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+Approximate Balancing Weights for Clustered Observational Study
+Designs∗
+Eli Ben-Michael†
+Lindsay Page‡
+Luke Keele§
+January 16, 2023
+Abstract
+In a clustered observational study, a treatment is assigned to groups and all units
+within the group are exposed to the treatment. We develop a new method for statisti-
+cal adjustment in clustered observational studies using approximate balancing weights,
+a generalization of inverse propensity score weights that solve a convex optimization
+problem to find a set of weights that directly minimize a measure of covariate imbal-
+ance, subject to an additional penalty on the variance of the weights. We tailor the
+approximate balancing weights optimization problem to both adjustment sets by de-
+riving an upper bound on the mean square error for each case and finding weights that
+minimize this upper bound, linking the level of covariate balance to a bound on the
+bias. We implement the procedure by specializing the bound to a random cluster-level
+effects model, leading to a variance penalty that incorporates the signal signal-to-noise
+ratio and penalizes the weight on individuals and the total weight on groups differently
+according to the the intra-class correlation.
+Keywords: Balancing Weights, Clustered Observational Study, Clustered Data
+∗This research is supported by the Institute of Education Sciences, U.S. Department of Education, through
+Grant R305D210014. The opinions expressed are those of the authors and do not represent views of the
+Institute or the U.S. Department of Education.
+One of the datasets used for this study was purchased
+with a grant from the Society of American Gastrointestinal and Endoscopic Surgeons. Although the AMA
+Physician Masterfile data is the source of the raw physician data, the tables and tabulations were prepared
+by the authors and do not reflect the work of the AMA. The Pennsylvania Health Cost Containment Council
+(PHC4) is an independent state agency responsible for addressing the problems of escalating health costs,
+ensuring the quality of health care, and increasing access to health care for all citizens. While PHC4 has
+provided data for this study, PHC4 specifically disclaims responsibility for any analyses, interpretations or
+conclusions. Some of the data used to produce this publication was purchased from or provided by the
+New York State Department of Health (NYSDOH) Statewide Planning and Research Cooperative System
+(SPARCS). However, the conclusions derived, and views expressed herein are those of the author(s) and
+do not reflect the conclusions or views of NYSDOH. NYSDOH, its employees, officers, and agents make
+no representation, warranty or guarantee as to the accuracy, completeness, currency, or suitability of the
+information provided here. This publication was derived, in part, from a limited data set supplied by the
+Florida Agency for Health Care Administration (AHCA) which specifically disclaims responsibility for any
+analysis, interpretations, or conclusions that may be created as a result of the limited data set.
+†Carnegie Mellon University, Pittsburgh, PA, Email: ebenmichael@cmu.edu
+‡Brown University, Providence, RI, Email: lindsay page@brown.edu
+§University of Pennsylvania, Philadelphia, PA, Email: luke.keele@gmail.com
+1
+arXiv:2301.05275v1  [stat.ME]  12 Jan 2023
+
+1
+Introduction
+In a study of comparative effectiveness, researchers seek to understand whether a treatment
+has a causal effect on an outcome of interest for a set of study units. In the causal infer-
+ence literature, treatment assignment is a critical element of the study design (Rubin, 2007,
+2008). Two key components of treatment assignment are whether the intervention is ran-
+domized or not and whether it is grouped or not. First, when interventions are randomly
+assigned, differences between treated and control groups can be interpreted as causal effects.
+In contrast, when subjects select their own treatments, stronger assumptions are needed
+to identify causal effects. Second, interventions may be assigned to individual units or to
+intact groups. For example, given that students are grouped in schools, a treatment may
+be assigned to all students in some schools and withheld from all students in other schools.
+When treatment is randomly assigned and grouped, the design is often called a clustered
+randomized trial (CRT) (Raudenbush, 1997; Hedges and Hedberg, 2007). In many cases,
+however, treatments are grouped but non-randomly assigned. This design is referred to as
+the clustered observational study (COS) (Pimentel et al., 2018; Page et al., 2020). COS
+designs differ from non-clustered observational studies, both because they rely on different
+identification assumptions and because they require different methods for analysis (Ye et al.,
+2022).
+In a COS design, as is true for any observational study, differences between treated and
+control outcomes may reflect initial differences in the treated and control groups rather
+than treatment effects (Cochran, 1965; Rubin, 1974). As such, analysis for COS designs
+requires statistical adjustment methods to account for observed confounders. Adjustment
+methods for COS designs, however, may need to remove treated and control differences in the
+distributions of covariates at the cluster level, the unit level, or perhaps both levels. Recent
+work has also shown that statistical adjustment in the COS design needs to reflect key
+substantive knowledge of how the clustered treatment is assigned and whether differential
+selection of clusters is present (Ye et al., 2022).
+In short, analysts require a statistical
+adjustment strategy that takes into account key aspects of the COS design.
+In this article, we develop an approximate balancing weights estimator tailored to the
+2
+
+COS context. Balancing weight estimators solve a convex optimization problem to find a set
+of weights that directly minimize a measure of covariate imbalance subject to an additional
+constraint or penalty on the complexity of the weights (Hainmueller, 2011; Zubizarreta,
+2015; Ben-Michael et al., 2021). Approximate balancing weights are a generalization of the
+standard inverse propensity score estimator. Approximate balancing weight methodology,
+however, requires a number of key innovations to be used in the COS context. First, we write
+separate objective functions with different balance measures and variance penalties for the
+two possible estimands in a COS. The first estimand adjusts for both cluster- and unit-level
+covariates while the second adjusts for cluster-level covariates only. For both estimands, we
+derive a bound on the mean square error of a general weighting estimator, and find weights
+that minimize this upper bound, showing that the level of balance between the treated and
+re-weighted control group gives a bound on the bias. Next, we write the variance penalty as
+a cluster-level random effects model, and we show that the variance penalty is comprised of
+two parts. The first component is the signal-to-noise ratio which measures the overall impact
+of the variance relative to bias. The second component is the intra-class correlation. These
+components correspond to hyper-parameters in the corresponding balancing optimization
+problem, and we develop a data-driven approach for selecting them. For cases where it is
+sufficient to condition on cluster-level covariates only, we show that it can be more efficient
+to further adjust for unit-level covariates when they are strongly predictive of the outcome
+or if the intra-class correlation is moderate to high. In applications with poor overlap or
+many covariates, it can be difficult to find weights that achieve good balance. For these
+cases, we develop two separate extensions: using an outcome model to perform additional
+bias correction, and adjusting the balancing optimization problem to find the maximally
+overlapping set between the treated and control group.
+Finally, we derive methods for
+variance estimation and show how to conduct inference based on asymptotic normality. In
+a series of simulations, we find that balancing weights are superior to multilevel matching
+— which is also tailored to the COS design — in terms of bias and variance reduction.
+We provide further comparisons with two empirical applications estimating the effects of
+Catholic schools and surgical training.
+Overall, we find that balancing weights produce
+superior balance relative to extant matching methods and have larger effective sample sizes.
+3
+
+Our article proceeds as follows. In Section 2, we review the details of the COS design.
+In Section 3 we develop approximate balancing weights for the COS design and discuss
+extensions and inference in Section 4. In Section 5, we evaluate our methods in a simulation
+study. In Section 6, we analyze the data from two COS designs: one from education and
+one from health services research. In Section 7, we conclude and discuss directions for future
+work.
+2
+The COS Design
+First, we review the formal aspects of the COS design and how the process of differential
+selection leads to different estimands and adjustment strategies. See Ye et al. (2022) for a
+detailed treatment of identification issues in the COS design.
+2.1
+Notation
+We consider a setup with m clusters, with nℓ units in cluster ℓ, and n = �m
+ℓ=1 nℓ total
+units. We denote the treatment status of cluster ℓ as Aℓ ∈ {0, 1}, n1 ≡ �m
+ℓ=1(1 − Aℓ)nℓ as
+the total number of treated units, and n0 ≡ n − n1 as the total number of control units.
