1 100 101 102 10−5 10−3 10−1 Magnitude Noisy data Original data LFaPP LS-Loewner Loewner-SVD Loewner-CUR Modified AAA 10−1 100 101 102 10−4 10−3 10−2 10−1 100 101 102 Frequency Relative error Noise LFaPP LS-Loewner Loewner-SVD Loewner-CUR Modified AAA Figure 3: Transfer function of a surrogate with enforced poles compared to the noisy and original transfer function values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The transfer function of a model r = 12 computed from Loewner- SVD is given for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' least-squares sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Application cases for the proposed methodology could include damping optimiza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Here, a family of parameterized interpolants could be used to find optimal positions for viscous dampers in a structural system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The Woodbury matrix identity We can expand the right part of (19), such that: ˆA = Λ − ˆBR ⇒ sIkm − ˆA = sIkm − Λ � �� � ˆ M + � �� ˆ W1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' ˆ Wk � �� � �� � ˆU �Im � ���� ˆT �Im · · Im � � �� � ˆV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' (33) The Woodbury matrix identity is as follows: � ˆM + ˆU ˆT ˆV �−1 = ˆM −1 − ˆM −1 ˆU � ˆT −1 + ˆV ˆM −1 ˆU �−1 ˆV ˆM −1, where ˆM, ˆU, ˆT and ˆV are conformable matrices: ˆM is n × n, ˆT is k × k, ˆU is n × k, and ˆV is k × n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' This can be derived using blockwise matrix inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' By denoting with Λs = sIkm − Λ, then the first Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 2023-01-13 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Gosea: Data-driven interpolation: challenges and solutions 17 100 200 300 400 500 600 700 800 900 1,000 10−6 10−2 102 Magnitude Noisy data Original data Loewner-SVD LFPP LFaPP LFaPP mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 100 200 300 400 500 600 700 800 900 1,000 10−3 10−2 10−1 100 101 102 103 Frequency [Hz] Relative error Noise Loewner-SVD LFPP LFaPP LFaPP mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Figure 4: Transfer function (top) and relative pointwise errors (bottom) as well as the added noise for reduced-order models of size r = 12 for the aircraft model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' transfer function of the fitted model is written: ˆH(s) = ˆC � sI3m − ˆA �−1 ˆB = ˆC � Λs + ˆU ˆV �−1 ˆB = ˆCΛ−1 s ˆB − ˆCΛ−1 s ˆU � Im + ˆVΛ−1 s ˆU �−1 ˆVΛ−1 s ˆB = ˆCΛ−1 s ˆB − ˆCΛ−1 s ˆB � Im + RΛ−1 s ˆB �−1 RΛ−1 s ˆB = ˆCΛ−1 s ˆB � Im − � Im + ˆX �−1 ˆX � = ˆCΛ−1 s ˆB � Im + ˆX �−1 , (34) where ˆX = RΛ−1 s ˆB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Hence, we arrive at (20) and the transfer function ˆH(s) can be written as follows: ˆH(s) = ˆCΛ−1 s ˆB � Im + RΛ−1 s ˆB �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' (20) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Pole placement as in [3] In order to enforce both prescribed poles and certain interpolation conditions in the ROM, we follow the derivations from [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' It is to be noted that this approach is intrusive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=', requires access to the system’s matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Hence, a descriptor model characterized in (generalized) state-space by the following equations ΣDes : � E ˙x(t) = Ax(t) + Bu(t), y(t) = Cx(t), (35) with corresponding transfer function HDes(s) = C(sE − A)−1B is considered to be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' For the (right) interpolation points λi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' , k (where interpolation is imposed), and the desired poles to be placed, denoted with ζj’s, the author in [3] starts by finding a row vector Cζ ∈ C1×n so that: Cζ � (λ1E − A)−1B · · · (λkE − A)−1B � = 01×k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' (36) Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 2023-01-13 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Gosea: Data-driven interpolation: challenges and solutions 18 Then, the next step is to choose projection matrices W, V ∈ Cn×k as WH = � �� Cζ(ζ1E − A)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' ˜C(ζkE − A)−1 � �� , V = �(λ1E − A)−1B · · · (λkE − A)−1B� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' (37) As explained in [3], the choice of WH above is explained by imposing the required poles for the reduced model, while V is chosen to match the interpolation conditions at the λi’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Moreover, using these notations, it follows that ˜CV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Next, put together the following matrices ˜E = WHEV, ˜A = WHAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Then, it follows that (s˜E − ˜A) loses rank when s ∈ {ζ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' , ζr}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' To show this, we simply write eT j (ζj ˜E − ˜A) = eT j WH(ζjE − A)V = Cζ(ζjE − A)−1(ζjE − A)V = CζV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' (38) Let Hζ(s) = Cζ(sE − A)−1B be a rational function in s and we note that ˆE and ˆA are a special type