+Each collection of treatment status vectors a = (a1, . . . , am) ∈ {0, 1}m is associated with
+a vector of potential cluster assignments, J(a) = (J1(a), . . . , Jn(a)) ∈ {1, . . . , m}n, where
+Ji(a) ∈ {1, . . . , m} denotes the cluster that unit i would belong to under overall treatment
+allocation a. We denote the observed cluster assignments as J = J(A). Each unit i has a
+potential outcome Yi(aJi(a), J(a)) corresponding to the outcome that would be observed if
+its associated cluster has treatment status aJi and the overall cluster assignment is J(A).
+Note that here we have followed Ye et al. (2022) and assumed that a unit’s potential outcome
+only depends on its own cluster’s treatment status and not the treatment status of other
+clusters, except through the potential cluster assignments. We further assume that potential
+outcomes are independent across clusters — i.e., they are independent for unit i and i′ if
+Ji(a) ̸= Ji′(a) — but may be dependent within clusters. We denote the observed outcomes
+as Yi(AJi, J). We also assume that we observe unit-level covariates Xi ∈ X and that the
+cluster assignments lead to potential cluster-level covariates, Wℓ(J((a))) ∈ W, which may
+include summaries of the unit-level covariates and so can depend on the potential cluster
+assignments. We let Wℓ = Wℓ(J((A))) denote the observed cluster-level covariates, and W
+4
+
+denote the collection of observed cluster-level covariates for all clusters. Taken together, our
+observed data consists of tuples (Xi, Ji, WJi, AJi, Yi).
+Next, we define several conditional expectations of the observed outcomes that we will use
+throughout. First, we denote mw(a, w) ≡ E[Yi | AJi = a, WJi = w] as the expected outcome
+for unit i, conditioned on its cluster having treatment status a and covariates w. Next, we
+use mwx(a, w, x) ≡ E[Yi | AJi = a, WJi = w, Xi = x] to denote the expected outcome if
+we add more information and additionally condition on unit-level covariates x. Next, let
+e(w) = P(Aℓ = 1 | Wℓ = w) denote the propensity score, the probability that a cluster is
+treated, conditioning on cluster-level covariates, and e(w, x) = P(Aℓ = 1 | Wℓ = w, Xi = x)
+denote the propensity score conditioning on cluster and unit-level covariates.
+2.2
+Estimand, Designs, and Assumptions
+Ye et al. (2022) delineate two primary COS designs. The first we denote as the Cluster-Unit
+Design (CUD), and the second we denote as the Cluster-Only Design (COD). Both designs
+focus on a common target causal estimand which is the average treatment effect for the
+treated units under the observed cluster assignments J,
+τ ≡ E [Yi(1, J) | AJi = 1]
+�
+��
+�
+µ1
+− E [Yi(0, J) | AJi = 1]
+�
+��
+�
+µ0
+.
+(1)
+This estimand measures the effect of treatment for units in treated clusters, keeping the
+cluster assignments fixed. The first term, µ1 can be written as the expectation of the observed
+outcomes among units in the treated clusters, µ1 = E[Yi | AJi = 1]. However, identifying and
+estimating µ0, the mean counterfactual outcome if those clusters had in fact been assigned
+to control, is more difficult. Next, we review the key assumptions required for identification
+of this target causal estimand under the two designs.
+2.2.1
+Cluster-Unit Design
+The key distinction between the identification strategies for these two designs is whether the
+units within the clusters respond to treatment assignment to clusters. That is, under the
+CUD, the mix of units within clusters is changed by the fact that clusters are treated. Ye
+et al. (2022) refer to such bias as differential selection. Differential selection could occur if the
+5
+
+student mix in a school changes in response to the school being treated. For example, parents
+may opt to enroll their child in a school implementing a whole-school curricular reform. For
+the CUD, we need to account for unit selection into the clusters, which can depend on the
+treatment assignments. Identification of µ0 under the CUD requires the following set of
+assumptions:
+• For two cluster assignment vectors j and j′ such that the treatment assignments are
+the same, aji = a′
+ji, and the cluster-level covariates are unchanged Wji = Wj′
+i for unit
+i, the potential outcomes are equal Yi(aji, j) = Yi(aj′
+i, j′).
+• Denote Yi(a, w) = Yi(aji = a, Wji = w). For every unit i, cluster ℓ, treatment value a,
+and cluster-level covariates value w, AJi ⊥ Yi(a, w) | WJi, Xi.
+• e(w, x) > 0 for all w, x ∈ W, X.
+Taken together, these assumptions form an ignorability assumption that includes both
+cluster-level covariates and unit-level covariates that drive treatment selection. Here, we
+identify µ0 as
+µ0 = E[mw(0, WJi, Xi) | AJi = 1] ≡ µ0wx.
+2.3
+Cluster-Only Design
+For the COD, treatment assignment is restricted to cases where either treatment assignment
+occurs after the unit-cluster pairing, or units are blinded to the cluster assignments before
+pairing. As such, treatment assignment only depends on cluster-level covariates. Identifica-
+tion of µ0 under the COD requires the following set of assumptions:
+• Cluster assignments are not affected by treatment: Jℓ(a) = Jℓ for all a ∈ {0, 1}m.
+• For every unit i, cluster ℓ, cluster assignment vector j, and treatment value a, AJi ⊥
+(J, Xi, Yi(a, j)) | WJi.
+• e(w) > 0 for all w ∈ W.
+Here, conditioning on the cluster-level covariates is sufficient to remove confounding. As
+such, under the COD, the estimated for the expected outcome conditional on the cluster
+6
+
+being assigned to the control condition and the cluster-level covariates is identified as:
+µ0 = E[mw(0, WJi) | AJi = 1] = µ0w.
+The CUD requires a stronger assumption about the form of the potential outcomes but
+a weaker assumption on the treatment selection process than the COD. However, the key
+distinction between these two identification strategies is the conditioning sets. For the CUD,
+we must condition on both unit and cluster-level covariates. For the COD, we only need
+to condition on cluster-level covariates. As such, the COD is a special case of the CUD
+without unit-level covariates. Below, we consider the implications of conditioning on unit-
+level covariates when the assumptions of the COD hold, and we show that while unit-level
+covariates are unnecessary for identification, they may be beneficial in terms of increased
+precision. Next, we tailor approximate balancing weight methods to each design.
+3
+Approximate Balancing Weights for the COS Design
+Recent work has developed matching methods—known as multilevel matching—that are
+specifically tailored to the COS design (Keele and Zubizarreta, 2017; Pimentel et al., 2018;
+Keele et al., 2021). Here, we outline approximate balancing weights as an alternative to mul-
+tilevel matching. Approximate balancing weights are a generalization of the standard inverse
+propensity score (IPW) estimator that solve a convex optimization problem to find a set of
+weights that directly minimize a measure of covariate imbalance subject to an additional
+constraint or penalty on the complexity of the weights (Hainmueller, 2011; Zubizarreta,
+2015). Like matching, and unlike IPW, balancing weights are designed to directly target
+covariate balance in the estimation process, as opposed using the estimated probability of
+being selected for treatment. Like weighting, and unlike matching, balancing weights use
+more of the available data and so often have larger effective sample sizes.
+Recent theoretical innovations have bolstered support for balancing weights. For example,
+it has been shown that balancing weights are implicitly estimates of the inverse propensity
+score, fit via a loss function that guarantees covariate balance (Zhao and Percival, 2016;
+Zhao, 2019; Wang and Zubizarreta, 2019; Chattopadhyay et al., 2020). Researchers have
+proposed non-parametric extensions that allow for flexible specifications of the outcome
+7
+
+model or propensity score (Hirshberg et al., 2019; Hazlett, 2019). Finally, recent literature
+on weighting allows these estimators to target different quantities such as sample overlap (Li
+et al., 2018). See Ben-Michael et al. (2021) for a general overview on balancing weights.
+Approximate balancing weights for the COS design, however, require the development of
+specific objective functions that are tailored to the identification conditions outlined above.
+Next, we develop objective functions that target measures of covariate imbalance and vari-
+ance penalties that are specific to COS designs. As we will see, one key challenge in develop-
+ing objective functions for COS designs is accounting for how clustering affects the variance
+of the weights. Below, we consider a generic weighting estimator of the form
+ˆµ0(γ) = 1
+n1
+m
+�
+ℓ=1
+(1 − Aℓ)
+�
+Ji=ℓ
+γiYi,
+where the weights γi are independent of the outcomes. We will first consider conditioning
+on cluster- and unit-level covariates to target µ0wx. Then we will specialize this estimator
+to only condition on cluster-level covariates and target µ0w.
+For both cases, we inspect
+the mean square error of the generic weighted average of control units’ outcomes, then find
+weights that minimize an upper bound. Under both settings, we will consider the finite
+sample estimands:
+˜µ0wx ≡ 1
+n1
+�
+Aℓ=1
+�
+Ji=ℓ
+mwx(0, WJi, Xi),
+and
+˜µ0w ≡ 1
+n1
+�
+Aℓ=1
+nℓmw(0, Wℓ),
+which converge to our main estimands.
+3.1
+Conditioning on cluster- and unit-level covariates
+To target µ0wx, we use the following decomposition for the estimation error of the weighting
+estimator:
+ˆµ0 (γ) − ˜µ0wx = 1
+n1
+�
+Aℓ=0
+�
+Ji=ℓ
+γimwx(0, Wℓ, Xi) − 1
+n1
+�
+Aℓ=1
+�
+Ji=ℓ
+mwx(0, Wℓ, Xi) + 1
+n1
+�
+Aℓ=0
+�
+Ji=ℓ
+γiεi.
+(2)
+To understand this decomposition, we compute the design-conditional bias and variance.
+First, since the weights are independent of the outcomes, under the ignorability condition
+8
+
+for the CUD, the conditional bias corresponds to the first term in the decomposition:
+E [ˆµ0 (γ) − ˜µ0wx | W , A, J, X] = 1
+n1
+�
+Aℓ=0
+�
+Ji=ℓ
+γimwx(0, Wℓ, Xi) − 1
+n1
+�
+Aℓ=1
+�
+Ji=ℓ
+mwx(0, Wℓ, Xi).
+(3)
+This bias is the post-weighting imbalance in a particular function of the cluster- and unit-
+level covariates: the conditional expectation function mwx(0, ·, ·). If we knew mwx, then
+we could remove all bias by ensuring that this function is balanced. Unfortunately we do
+not know it, or else we would be able to perfectly impute the missing potential outcomes.
+Instead, following Hirshberg et al. (2019) and Ben-Michael et al. (2021), we consider a class
+of potential models Mwx and consider the worst case bias over this class:
+imbalanceMwx(γ) = max
+m∈Mwx
+�����
+1
+n1
+�
+Aℓ=0
+�
+Ji=ℓ
+γim(0, Wℓ, Xi) − 1
+n1
+�
+Aℓ=1
+�
+Ji=ℓ
+m(0, Wℓ, Xi)
+����� .
+Here, since we are conditioning on both unit and cluster-level covariates, the model class
+Mwx can be relatively complex. For example, it might include interactions between WJi and
+Xi.
+Next, the design-conditional variance corresponds to the second term:
+V unit ≡ Var (ˆµ0 (γ) | W , A, J, X) = 1
+n2
+1
+�
+Aℓ=0
+��
+Ji=ℓ
+γ2
+i Var (εi) +
+�
+Ji=ℓ
+�
+Jk=ℓ,k̸=i
+γiγjCov(εi, εk)
+�
+,
+(4)
+where εi = Yi − mwx(0, WJi, Xi) is the residual in the observed outcomes after conditioning
+on cluster- and unit-level covariates. Note that due to potential correlation in the outcomes
+within clusters, the variance term includes a cross-term involving the product of pairs of
+weights on each unit, in addition to the typical squared weight term used in settings where
+the weights are independent. Taken together, we try to find weights that minimize an upper
+bound on the design-conditional mean square error:
+min
+γ
+imbalanceMwx(γ)2 + 1
+n2
+1
+�
+Aℓ=0
+��
+Ji=ℓ
+γ2
+i Var (εi) +
+�
+Ji=ℓ
+�
+Jk=ℓ,k̸=i
+γiγjCov(εi, εk)
+�
+.
+(5)
+Implementing the optimization of Equations (9) requires making several modeling choices.
+9
+
+First, we must choose the model class for the conditional expectation function Mwx. Second,
+we must specify the variances and covariances of the residuals. In making these choices, we
+will attempt to strike a balance between flexibility and practicality.
+We begin by selecting the model class for Mwx. In the non-clustered treatment assign-
+ment setting, many model classes have been considered, from sparse models (Zubizarreta,
+2015) to models with expanding basis functions (Wang and Zubizarreta, 2020) to reproduc-
+ing kernel Hilbert spaces (Hirshberg et al., 2019). See Ben-Michael et al. (2021) for a recent
+review and discussion on the difficulty of balancing these model classes. To target µ0wx in
+the COS design, we consider a model class that is linear in a transformation of the covariates
+and incorporating both cluster- and unit-level covariates with L2-bounded coefficients and
+an unbounded intercept, i.e.:
+Mwx = Ψwx ≡ {α + β · ψ(w, x) | ∥β∥2 ≤ Cwx, α ∈ R}.
+These model classes naturally allow for a non-parametric extension to reproducing kernel
+hilbert spaces with infinite-dimensional transformations via the “kernel” trick, e.g., defining
+a kernel k((w1, x1), (w2, x2)) = ψ(w1, x1)·ψ(w2, x2). Notably, these transformations, ψ(w, x),
+can include interactions between the two levels of variables, allowing for the cluster context
+to affect the relationship between the outcome and the unit-level covariates.
+To specify the variances and covariances of the residuals, we use a random effects model.
+Momentarily abusing notation, we write the residual between unit i’s outcome and its ex-
+pected control outcome conditional on cluster- and unit-level covariates as Yi−mwx(WJi, Xi) =
+δJi + εi, with a cluster-level random effect δJi and an independent unit-level residual εi. We
+parameterize the variance with two terms: Var(δJi) = σ2ρ, and Var(εi) = σ2(1 − ρ). Under
+the random-effects model, σ2 represents the total residual variance, and ρ is the intra-class
+correlation (ICC), which is a well-known measure of relatedness within clusters. Under this
+random effects model, the variance is
+σ2
+n2
+1
+�
+Aℓ=0
+�
+�(1 − ρ)
+�
+Ji=ℓ
+γ2
+i + ρ
+��
+Ji=ℓ
+γi
+�2�
+� .
+10
+
+We then find the weights, γi that solve the following optimization problem:
+min
+γ
+�����
+1
+n1
+�
+Aℓ=0
+�
+Ji=ℓ
+γiψ(Wℓ, Xi) − 1
+n1
+�
+Aℓ=1
+�
+Ji=ℓ
+ψ(Wℓ, Xi)
+�����
+2
+2
++ σ2
+C2
+wx
+1
+n2
+1
+�
+Aℓ=0
+�
+�(1 − ρ)
+�
+Ji=ℓ
+γ2
+i + ρ
+��
+Ji=ℓ
+γi
+�2�
+�
+subject to
+�
+Aℓ=0
+�
+Ji=ℓ
+γi = n1
+and
+L ≤ ¯γℓ ≤ U,
+(6)
+where we have included some additional optional upper and lower bounds on the weights.