of diagonally scaled Cauchy matrices, with the following exact definition: ˜Ei,j = −Cζ(ζiE − A)−1B − Cζ(λjE − A)−1B ζi − λj = − Hζ(ζi) ζi − λj ˜Ai,j = −ζiCζ(ζiE − A)−1B − λjCζ(λjE − A)−1B ζi − λj = −ζiHζ(ζi) ζi − λj (39) From the definition in (39), it follows that ˜E = −D ˜BCζ,λ, where D ˜B = diag( ˜B) is a diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Similarly, it follows that ˜A = −ZD ˜BCζ,λ, where Z = diag(ζ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' , ζk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Next, we write the other projected quantities as ˜B = WHB = �Hζ(ζ1) · · Hζ(ζk)�T , ˜C = CV = �H(λ1) · · H(λk)� (40) Hence, the reduced-order linear descriptor system Σpp : (˜E, ˜A, ˜B, ˜C) matches k interpolation conditions and has the required poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Next, we show that this model can be written equivalently in the AF format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' We first note that ˆC = ˜C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' For next step, provided that the matrix ˜E is non-singular, we remove it by incorporating it into the other matrices, as: ˘A = ˜E −1 ˜A, ˘B = ˜E −1 ˜B, ˘E = Ik, ˘C = ˜C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' We note that the two realizations of the interpolatory ROM, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=', ( ˆA, ˆB, ˆC) in (21) and ( ˘A, ˘B, ˘C) introduced above, are actually identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' The reason for this is that ˘C = ˆC and the two ROMs match the same k moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Hence, it also follows that ˘B = ˆB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Now, since ˘B = ˜E −1 ˜B and ˜E = −D ˜BCζ,λ, we can write that ˆB = −(D ˜BCζ,λ)−1 ˜B = −C−1 ζ,λD−1 ˜B ˜B = −C−1 ζ,λ1k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' (41) Hence, the above choice of vector ˆB in (21) imposes the required poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Antoulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Approximation of Large-Scale Dynamical Systems, volume 6 of Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Des.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' SIAM Publications, Philadelphia, PA, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' doi:10.' metadata={'source': 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+page_content=' scl, 54(4):361–374, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='sysconle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Antoulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Polplatzierung bei der Modellreduktion (On pole placement in model reduction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' at-Automatisierungstechnik, 55(9):443–448, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='1524/auto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content='443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 2023-01-13 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Gosea: Data-driven interpolation: challenges and solutions 19 [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Antoulas and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Anderson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' On the scalar rational interpolation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' IMA J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' Control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} +page_content=' 2023-01-13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'} diff --git a/HdAzT4oBgHgl3EQfHvsI/content/tmp_files/2301.01048v1.pdf.txt b/HdAzT4oBgHgl3EQfHvsI/content/tmp_files/2301.01048v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7b467ec94e6e4b78feebb5dc6a48d7dfe92e8bc9 --- /dev/null +++ b/HdAzT4oBgHgl3EQfHvsI/content/tmp_files/2301.01048v1.pdf.txt @@ -0,0 +1,1483 @@ +1 +A Theory of I/O-Efficient Sparse Neural Network +Inference +Niels Gleinig, Tal Ben-Nun, Torsten Hoefler +{niels.gleinig, talbn, torsten.hoefler}@inf.ethz.ch +ETH Zürich +Abstract—As the accuracy of machine learning models in- +creases at a fast rate, so does their demand for energy and +compute resources. On a low level, the major part of these +resources is consumed by data movement between different +memory units. Modern hardware architectures contain a form +of fast memory (e.g., cache, registers), which is small, and a slow +memory (e.g., DRAM), which is larger but expensive to access. We +can only process data that is stored in fast memory, which incurs +data movement (input/output-operations, or I/Os) between the +two units. In this paper, we provide a rigorous theoretical analysis +of the I/Os needed in sparse feedforward neural network (FFNN) +inference. We establish bounds that determine the optimal +number of I/Os up to a factor of 2 and present a method that uses +a number of I/Os within that range. Much of the I/O-complexity is +determined by a few high-level properties of the FFNN (number +of inputs, outputs, neurons, and connections), but if we want to +get closer to the exact lower bound, the instance-specific sparsity +patterns need to be considered. Departing from the 2-optimal +computation strategy, we show how to reduce the number of I/Os +further with simulated annealing. Complementing this result, we +provide an algorithm