+This objective function implements the general balancing weights problem in Equation (5)
+with the model class Ψwx and a random-effects model. The variance penalty includes two
+hyperparameters. First, there is the noise to signal ratio, σ2/C2
+w, that measures the overall
+impact of the variance relative to the bias in the MSE. If this ratio is small, then better
+balance will be prioritized over lower variance; if the ratio is large then the opposite is true.
+The second hyperparameter is the ICC, which determines the level of penalization for each
+cluster. If the ICC is small and units’ outcomes are nearly uncorrelated, then the weight on
+each unit will be penalized the same. Conversely, if the ICC is large then the outcomes are
+very correlated within clusters, the penalization focuses on the total weight assigned to each
+cluster instead. We discuss setting these hyperparameters below. Note that this specification
+of the variance penalty is equivalent to Rubinstein et al. (2022), who consider region-level
+policy analysis via weighting.
+The weights are also constrained to sum to the total number of treated units, and to be
+bounded between a lower bound L and an upper bound U. The former constraint comes
+from the possibility of an unbounded intercept in the model class Φwx; by ensuring the
+sum constraint we can be sure that the estimator is invariant to constant shifts in the
+outcome. The latter constraint acts as a form of regularization, and allows us to perform
+typically post-hoc adjustments such as weight trimming directly when finding our weights.
+If L = 0 and U = ∞, the estimator will be restricted from extrapolating away from the
+support of the control data (see Ben-Michael et al., 2021, for further discussion on the role
+of extrapolation). If we additionally set U to be non-infinite, we can prevent any weights
+from becoming extremely large, at the cost of balance.
+11
+
+3.2
+Conditioning on cluster-level covariates only
+Next, we consider estimating µ0w, which does not condition on unit-level covariates. This
+weighting estimator ignores the unit-level information, with weights that are constant within
+clusters, i.e. γclus
+i
+= ¯γJi for all units i. Here, the results are a special case of the CUD,
+removing the unit-level covariates from the optimization procedure. The design-conditional
+bias and variance in this special case are:
+E
+�
+ˆµ0
+�
+γclus�
+− ˜µ0w | W , A, J
+�
+= 1
+n1
+�
+Aℓ=0
+nℓ¯γℓmw(0, Wℓ) − 1
+n1
+�
+Aℓ=1
+nℓmw(0, Wℓ),
+(7)
+V clus ≡ Var
+�
+ˆµ0
+�
+γclus�
+| W , A, J
+�
+= 1
+n2
+1
+�
+Aℓ=0
+¯γ2
+ℓ Var
+��
+Ji=ℓ
+ei | J
+�
+.
+(8)
+Here, the bias only depends on imbalance in a function of the cluster-level covariates, mw.
+As above, we consider a model class Mw and upper bound the bias by worst-case imbalance:
+imbalanceMw(¯γ) ≡ max
+m∈Mw
+�����
+1
+n1
+�
+Aℓ=0
+nℓ¯γℓm(0, Wℓ) − 1
+n1
+�
+Aℓ=1
+nℓm(0, Wℓ)
+����� .
+Comparing the variances V clus and V unit, we see that restricting to cluster-level covariates
+only removes the cross term between weights in the same cluster. So V clus depends principally
+on the sum of the squared weights, weighted by the total variance of the outcomes in each
+cluster.
+Tailoring Equation (5) yields the following optimization problem to control the
+MSE:
+min
+¯γ
+imbalanceMw(¯γ)2 + 1
+n2
+1
+�
+Aℓ=0
+¯γ2
+ℓ Var
+��
+Ji=ℓ
+ei | J
+�
+.
+(9)
+To implement this optimization procedure, we will use the same model class and variance
+model as above, removing the unit-level covariates. For the model class, we use the set of
+models that are linear in transformations of the cluster-level covariates:
+Mw = Φw ≡ {α + β · φ(w) | ∥β∥2 ≤ Cw, α ∈ R}.
+For the variance model, we again assume that the residual decomposes into a cluster-level
+12
+
+residual and an independent unit-level residual: Yi − mw(0, WJi) = ei + dJi, with Var(ei) =
+s2(1 − r) and Var(dℓ) = s2r. Here, s2 represents the total variance in the residual, and r is
+the ICC. Now, the variance can be written as
+s2
+n2
+1
+�
+Aℓ=0
+¯γ2
+ℓ ((1 − r)nℓ + rn2
+ℓ).
+Putting together the pieces, we find weights ¯γ that solve the following optimization problem
+min
+¯γ
+�����
+1
+n1
+�
+Aℓ=0
+nℓ¯γℓφ(Wℓ) − 1
+n1
+�
+Aℓ=1
+nℓφ(Wℓ)
+�����
+2
+2
++ s2
+C2
+w
+1
+n2
+1
+�
+Aℓ=0
+¯γ2
+ℓ ((1 − r)nℓ + rn2
+ℓ)
+subject to
+�
+ℓ
+�
+Aℓ=0
+nℓ¯γℓ = n1 and
+L ≤ ¯γℓ ≤ U.
+(10)
+The cluster-level balancing problem in Equation (10) and the unit-level problem in Equa-
+tion (6) share many similarities, including the constraints on the weights and the noise-to-
+signal ratio.1 However, the key difference is whether the weights can differ within clusters in
+order to balance the additional unit-level covariates. This is most apparent in how the ICC
+affects the variance penalty. When only cluster-level covariates are included, the ICC only
+appears in the role that the cluster size nℓ has in the variance penalty. When weights are
+allowed to differ by unit, however, this creates more options in how the weights are penalized.
+3.3
+Hyperparameter selection
+Both of the objective functions specified above include two terms that we consider hyper-
+parameters: the overall noise to signal ratio, (s2/C2
+w or σ2/C2
+wx) and the ICC. The noise
+to signal ratio governs the bias-variance tradeoff. As it approaches zero, more and more
+emphasis is placed on achieving covariate balance; in the (unreasonable) noiseless limit, we
+would not need or want to penalize the variance. On the other hand, as the noise to signal
+ratio increases and the covariates are less predictive of the outcome, the optimization will
+prioritize variance more. In contrast, the ICC term determines the form of the variance
+penalty. In both cases, when the ICC is 0, the weights on each unit are penalized separately
+since units’ outcomes are independent. When the ICC is 1, units’ outcomes completely move
+1The noise-to-signal ratio will in general be different when unit-level covariates are included.
+13
+
+together within a group, in which case the total weight placed on the group is penalized.
+Intermediate values of the ICC correspond to a mixture of these penalization schemes.
+Note that the particular values of the noise to signal ratio and the ICC only enter the
+estimator through the variance penalties on the weights in Equations (6) and (10). This is
+why we view them as hyperparameters in the optimization problem rather than parameters
+to be estimated. However, we can use the data at hand to guide our choice of these hyper-
+parameters. We do so by regressing the outcome on the covariates with a random intercept
+model. The random intercept model estimates cluster- and unit-level variance terms that
+can be used as an estimate for the ICC. To estimate the signal-to-noise ratio, we take the
+ratio of the estimated residual variance and the squared sum of the estimated regression
+coefficients from the fitted model. Below, we use this heuristic to choose the hyperparame-
+ters in our simulation studies and empirical analyses. Note that this procedure can induce a
+dependence between the outcomes and the weights through the hyper-parameters, which can
+be avoided via sample splitting. Other forms of hyperparameter selection are possible, in-
+cluding cross-validation style approaches that evaluate balance on a held-out sample (Wang
+and Zubizarreta, 2020).
+3.4
+When should we include unit-level covariates in the Cluster-Only Design?
+Given that under the COD it is sufficient to only include cluster-level covariates, one natural
+question for this design is whether it is useful to also include unit-level covariates. Including
+unit-level covariates may decrease the variance: balancing these covariates is akin to adjust-
+ing for baseline covariates that predict the outcome in randomized experiments. On the other
+hand, balancing unit-level covariates may lead to more extreme weights, and, depending on
+the ICC, this could lead to higher variance.