that constructively generates networks with +maximum I/O-efficiency for inference. We test the algorithms and +empirically verify our theoretical and algorithmic contributions. +In our experiments on real hardware we observe speedups of up +to 45× relative to the standard way of performing inference. +Index Terms—Neural Network Inference, Sparse Neural Net- +works, I/O-Complexity, Simulated Annealing. +I. INTRODUCTION +Almost all modern computing systems deploy different +memory units: Some are fast but small and some are large but +slow. Data can only be processed in fast memory. As the fast +memory is very limited in size, data needs to be moved between +fast and slow memory over the course of a computation. These +data movements are called I/Os (abbreviating input/output- +operations). They are very expensive in time and energy. For +example, on an NVIDIA V100 GPU, loading values from +global memory costs up to 514.5× more clock cycles than +one fused-multiply-add (FMA) operation between two 32-bit +floating point values (or 14× if stored on an L1 cache) [1]. +In deep learning, and scientific computing in general, I/Os +account for the major part of time and energy expenses [2]. +Hence, it is important to use I/Os efficiently by performing +computations in a way that makes use of “data locality” and +“data reuse”. We need to schedule the computational steps in +such a way that accesses to the same data are close together, +sparing eviction of values from fast memory (e.g., cache) and +reading them again. Therefore, optimizing the implementation +of DNN inference [3] and training [4] often requires tailor- +made implementations for the available memory architectures +in order to reach high utilization. +In this paper, we analyze and optimize I/Os with a theoretical +model that applies across the wide variety of hardware in which +this phenomenon occurs. We consider exclusively sparse Feed- +Forward Neural Networks (FFNNs) without shared connections. +This model contains, but is not limited to, pruned Multi-Layer +Perceptrons (MLP). We assume that the FFNN is given as a list +of weighted edges in a Directed Acyclic Graph (DAG) together +with one additional value for each vertex (being the input-value +for input-neurons and the bias for non-input neurons). In this +setting, the primary way to obtain I/O-efficiency is to “use” +the connections in an efficient order. That is, when we use +a connection, the value of the input neuron and the value of +the partial sum of the output neuron should ideally already be +in fast memory. Therefore, we need to find an order of the +connections, in which the connections that have neurons in +common, are close together. +Finding an order which is optimal with respect to this +property is a combinatorial problem. We will show that if +we use a traditional computational order, which corresponds +to a natural interpretation of FFNN inference (for example, +seeing it as a sequence of matrix vector multiplications) we +can be far from optimal: The number of read-I/Os could be +up to twice the optimal number, and the number of write-I/Os +could differ from the optimal number by an arbitrarily large +factor. +Computations in ML have some special features that we can +use when aiming for I/O-efficiency. Unlike the research on +I/O-efficient computing outside of ML, there are more degrees +of freedom that we can exploit. For example, there are several +acceptable solutions to a given problem: Two different FFNNs +may produce different outputs, yet both can have a satisfying +accuracy. It thus makes sense to adapt the neural architecture +to the hardware. On the other hand, there is additional freedom +on the side of the hardware, as it is not uncommon to search +or even build hardware for specific ML applications [5], [6], +[3]. +Hence, our investigation is directed towards answering the +following questions: (1) If we are given an FFNN and a fast +memory of a given size, what is the minimal number of I/Os +needed to perform inference on this FFNN? (2) For a given +FFNN, what is the smallest memory size with which we can +perform inference with a minimal number of I/Os (i.e., we can +avoid having to read and write intermediate results due to +lack of memory)? and (3) For a fast memory of a given size, +arXiv:2301.01048v1 [cs.DC] 3 Jan 2023 + +2 +what are the FFNN architectures on which we can perform +inference with a minimal number of I/Os? +A. Background and related work +a) I/O-complexity: There is an extensive line of theoret- +ical research on I/O-efficient algorithms outside of machine +learning. Hong & Kung’s seminal work [7] is among the +first to investigate I/O-efficient algorithms formally. This work +introduced a theoretical model called the red-blue pebble game +and used this model to show how standard matrix multiplication +or FFT should be scheduled to use an asymptotically minimal +number of I/Os. This model has been extended and altered in +various ways. These extensions have been used to establish +lower bounds and matching upper bounds for problems such +as sorting, permutation, and matrix transposition [8]; as well as +sparse matrix dense vector multiplication [9], and various graph +algorithms [10]. However, these solutions are typically only +optimal up to constant multiplicative factors and often these +factors are large or not exactly known. In fact, it has been shown +that finding exactly optimal solutions to the red-blue pebble +game (corresponding to computation schedules with an exact +minimal number of data movements) is a PSPACE-complete +problem [11]. Furthermore, this problem has been hypothesized +to be even hard to approximate [11], since inapproximability +is known for related pebble games [12], [13]. +There are also bounds on the I/O-complexity of general +computations with large proportions of inputs [14]. These +bounds apply to the problem considered in our paper. However, +they are less fine-grained (they do not distinguish between +read- and write-I/Os as we do) and generally less tight as the +ones that we establish in this paper. +In ML-specific algorithms, I/O-efficiency has been practically +identified as a crucial performance factor. Performance model- +ing of DNNs is usually derived from operation counts [15] and +sums of fixed values obtained from benchmarking individual +layers [16], [17]. As for I/O complexity of DNN evaluation, +the only theoretical work apart from this one, to the best of +our knowledge, is given by Demmel & Dinh [18], who provide +communication bounds for convolutional and pooling layers. +b) Network pruning and sparsity: DNN pruning (i.e., +removing certain edges or nodes from a network while +preserving accuracy) has been extensively studied over the last +decades [19], [20], [21]. Pruning is of much practical interest +for edge computing devices, as it allows to drastically reduce +the memory footprint and number of floating point operations. +Dense (fully-connected) layers have been a prime target for +pruning, as they tend to account for a major proportion of the +memory size. For example, in AlexNet [22], the final dense +layers contain 90% of the parameters. Also transformers [23], +which are currently very popular and achieve state-of-the-art +results in NLP tasks, include FFNNs that comprise a large +proportion of the overall size. For example, BERT [23] includes +several FFNNs of depth 2 and weight matrices of dimensions +1024 × 4096 and 4096 × 1024 and the major part of BERTs +parameters and compute time for inference come from these +FFNNs [24]. +There are different ways to prune networks, e.g., by a given +weight threshold [25], [26], the top-k values [27], [28], or +by way of gradual weight elimination [19]. Regardless of the +method, pruning dense layers leads to sparse and unstructured +networks, for which data-locality is more difficult to obtain. +c) Hardware Architectures: There has been work on +hardware-based optimizations for inference, especially for +the class of sparse FFNNs, where Sze et al. [3] provide a +detailed survey. Specific approaches include EDEN [29], which +uses approximate DRAM for improving energy-efficiency; +Bhardwaj et al. [30], who consider communication in inference +in a distributed setting; and EIE [5] and Eyeriss [31] runs +inference on a compressed CNN directly, using accelerated +sparse matrix-vector multiplications and run-length encoding, +respectively. While these approaches improve certain aspects of +DNN processing, as the network size grows, such architectures +ultimately also resort to multi-level memory hierarchies, which +our approach addresses. +B. Contributions +Our main theoretical contributions are a collection of +theorems and propositions that analyze the I/Os in FFNN +inference. Theorem 1 provides bounds on the number of +I/Os that are needed to perform inference on a given FFNN- +architecture. In Proposition 1, we show that these bounds are +optimal in the sense that none of them could be tightened by +multiplying with any factor other than 1. The lower and upper +bounds on the total number of I/Os differ by a multiplicative +factor of at most 2. The proof of Theorem 1 is constructive +and shows how a computation can be done to use a number of +I/Os within this 2-optimal range. Departing from 2-optimality, +our goal is to get closer to the exact optimum by reordering +the connections beneficially. We approach this problem with +Simulated Annealing. In Section V we present a method for +generating FFNNs, which according to Theorem 2 completely +characterizes the architectures on which we can perform +inference with a minimal number of I/Os for a given memory +size (i.e., exactly matching the lower bound). Hence, this +method can be used as a powerful tool to co-design neural +networks and hardware. As a corollary of this theorem we +obtain an upper bound on the smallest possible memory size +that allows us to perform inference with maximal I/O-efficiency +(that is, without having to write or read any temporary values). +We test the algorithms on random FFNNs across a wide range +of parameter settings (density, depth, width, memory sizes). +II. MODEL AND NOTATIONS +We assume that initially all FFNN parameters and all input +values are laid out in slow memory. These data are exactly the +union of the following three types of data: +• The weighted connections. +• One bias value for each non-input neuron. +• One input value for each input neuron. +Each connection is described by an independent parameter +triple (i, j, wij), where i is the input neuron, j the output +neuron, and wij the weight (there is no reuse or sharing of +weights). We denote the number of connections as W, the +number of neurons N, the number of input