+To characterize this trade off, we consider the ratio of the design conditional variance
+with and without including unit-level covariates, V unit and V clus, respectively.
+In cases
+with excellent balance in both cluster- and unit-level covariates, this will give a measure
+of the expected difference in the overall mean squared error. To simplify the problem, we
+restrict our attention to the cluster-level random effects model, and assume that the ICC is
+unchanged when conditioning on unit-level covariates (i.e., ρ = r). Under these assumptions,
+14
+
+the variance ratio is
+V unit
+V clus = σ2
+s2 deff(ρ),
+(11)
+where
+deff(ρ) = (1 − ρ) �
+Aℓ=0
+�
+Ji=ℓ γ2
+i + ρ �
+Aℓ=0
+��
+Ji=ℓ γi
+�2
+(1 − ρ) �
+Aℓ=0 nℓ¯γ2
+ℓ + ρ �
+Aℓ=0 (nℓ¯γℓ)2
+is the design effect of including unit-level covariates.
+The first term, σ2/s2 < 1 represents the reduction in total variance in the residuals by
+conditioning on unit-level covariates, and serves to reduce the variance ratio by incorporat-
+ing additional information. The more predictive the unit-level variables are, the lower we
+expect this ratio to be. Counteracting this, we have the design effect deff(ρ). Rearranging
+Equation (11), we see that in order for it to be beneficial to include unit-level covariates,
+the variance after conditioning on unit-level covariates must decrease by at least the design
+effect, σ2 < s2deff(ρ).
+For any given value of ρ, we expect the cluster-level weights to be less extreme, because
+they do not have to additionally balance the unit-level covariates, so typically deff(ρ) ≥ 1.
+Whether the sum of the squared weights on the units or on the clusters matters more to the
+design effect depends on the ICC. When ρ is small, there is little effect of clustering on the
+variance, and so we have a standard tradeoff: including unit-level covariates will increase
+efficiency as long as the effective sample size does not decrease by more than 1 − σ2/s2.
+Conversely, when ρ is large, the sum of the squared weights on the clusters will dominate,
+in which case the unit-level weights can differ substantially from their average cluster-level
+weight without incurring much extra variance. In this case including unit-level covariates can
+improve efficiency even if they are only somewhat predictive. Hedges and Hedberg (2007)
+reported that from 41 clustered randomized experiments in education, the ICCs range from
+0.07 to 0.31, with an average value of 0.17. Small et al. (2008) report that ICCs in the
+range of .002 to 0.03 in public health interventions that target clusters such as hospitals or
+clinics. Thus, in education settings — where ICCs are larger, and we typically have strongly
+predictive covariates such as prior test scores — it will likely be beneficial to include unit-
+level covariates. In contrast, for public health settings it may depend much more on the
+particular context and the set of unit-level covariates available.
+15
+
+4
+Extensions and Inference
+4.1
+Dealing with poor balance
+When there are a large number of cluster-level covariates relative to the total number of
+clusters — or conversely, when there is a small number of clusters overall — it may be
+difficult to find weights that achieve good balance.
+Often this occurs when the overlap
+between the treated and control cluster covariate distributions is limited (Keele et al., 2022).
+This is true for the application in Section 6.1 below. We consider two different approaches
+to account for this. First, we outline using an outcome model to correct for the bias due
+to remaining imbalance. Second, we consider changing the estimand by also weighting the
+treated units to find a maximally sized overlapping set between the treated and control units.
+4.1.1
+Bias correction with an outcome estimator
+Bias correction, sometimes also called augmentation, is a popular approach to reducing bias
+due to imbalance.
+We describe the bias-correction procedure when only including both
+cluster- and unit-level covariates; restricting to cluster-level covariates alone will be analo-
+gous. First we estimate the conditional expectation function to get estimates ˆmwx(0, WJi, Xi).
+There are many possible estimation strategies, one choice being regularized regression. Then
+we perform bias-correction by estimating µ0 as
+ˆµbc
+0 (γ) ≡ ˆµ0(γ) + 1
+n1
+�
+Aℓ=1
+�
+Ji=ℓ
+ˆmwx(0, Wℓ, Xi) − 1
+n1
+�
+Aℓ=0
+�
+Ji=ℓ
+γi ˆmwx(0, Wℓ, Xi).
+(12)
+If we compare this to the bias in Equation (2), we see that ˆµbc
+0 (ˆγ) uses the estimated model to
+estimate the bias due to imbalance after weighting. Then it attempts to remove the bias by
+subtracting the estimated bias off of the estimate. In the bias-variance decomposition above,
+bias correction changes the imbalance measure to be over the worst-case model error ˆm−m,
+which will generally be smaller than the imbalance in the worst-case model (Hirshberg and
+Wager, 2021). This bias correction is akin to what has been proposed for matching estimators
+(Abadie and Imbens, 2011) and augmented IPW (Robins et al., 1994), and has been used
+with a variety of balancing weights estimators (e.g. Athey et al., 2018; Ben-Michael et al.,
+2021).
+16
+
+4.1.2
+Subset Weights: Finding a maximally overlapping set
+One issue with bias correction based on an outcome model is that the additional use of
+modeling can lead to extrapolation away from the control units’ data (Ben-Michael et al.,
+2021). One alternative to outcome modeling that can be attractive, especially if overlap is
+limited, is to focus on a more limited estimand than the ATT. Specifically, one such estimand
+is the average treatment effect for the overlap population (ATO). The ATO corresponds
+to the treatment effect for the marginal population that might or might not receive the
+treatment of interest rather than a known, a priori well-defined population such as the
+treated group. Li et al. (2018) developed model-based overlap weights for the ATO that
+continuously down-weight the units in the tails of the propensity score distribution. We
+develop an analog with balancing weights for the COS setting.
+To do so, we will weight the treated units as well as the control units, leading to an
+estimator
+ˆτ(γ) = 1
+n1
+�
+Aℓ=1
+�
+Ji=ℓ
+γiYi − 1
+n0
+�
+Aℓ=0
+�
+Ji=ℓ
+γiYi.
+(13)
+By weighting the treated units as well as the control units, we are no longer estimating
+the expected effect among the treated units. We are instead trimming the estimand. To
+understand what the new estimand is, consider the design conditional expectation:
+E [ˆτ(γ) | W , J, X] = 1
+n1
+�
+Aℓ=1
+�
+Ji=ℓ
+γimwx(0, Wℓ, Xi)− 1
+n0
+�
+Aℓ=0
+�
+Ji=ℓ
+γimwx(0, Wℓ, Xi)+ 1
+n1
+�
+Aℓ=1
+�
+Ji=ℓ
+γiτ(Wℓ, Xi),
+where τ(w, x) = E[Yi(1, J)−Yi(0, J) | WJi = w, Xi = x] is the conditional average treatment
+effect (CATE). As before, we want to find weights that balance the conditional expected con-
+trol outcome. Then ˆτ(γ) will be an unbiased estimator of a weighted average of conditional
+treatment effects for the treated group. We will once again try to find weights that minimize
+the worst case balance across a model class Mwx, now also weighting the treated units:
+imbalanceo
+Mwx(γ) = max
+m∈Mwx
+�����
+1
+n1
+�
+Aℓ=0
+�
+Ji=ℓ
+γim(0, Wℓ, Xi) − 1
+n0
+�
+Aℓ=1
+�
+Ji=ℓ
+γim(0, Wℓ, Xi)
+����� .