neurons I, and the +number of output neurons S. In our theoretical analysis, we + +3 +assume the connections (the entire triples describing them), the +numerical values at the neurons (biases, partial sums, outputs +of the activation functions), and all other numerical values +involved in the computation are data types of the same size. +Hence, the entire problem size is given by the total number of +weights, biases, and input neurons: W +(N −I)+I = W +N. +We have a fast memory that can hold at any given time a number +of values (of this data type) that is given by the parameter M, +where we assume M ≥ 3. In this model the single parameter +M defines the whole architecture: fast memory of size M +and slow memory of unlimited size. We can perform arbitrary +computations “for free” on data that is stored in fast memory, +including application of activation functions to those values. +In particular, if a connection, the value of its input neuron, +and the partial sum of its output neuron (which we consider +initially the bias) are stored in fast memory, then we can add +the product of the input value and the connection weight to +the partial sum of the output neuron (and we assume that no +additional memory is required for this operation). If we want +to do a computation that requires a value that is currently only +stored in slow memory, we first need to move this value from +slow to fast memory. This movement counts as 1 read-I/O. +Furthermore, if our memory is full, we first need to free up +space before we can read new values. To do this, we can +delete values. However, if we need a value again in future +computation steps or if it is the value of an output neuron, +we have to write the value to slow memory before deleting +it. Deletions are for free whereas each write operation counts +as 1 write-I/O. Initially, all N + W parameters are stored in +slow memory. The number of I/Os of a computation is the sum +of read-I/Os and write-I/Os. The goal is to perform inference +computation, namely compute and store (i.e., write back to +slow memory) the value of all output-neurons, with as few +I/Os as possible. We formally define the optimum that we aim +for: +Definition 1. For a given FFNN N and fast memory size +M, we let I/Os(N, M) ∈ N denote the minimum number +of overall I/Os that are needed to perform inference on +N with a memory size M, where the minimum is taken +over all possible computation strategies (i.e., sequences of +computation-, read-, and write- steps, that solve this problem). +Let rI/Os(N, M) ∈ N denote the minimum number of read- +I/Os and wI/Os(N, M) ∈ N denote the minimum number of +write-I/Os needed. +A. How can we describe inference-computations in this model? +We introduce the concept of eviction policy, and show that +a computation corresponds to an eviction policy together with +a topological order of the connections. An eviction policy is a +set of rules or instructions that specify how we evict data from +fast memory. That is, when the fast memory is full and we +need to evict a value to create free space, the eviction policy +determines which of the values to evict. LRU (least-recently- +used) is the eviction policy defined by always evicting the +value that has been used least recently among all values in +fast memory. RR (round-robin) is the eviction policy defined +by having a pointer specifying the value to be evicted next +and moving this pointer one place to the right whenever we +evict a value (and moving the pointer again to the first place +of the memory when we reach the end). The eviction policy +MIN (also known as Belady’s optimal replacement algorithm) +is defined by always evicting the value that will be referenced +farthest in the future (if there are values that will not be used +again, any of those is evicted). It has been shown that MIN +uses the minimal number of I/Os for a given sequence of +computation steps [32]. Notice that while it is difficult to +implement MIN eviction policies for general computations, +in the case of FFNN inference it is trivial to implement it +offline once we fixed a topological order in which we process +the weights (always evict the activations adjacent to weights +farthest away in the given topological order). +We assume that an efficient eviction policy is provided. That +is, when we evict a value that is either (1) already stored +in slow memory (for example, when we read the value of a +computed neuron to use it for its outgoing connections, but do +not change it), or (2) a value that we will not need again in the +future (a computed non-output-neuron has been used already +for computing all neurons that depend on it), we simply delete +it from fast memory, without spending a write-I/O. +Now, let e1, e2, . . . , eW be a topological order of the +connections of the neural network (that is, whenever ei and +ej are connections for which the output-neuron of ei is the +input-neuron of ej, we have i < j). Together with an eviction +policy, a topological order of the connections gives rise to an +inference-computation in a natural way, shown by Algorithm +1. +Algorithm 1 Inference Algorithm +1: Input: A topological order of the connections e1, . . . , eW +2: Output: Values of the output neurons +3: for i = 1 to W do +4: +Read the connection ei = (a, b, w); +5: +if value of the input neuron, na, is not in fast memory +then +6: +Read na (possibly, first evicting one value, if fast +memory is full); +7: +end if +8: +if partial sum of the output neuron, nb, is not in fast +memory then +9: +Read nb (possibly, first evicting one value, if fast +memory is full); +10: +end if +11: +Update nb = nb + w · na +12: +if there is no connection after ei with output-neuron nb +then +13: +Apply the activation function: nb = f(nb); +14: +end if +15: end for +Notice that this pseudocode generalizes all “standard +ways” of performing inference. For example, matrix-vector- +multiplication based inference would correspond to orders that +start with all connections from the first layer, followed by +the connections of the second layer and so on. Yet, it also + +4 +allows us to perform inference on FFNN-architectures given +by any possible DAG (including those with very “chaotic” skip +connections) and not just those that are layered. +As for optimizing I/Os, it gives us more flexibility, as it +allows us to employ any possible topological order of the +connections, including those that do not correspond to layer- +after-layer computations. For example, it allows us to start +computing neurons of the layers i+1, i+2, . . . even when not +all neurons of the i-th layer have been computed. This can save +I/Os, because when we finish computing a neuron from the +i-th layer, we can directly reuse it to start computing neurons +of the (i + 1)-st layer, instead of storing it and reading it again +later. Since for each connection we perform a computation for +which we need the value of the input neuron and the partial +sum at its output neuron, we would like to find an order in +which connections that have neurons in common, are clustered +together. +III. BOUNDS FOR INFERENCE +In this section we present generic bounds on the I/O- +complexities of inference. They serve as guidelines for the +more instance-specific optimizations that we consider later on. +Theorem 1. Let N be a connected FFNN and assume M ≥ 3. +Then the optimal number of I/Os for inference satisfies +W + N + S ≤ I/Os(N, M) ≤ 2 · (W + N − I). +(1) +The optimal number of read-I/Os for this problem satisfies +W + N ≤ rI/Os(N, M) ≤ 2 · W + N − I. +(2) +The optimal number of write-I/Os for this problem satisfies +S ≤ wI/Os(N, M) ≤ N − I. +(3) +Proof. The weights, biases and input values have a total size +of N + W. Since we cannot perform inference without having +read all of these data at least once, we obtain the lower bound +for the number of read-I/Os. Likewise, we cannot perform +inference with less than S write-I/Os, because (by definition +of the inference problem) we need to write the values of all S +output-neurons. Hence, the lower bound for the total number +of I/Os follows by adding the lower bounds for the read- and +write-I/Os. +Now it remains to show that we can do inference using +no more than N + 2 · W − I read-I/Os, N − I write-I/Os +and 2 · (N + W − I) overall I/Os. To achieve this, we fix +a topological order of the non-input neurons: n1, . . . , nN−I. +Then, we reorder the connections in such a way that their +output neurons appear in the order of this topological order +(notice that this is also a topological order of the connections; +see Figure 1 for an illustration of this association of topological +orderings). Notice that this order is naturally partitioned into +intervals: It begins with an interval of connections ending in +n1, followed by the connections ending in n2 and so on. +We now show that performing inference in this order with +our inference-algorithm (Algorithm 1 from the main paper) and +a MIN eviction policy, costs at most 2W + N − I read-I/Os +and N − I write-I/Os. As we start reading the connections, we +Fig. 1: Associated topological orderings of neurons and +connections. +spend 1 read-I/O to read the bias of n1 and then at most 2 read- +I/Os for each of the connections that end on n1 (1 for reading +the input neuron and 1 for reading the connection itself). Once +we passed through this interval of connections that end on n1, +we apply the activation to finish the computation of neuron n1. +Then we continue with the interval of connections that end +on n2. Also in this interval of connections as well as in all +following intervals of connections, we spend 1 read-I/O for +the bias of the output-neuron and at most 2 on each of the +connections. Since there is exactly one interval of connections +for each of the N − I non-input neurons, we spend at most +N − I read-I/Os to read the biases and at most additional +2W for the connections and their input-neurons, adding up to +2W + N − I read-I/Os. +Now we count the write-I/Os deployed in this computation. +Since all connections that end in the same neuron follow +each other, we always compute neurons without having to +write temporary values (once we start computing a neuron all +of the following computation steps are also directed towards +computing this neuron, and we only start computing another +neuron once the previous is finished). From this it follows, that +if we spend a write-I/O on some neuron, then we are writing a +fully computed neuron value to slow memory. And hence, if we +read and evict this value again, we do not spend another write- +I/O on this value (since the fully computed value is already +stored in slow memory, the efficient eviction policy will evict +this value