+17
+
+The design-conditional variance of ˆτ(γ) is
+Var (ˆτ(γ) | W , A, J, X) =
+m
+�
+ℓ=1
+�Aℓ
+n1
++ 1 − Aℓ
+n0
+�2 ��
+Ji=ℓ
+γ2
+i Var (εi) +
+�
+Ji=ℓ
+�
+Jk=ℓ,k̸=i
+γiγjCov(εi, εk)
+�
+,
+where we generalize the definition of the residual to be εi = Yi−mwx(AJi, WJi, Xi). With this,
+we once again try to find weights that minimize the imbalance and the variance. Focusing on
+the specialization to the constrained linear transformation model class Ψwx and the random
+effects variance model, we find weights γi that solve:
+min
+γ
+�����
+1
+n0
+�
+Aℓ=0
+�
+Ji=ℓ
+γiψ(Wℓ, Xi) − 1
+n1
+�
+Aℓ=1
+�
+Ji=ℓ
+γiψ(Wℓ, Xi)
+�����
+2
+2
++ σ2
+C2
+wx
+m
+�
+ℓ=1
+�Aℓ
+n1
++ 1 − Aℓ
+n0
+�2
+�
+�(1 − ρ)
+�
+Ji=ℓ
+γ2
+i + ρ
+��
+Ji=ℓ
+γi
+�2�
+� ,
+subject to
+�
+Aℓ=0
+�
+Ji=ℓ
+γi = n0
+and
+�
+Aℓ=1
+�
+Ji=ℓ
+γi = n1
+and
+L ≤ ¯γℓ ≤ U.
+(14)
+This optimization problem tries to find weights on both treated and control units such
+that the weighted average of their transformed covariates are similar. Hereafter, we refer
+to these weights as subset weights, since they weight the subset of the data for which the
+covariate distributions overlap. The variance penalty in Equation (14) serves to ensure that
+this weighted subset is pushed towards having a larger effective number of units. Finally,
+Equation (14) is also related to the Lagrangian dual of the SVM problem. Tarr and Imai
+(2021) show that the SVM dual minimizes the same measure of imbalance, but includes a
+penalty to ensure that the sum of the weights is not small, leading to a different measure of
+the “size” of the overlapping set.
+4.2
+Variance Estimation and Uncertainty Quantification
+We now turn to constructing asymptotically valid confidence intervals for µ0 by estimating
+the variance of ˆµ0(ˆγ) − µ0 and relying on asymptotic normality. When estimating the vari-
+ance, it is important to account for dependence within clusters. One method for variance
+estimation in this context is the cluster-robust sandwich estimator (Huber, 1967; White,
+18
+
+1980), which we adapt here. Focusing on the CUD first, we estimate the conditional expec-
+tation function ˆmwx(0, w, x), then compute unit-level residuals ˆεi ≡ Yi − ˆmwx(0, WJi, Xi).
+This leads to a plug-in estimator for the variance:
+ˆV unit ≡ 1
+n2
+1
+�
+Aℓ=0
+��
+Ji=ℓ
+γ2
+i ˆε2
+i +
+�
+Ji=ℓ
+�
+Jk=ℓ,k̸=i
+γiγjˆεiˆεk
+�
+.
+Assumption ?? in the Supplementary Materials lists regularity conditions for asymptotic
+inference, based on conditions from Hansen and Lee (2019). Importantly, these regularity
+conditions allow the cluster sizes to grow, but limit the number of clusters.
+Here we highlight the dependence of the variances on the sample size by indexing them by
+n. To state the asymptotic normality result below, define µl ≡ �
+Ji=ℓ
+�
+Jk=ℓ ˆγiˆγkE [εi ˆm(0, Wℓ, Xi) | W , A, J, X]
+as the covariance between the true residuals and the estimated model predictions in cluster
+ℓ.
+Theorem 1. Under the regularity conditions in Assumption ??, with constant lower and
+upper bounds L and U in Equation (6), if imbalanceMwx(ˆγ) = op
+�
+1/
+�
+V unit
+n
+�
+, then
+1
+�
+V unit
+n
+(ˆµ(ˆγ) − ˜µ0wx) ⇒ N(0, 1).
+Furthermore, if supw,x | ˆm(0, w, x) − m(0, w, x)| = op(1) and
+1
+n2
+1
+�
+Aℓ=0 µl → 0 then
+ˆV unit
+V unit
+n
+→ 1
+in probability, and consequently,
+1
+�
+ˆV unit
+n
+(ˆµ(ˆγ) − ˜µ0wx) ⇒ N(0, 1).
+Theorem 1 implies that we can construct approximate 1 − α confidence intervals as
+ˆµ0(ˆγ)±z1−α/2
+�
+ˆV unit, where z1−α/2 is the 1−α quantile of the standard normal distribution.
+The key assumption is that the weights ˆγ can achieve good balance in the sense that the
+worst-case imbalance imbalanceMwx(ˆγ) converges to zero faster than 1/
+√
+V unit. This rate
+depends on the correlation within clusters. If units are uncorrelated, it will scale with the
+total number of units; if units are perfectly correlated then it will scale with the number
+of clusters; see Hansen and Lee (2019) for further discussion on rates of convergence with
+19
+
+clustered data. Whether the weights can actually achieve this level of balance depends on
+the model class. Hirshberg et al. (2019) show that for Reproducing Kernel Hilbert Spaces
+with i.i.d. data, the imbalance will be small enough, while Hirshberg and Wager (2021)
+show that the bias-corrected estimator will have small enough bias in more general settings.
+See Ben-Michael et al. (2021) for further discussion. Finally, Theorem 1 assumes that the
+estimated model ˆm is consistent and that the covariance between the true residuals and
+the estimated model predictions converges to zero, ensuring that the variance estimator is
+consistent. The latter can be guaranteed by using sample-splitting or cross fitting, or by
+using models with restricted complexity.
+Below, we estimate ˆm via regularized weighted least squares on the control units, using
+the weights from Equation (10) above. If we only include an intercept and exclude all of the
+covariates in the regression, then these variance estimates are equivalent to the cluster-robust
+sandwich estimates of a weighted mean. However, this approach may be conservative as it
+disregards the variance reduction due to balancing the covariates.
+Finally, by including a separate regression for the treated units, we can also compute a
+plug in estimate of the variance for the maximal overlap estimator ˆτ(γ) above. In addition, to
+construct a variance estimate for the COD we can follow the same procedure: (i) estimate the
+conditional expectation function ˆmw(0, w); (ii) estimate the residuals ˆεi ≡ Yi − ˆmw(0, WJi);
+and (iii) create a plug-in estimate for the variance:
+ˆV clus ≡ 1
+n2
+1
+�
+Aℓ=0
+¯γ2
+ℓ
+�
+Ji=ℓ
+�
+Jk=ℓ
+ˆeiˆek.
+5
+Simulation Study
+Next, we conduct a simulation study to understand the performance of balancing weights
+using a simulation design from Keele et al. (2021) developed to evaluate multilevel matching.
+First, we describe the data generation process (DGP), which we alter slightly to manipu-
+late the level of overlap between the treated and control distributions. The DGP partially
+depends on empirical data from a summer school reading intervention that follows the COS
+template. In the data, there are 18 treated schools with 1,367 students, and 26 control
+20
+
+schools with 2,060 students. There are 5 student-level variables and 9 school-level variables.
+The student-level variables are reading and math test scores and indicator variables for race,
+ethnicity, and sex. The school-level covariates include the percentage of students who re-
+ceive free/reduced price lunch, who are English language learners, and who are proficient in
+math and reading based on state standardized tests. The school-level covariates also include
+the share of teachers who are novice (e.g., in their first year), the rate of year-to-year staff
+turnover, and student average daily attendance.
+For this DGP, we first fit a school-level propensity score model where we regressed the
+observed treatment indicator on the following set of school level variables: the percentage
+of students receiving free/reduced price lunch, the percentage of students who are English
+language learners, the percentage of teachers who are novice, and student average daily atten-
+dance. We denote this estimated propensity score as ˆe(w). We define the latent probability
+of treatment as:
+Z∗ = (ˆe(w)/c) + Unif(−.5, .5)
+where c controls the level of overlap between treated and control clusters. Observed treatment
+status is generated via the following model: Zj = 1(Z∗ > 0.25).