by deleting it). Therefore, we spend at most one write- +I/O for each of the non-input neurons. Since there are N − I +non-input neurons, we have overall at most N − I write-I/Os. +Adding the read- and write-I/Os, we conclude that the total +number of I/Os is at most N+2·W−I+N−I = 2·(N+W−I). +Notice that in both Inequalities 1 and 2, the term on the +right hand side is at most twice as large as the term on the +left hand side. The next Proposition establishes that none of +the generic bounds given by this theorem could be tightened + +4 +C +6 +35 +any further by multiplying it with a constant other than 1. +Proposition 1. As generic bounds that depend only on W, N, I, +and S, the bounds in Theorem 1 are tight: For each one of +them and for any ϵ > 0, there are instances for which the true +value differs from the bound by a multiplicative factor that lies +in [1 − ϵ, 1 + ϵ]. +To prove this Proposition, we need to show that for each of +the bounds of Theorem 1, there are instances that are arbitrarily +close to the bound (close in terms of multiplicative factors). +The first lemma establishes that there are instances that +exactly attain all lower bounds of Proposition 1 (and hence, +are obviously arbitrarily close). +Lemma 1. On any FFNN N = [L1, L2, . . . , Ld] in which any +two consecutive layers Li, Li+1 have together at most M − 1 +neurons, we can do inference, with a number of read-, write-, +and total I/Os that matches the lower bounds given by Theorem +1. +Proof. We prove this by describing a computation strategy that +achieves this. +Read the |L1| input values into fast memory. Also initialize +in fast memory |L2| partial sums with the biases of the neurons +in L2. Since |L1| + |L2| ≤ M − 1, we have in fast memory +free space for at least one more value. We use this free space +to iterate over the weights between L1 and L2 and perform +the following steps for each one of them: +1) Read the weight. +2) Multiply it with the value of its input neuron (which is +already in fast memory). +3) Add this product to the partial sum of its output neuron. +4) Delete the weight from fast memory. +Once this iteration is finished, we apply the activation function +to the partial sums of L2, which finishes the computation of the +neurons in this layer. Now we do not need the values from L1 +anymore and can delete them from fast memory. This gives us +enough free space to read the biases of L3, and compute this +layer in the same way. We proceed like this, computing layer +after layer. Once we finish the computation of the neuron-vales +of the last layer, we write those S values back to slow memory. +Notice that we read each of the N −I biases, I input values, +and W weights exactly once. Hence we use N + W read-I/Os. +We only use write-I/Os to write the S output values. Hence, +altogether we use N + W + S I/Os. +Alternatively, we could have proved the previous Lemma +by verifying that an FFNN with the mentioned properties, +can be constructed with the Compact Growth method that we +introduce later. The next lemma shows that there are instances +that use a number of read- and total I/Os that are arbitrarily +close to their upper bound. +Lemma 2. For every ϵ > 0, there exist FFNNs N (of arbitrary +size) that have I/Os(N, M) > (1 − ϵ) · 2 · (W + N − I) and +rI/Os(N, M) > (1 − ϵ) · (2 · W + N − I). +Proof. This is the case for architectures, where a large +proportion of the neurons are input neurons. For example, +in a tree with I input neurons, all connected to a single +output neuron, we have I/Os(N, M) = ·2 · (W + N − I) +and rI/Os(N, M) = 2 · W + N − I. +The next lemma shows that there are instances that use +a number of write-I/Os that is arbitrarily close to the upper +bound. +Lemma 3. For every ϵ > 0, there exist FFNNs N (of arbitrary +size) that have wI/Os(N, M) > (1 − ϵ) · (N − I). +Proof. Notice that this inequality is satisfied by any FFNN +for which all neurons are either input or output neurons. But +to give less trivial examples, we will construct FFNNs with +hidden neurons that satisfy this inequality. +For M, h, w ∈ N+, consider the FFNN that has I input +neurons, S output neurons, and one hidden layer with h neurons. +Clearly, this FFNN requires at least S write-I/Os. So, for any +parameter configuration I, h, S with S > h·(1−ϵ)/ϵ, we have +wI/Os(N, M) > (1 − ϵ) · (N − I). +The three previous lemmata provide the tight extremal cases +that bring us into the position to prove Proposition 1. Although +the constructed architectures in the proofs of these lemmata +were “artificial”, they suffice to show that generic bounds that +only depend on I, S, N, and W, cannot be tighter (tighter by a +constant multiplicative factor other than 1) than our bounds. Yet, +notice that, despite being “artificial”, these extremal cases are +more than just “a few cornercases”. In fact, the constructions +in each of these proofs could have been chosen to be arbitrarily +large. +Proof of Proposition 1. Lemma 1 shows that the lower bound +on the number of read-, write-, and total I/Os is tight. Lemma +2 shows that the upper bound on the number of read-I/Os and +total I/Os is tight. Lemma 3 shows that the upper bound on +the number of write-I/Os is tight. +Notice that none of the bounds depends on M. This is +surprising, because for most computational problems, I/O- +complexities depend strongly on the memory