+Next, we fit an outcome model for the observed data. Here, we regressed reading scores
+on the student-level covariates with the basis expanded to include interactions between race
+and test scores. After model fitting, we save ˆβ0, the intercept from this regression. We use
+τ to denote the true treatment effect estimate in the simulation, and set it to be 0.3 of a
+standard deviation of the raw outcome measure. We denote student-level reading scores with
+Rij, student-level math scores as Mij, and the percentage of students proficient in math and
+reading in each school with Pj. Next, we generate potential outcomes under control as:
+y0 = ˆβ0 + 2.5Rij + 2.5Mij + 1.9Pj + v1,
+where v1 is a draw from a normal distribution that is mean zero with a standard deviation
+of 12. We selected these parameter values so that misspecification would produce a bias of
+approximately 0.3 standard deviations on a standardized scale. A bias of this magnitude is
+large enough to completely obscure the true treatment effect. Next, we generated potential
+21
+
+outcomes under treatment as y1 = y0 + τ, and we generated simulated outcomes as ˜Yij =
+Zjy1 + (1 − Zj)y0. Note that in this DGP, selection into treatment at the school level is only
+a function of school-level covariates, but the potential outcomes under control are a function
+of the student-level test scores and, to a lesser extent, the overall quality of the school as
+measured by the percentage of proficient students. This DGP also induces a correlation
+within clusters. The average ICC across simulations was 0.29 with a standard deviation of
+0.04. We judge this to be a common data structure in educational COS designs. Using this
+DGP, we conduct two different simulation studies.
+5.1
+Simulation study 1
+In the first study, we conduct a comparative analysis of adjustment methods for COS designs.
+First, we produce a naive estimate of the treatment effect as the difference in means without
+any covariate adjustment, and an estimate using multilevel matching as implemented in
+Pimentel et al. (2018). Next, we implement our balancing weights in two ways: first, as
+balancing weights by solving Equation (6) and second, as subset balancing weights by solving
+Equation (14).
+We set the hyperparameters via the heuristic in Section 3.3, using the
+estimated coefficients and variance components from a multilevel regression model. In this
+simulation, we focus on how the performance changes as we vary the level of overlap by
+setting c to values of 1, 2.5, 7.5, and 10. When c = 1 this will induce poor overlap, and when
+c = 10 there is excellent overlap, with intermediate values increasing the level of overlap. For
+each scenario, we repeated the simulation 1,000 times, and we report the bias and root mean-
+squared error (RMSE). In our results, we standardize the bias, dividing it by the standard
+deviation of the control group’s outcomes from the original data.
+In the first panel of Figure 1, we plot the bias as a function of overlap for all four
+estimation methods.
+We observe that matching and both types of weights reduce bias
+compared to an unadjusted estimate. However, when overlap is poor, both sets of weights
+remove substantially more bias than multilevel matching.
+The performance of matching
+could be improved by trimming treated observations. However, with the balancing weights,
+we can reduce the bias compared to matching without having to change the estimand. The
+subset weights also alter the estimand, but allow for an unbiased treatment effect estimate.
+22
+
+−0.15
+−0.10
+−0.05
+0.00
+Poor Overlap
+Low
+Medium
+Good Overlap
+Degree of Overlap
+Standardized Bias
+0
+5
+10
+15
+Poor Overlap
+Low
+Medium
+Good Overlap
+Degree of Overlap
+RMSE
+Unadjusted
+ML Matching
+Weighting
+Subset Weighting
+Figure 1: Bias and RMSE for matching and two different weighting estimators by overlap
+condition.
+We next compare the methods in terms of RMSE. Both matching and weighting discard
+data. Matching discards some of the control schools and units, while weighting gives some
+of the control schools and students zero weight. One open question is whether one method
+or the other does so in a more efficient fashion. In the second panel of Figure 1, we plot the
+RMSE as a function of overlap for all the estimation methods. In this scenario, the difference
+in performance between matching and balancing weights is clear. While the subset weights
+have the lowest RMSE, the difference between the two weighting methods is minor. However,
+both forms of balancing weights outperform matching across all overlap scenarios by nearly a
+factor of 10. Balancing weights thus manage to reduce bias while also retaining a much larger
+effective sample size, which improves efficiency. Overall, we find that balancing weights are
+a clear improvement over matching. Balancing weights have lower bias than matching when
+overlap is poor and also are considerably more efficient.
+5.2
+Simulation study 2
+In the second simulation, we focus on the performance of the proposed plug-in variance
+estimator, relative to using the standard weighted cluster-robust sandwich estimator, for the
+balancing weights solving Equation (6). In this simulation, we fixed the overlap parameter
+at 10 and vary the number of clusters. We control the number of clusters by resampling
+clusters with replacement from the original data, and then generate outcomes and treatments
+following the DGP above. We used cluster sample sizes of 50, 100, 150, 200 and 250. For
+23
+
+4
+6
+8
+10
+12
+50
+100
+150
+200
+250
+Number of Clusters
+SEs
+20
+30
+40
+50
+50
+100
+150
+200
+250
+Number of Clusters
+Confidence Interval Length
+Plug−in SE
+Sandwich SE
+Figure 2: Comparative performance for two different variance estimators.
+each scenario, we repeated the simulation 1,000 times, and we report the average standard
+error estimate and the length of the 95% confidence interval.
+Figure 2 shows the results for the good overlap scenario. In the first panel, we compare
+the magnitude of the two variance estimates. As we expect, our proposed plug-in method
+produces smaller standard errors by removing variance due to the covariates. The difference
+between the two methods is largest when the number of clusters is small. These smaller
+standard errors also produce narrower confidence intervals. We also measured nominal cov-
+erage of the confidence intervals for both methods and found that both methods led to
+over-coverage of the true effect. This result is consistent with the general finding that the
+sandwich variance estimator for weighting methods is conservative. The pattern from the
+good overlap scenario is very similar for the other two overlap settings, so we report those
+results in the supplementary materials.
+6
+Applications
+Next, we present results from two different empirical applications. The first, from education,
+compares the performance of Catholic and public schools. The overlap between Catholic
+and public schools is known to be poor (Keele et al., 2022), which allows us to study the
+performance of the subset weights in a context for which they were designed. The second ap-
+plication, from health services research, compares different residency programs for surgeons.
+Here, overlap is much better, but the sample sizes may prove computationally challenging
+24
+
+for multilevel matching.
+6.1
+Catholic Schools
+Our analysis is a replication Keele et al. (2022), which used multilevel matching and high-
+lighted the limited amount of overlap between Catholic and public schools. Here, we compare
+results based on our proposed weighting methods to those based on multilevel matching. The
+data are a public release of the 1982 High School and Beyond survey and includes records for
+7,185 high school students from 160 schools. Of these schools, 70 are Catholic schools and are
+thus considered treated in this application, while the remainder are public high schools and
+thus serve as a reservoir of controls. The average number of students sampled per Catholic
+school is approximately 50 and ranges from 20 – 67, while the average number of students
+sampled per public school is 41 and ranges from 14 – 61. The data contain covariates on
+both students and schools. Student-level covariates are: an indicator for whether or not the
+student is female; an indicator for whether a student belongs to a particular racial/ethnic
+group; and a scale for socioeconomic status (SES). Three of the school-level measures are
+school-level averages of these student-level measures. Three additional school-level covariates
+are: total enrollment; the percentage of students on an academic track; and a measure of
+disciplinary climate. The disciplinary climate variable is a composite measure created from a
+factor score on measures of the number of attacks on teachers, fights, and other disciplinary
+incidents. This variable ranges from -1.7 to 2.7.