size1. As we +increase M, we gain more flexibility that we can use to avoid +I/Os. Also surprising is the fact that these bounds do not depend +on specific properties of the network, except for its sizes. This +does not mean that the overall I/O-complexity does not depend +on these quantities (memory size and FFNN architecture), +but from the tightness of our bounds we conclude that the +dependence is only moderate. +For write-I/Os, this is very different. The bounds for write- +I/Os given by Theorem 1 are less tight. In fact, they can +be arbitrarily loose (despite being optimal in the sense of +Proposition 1). The problem for write-I/Os is that, it is simply +not possible to give interesting bounds that only depend on +W, N, I, and S but no other factors such as the connection +order, the memory size M, and the FFNN architecture (this +impossibility is implied by Proposition 1). The following +Proposition shows that inference in a layer-after-layer fashion +can be arbitrarily more expensive than using an optimal order. +1For example, in the case of matrix multiplication, the dependency on M +is of the order O(1/ +√ +M) [7]. + +6 +Proposition 2. For every c ∈ N and every memory size M ∈ N, +there exists an FFNN NM,c, such that performing inference +in a layer-after-layer fashion, requires at least c times more +write-I/Os than optimal. +Proof. Consider a sparse FFNN that has c+2 layers: 1 neuron +in the input layer, 2M neurons in each of the c hidden layers, +and one neuron in the output layer. Let each hidden neuron +have exactly one incoming and one outgoing connection, let the +input-neuron be connected to each neuron of the first hidden +layer, and let the output-neuron have incoming connections +from all neurons of the last hidden layer (in other words, this +FFNN has 2M chains of neurons of length c + 2 that meet in +the input and output neuron). If we perform inference on this +FFNN in a layer-after-layer fashion, we need the values of all +2M neurons of each hidden layer to compute the next layer. +Since our fast memory has only capacity for M neurons, we +will need to store the other M neurons, requiring at least M +writes for each hidden layer. Hence, if we perform inference +on this FFNN in a layer-after-layer fashion, we would need to +use at least M × c write-I/Os. Yet, if we compute chain after +chain, a single write-I/O would suffice. +IV. CONNECTION REORDERING: ADAPTING THE ORDER TO +THE FFNN AND HARDWARE +In this section we introduce Connection Reordering, which is +a method to optimize the topological order of the connections +for a given FFNN architecture and memory size M. This +method depends on the following hyperparameters: The win- +dow size ws ∈ N, the cooling rate σ ∈ R, and the number +of iterations T ∈ N. The high-level idea of this method is +based on Simulated Annealing [33]: Over T iterations, we +perform random changes to the topological order (we call this +creating neighbors) and either retain or discard the changes +with a probability that depends on the quality of the old and +the new order (called updating). Now, we fully specify this +method by describing the processes of creating neighbors and +updating. +A. Creating neighbors +We start with a topological order of the connections +e1, e2, . . . , eW and let OldI/Os ∈ N denote the number of I/Os +used in this topological order. The number of I/Os obviously +depends on the memory size and eviction policy, but those are +fixed throughout the execution of this algorithm and hence the +number of I/Os depends only on the topological order of the +connections. +First, we choose uniformly at random one connection ei. We +let w be an integer that we choose uniformly at random from +{0, 1, . . . , ws − 1}. We consider the window of connections +ei, ei+1, . . . , emin(i+w,W ). We choose the direction in which +we move the connections from this window: Either left or right +with probability 0.5. +a) Case 1: Moving to the left: If we move the connections +to the left, we start moving the leftmost connection ei of the +window. We move ei to the left until we encounter another +connection es that has the same input neuron as ei, or whose +output neuron is equal to the input neuron of ei (if we +never encounter such a connection, we move ei to the very +beginning of the order). We insert ei right next to es so that +. . . , es, ei, es+1, . . . is the new order. Note that this is again +a topological order. Then we continue moving the second +leftmost connection in the window in the same fashion. We +continue like this until we moved all connections from this +window. +b) Case 2: Moving to the right: If we move the connec- +tions to the right, we start moving the rightmost connection of +the window. We move it until we encounter another connection +ez, which has the same output neuron as ei or whose input +neuron is equal to the output neuron of ei. We insert ei right +before ez so that . . . , ez−1, ei, ez, . . . is the new order. Then +we continue with the second rightmost connection from the +window. We continue like this until we moved all connections +from this window. +B. Updating +After we moved all connections from the window, we have +a new topological order. We denote it ˜e1, ˜e2, . . . , ˜ +eW . We +measure how many I/Os are used to perform inference in this +new order and call this number newI/Os. If newI/Os