+In our analysis, we computed COS balancing weights for the ATT and subset weights for
+the overlapping set. Using a random effects model, we estimated the two components of the
+hyperparameter: the estimated ICC is 0.036, and the estimated signal-to-noise ratio is 1.2.
+We also include two of the multilevel matches implemented in Keele et al. (2022). The first
+of these matches retains all the Catholic schools, and the second trimmed 10 Catholic schools
+to improve the balance and increase overlap. Table 1 contains a comparison of how well each
+method balanced the baseline covariates as measured by the standardized difference: the
+difference in weighted/matched Catholic and public schools means divided by the pooled
+standard deviation before adjustment. The lack of overlap is apparent in the size of the
+standardized differences in the school-level covariates; several of the standardized differences
+25
+
+are larger than 0.50, and three exceed 1. We observe that balancing weights outperform
+matching in terms of balance. While the standardized differences for the COS balancing
+weights are still fairly large for two covariates, they are less than half of those obtained via
+multilevel matching. Moreover, if we are willing to alter the estimand, the subset weights
+are able to nearly exactly balance the Catholic and public school distributions.
+Table 1: Balance Table Comparing Catholic and Public Schools on Baseline Covariate Distribu-
+tions.
+Unweighted
+Balancing
+Subset
+Matching
+Matching –
+Weights
+Weights
+Trimmed
+Student SES
+0.48
+0.00
+0.00
+0.33
+0.13
+% Students Minority
+-0.14
+0.26
+0.00
+0.24
+0.22
+% Students Female
+-0.00
+-0.10
+-0.00
+0.04
+-0.09
+Enrollment
+-0.80
+-0.17
+-0.01
+-0.58
+-0.66
+% Students on Academic Track
+1.52
+0.32
+0.00
+1.27
+0.90
+Disciplinary Climate Scale
+-1.64
+-0.47
+-0.01
+-0.92
+-0.92
+School SES Average
+1.17
+0.15
+0.00
+0.79
+0.31
+Note: Cell entries are standardized differences, the difference in means divided by the pool
+standard deviation.
+To better understand the differences between the COS weights and the COS subset
+weights, we provide descriptive statistics on the largest weights. For the balancing weights,
+the vast majority of the weights are between 0 and 1, but there are 63 weights that are larger
+than 10, and the largest weight has a value of 92. For the COS subset weights the largest
+weight is 14. This should reduce the likelihood of unstable behavior in the treatment effect
+estimates due to large weights. Next, we measure the effective sample sizes to understand
+the loss of information due to weighting. In our data, before adjustment there are 5,273
+students. The effective sample size for the balancing weights is 1,710, and 1,036 for the
+subset weights. For comparison, the effective sample size for the match with all treated
+schools is 570, and 392 for the match that trimmed treated schools.
+Next, we review the point estimates for the Catholic school effect. Note that the outcome
+is a standardized test score, so estimates are measured in standard deviations. First, the
+unadjusted effect is 0.43 with a 95% CI of (0.31, 0.55). Next, the Catholic school effect
+estimated by balancing weights is -0.08. The 95% confidence interval using the sandwich
+26
+
+variance estimator is (-0.76, 0.59), and the confidence interval based on our proposed plug-in
+estimator is (-0.47, 0.31). The Catholic school estimate based on subset weights is 0.07.
+The 95% confidence interval using the sandwich variance estimator is (-0.29, 0.44), and
+the confidence interval based on our proposed plug-in estimator is -0.08–0.24. Note that
+these estimates are not directly comparable, as they target different estimands, but we
+observe in both cases small point estimates with confidence intervals that include zero.
+Notably, the plug-in variance estimator produces shorter confidence intervals consistent with
+the simulation results. For comparison, we next report the estimates based on multilevel
+matching. The estimate based on multilevel matching is 0.26 (95% CI: 0.12, 0.40), and the
+estimate based on matching with trimming is 0.13 (95% CI: -0.06, 0.31). The estimates based
+on matching are closer to those based on weighting if we include additional bias reduction via
+outcome modeling. This suggests that weighting was much more successful than matching
+at removing bias without reference to outcomes.
+Finally, while the subset balancing weights were able to balance the data better even with
+poor overlap, we had to change the estimand. To aid in the interpretation of this estimand,
+we plot the covariate means for Catholic and public schools before and after subset weighting
+in Figure 3. That is, we compare the raw Catholic school means to the weighted Catholic
+school means, and the same for public schools. We observe that Catholic and public school
+do not differ much in terms of gender and racial mix. While Catholic and public schools differ
+some in terms of SES, the key difference is in terms of disciplinary climate. For this covariate,
+we observe the largest difference between the weighted and unweighted estimates for both
+Catholic and public schools. In the overlap population of schools, Catholic schools have a
+much stricter disciplinary climate and public schools have a more permissive disciplinary
+climate.
+6.2
+Surgical Training
+One important question in health services research is whether certain aspects of surgical
+training have an effect on patient outcomes (Asch et al., 2009; Bansal et al., 2016; Za-
+heer et al., 2017; Sullivan et al., 2012).
+Sellers et al. (2018) studied whether surgeons
+from university-based residency programs produce superior patient outcomes than surgeons
+27
+
+% Students Female
+% Students Minority
+% Students on Academic Track
+Disciplinary Climate Scale
+School SES Average
+Student SES
+−0.4
+0.0
+0.4
+Means
+Covariates
+Contrast
+Unweighted − Catholic
+Subset Wgt. − Catholic
+Unweighted − Public
+Subset Wgt.
+Figure 3: Covariate means for Catholic and Public schools before and after subset weighting.
+trained in community-based residency programs. The original study used data based on all-
+payer hospital discharge claims from New York, Florida and Pennsylvania from 2012–2013.
+In the data, surgeons were classified as having attended a university-based (UBR) or non-
+university based (NUBR) residency based on the residency program listed in the American
+Medical Association Masterfile. The data contain covariates for surgeons, including age, sex,
+and years of training completion, and covariates for patients, such as sociodemographic and
+clinical characteristics including 31 comorbidities based on Elixhauser indices (Elixhauser
+et al., 1998). The primary outcome was postoperative complications. Complications were
+identified using ICD-9 diagnosis codes and collapsed into a binary variable indicating the
+development of 1 or more complications. They compared surgeon performance between UBR
+and NUBR surgeons for patients that underwent one of 44 common operations performed
+by general surgeons in an inpatient setting (Sellers et al., 2018).
+In this application, there are 498 treated surgeons and 1,201 control surgeons. Overall,
+there are 86,305 patients operated on by UBR surgeons, and 193,307 patients operated on by
+NUBR surgeons. The number of patients treated by each surgeon varied from five to 1,074
+over the two-year period. In the UBR application, standardized differences before weighting
+are relatively small–one indication that overlap is good for this data set. As such, we only
+target the ATT estimand and do not use the subset weights. We again set hyperparameter
+28
+
+values based on estimates from a random effects model. In this data, the estimated ICC is
+0.016, and the noise to signal ratio is 0.221. Figure 4 contains a balance plot for the subset
+of covariates with the largest imbalances. For each of these covariates, weighting improves
+any imbalance relative to the unadjusted data, giving close to exact balance.
+In the full data, there are 279,611 patients. After weighting, the effective sample size is
+262,672. Here, the loss of sample size is relatively small. We also attempted to implement a
+multilevel match on a desktop computer with 32 GB of RAM. However, we received an error
+that R was unable to allocate enough memory to complete the match. For the balancing
+weights, we are able to estimate weights is less than a minute. As such, balancing weights
+have clear computational advantages for COS applications with larger sample sizes.
+Next, we estimate treatment effects. First, we estimate the unadjusted treatment effect
+via a regression model with clustering at the surgeon level. We find that for UBR surgeons,
+the estimated percentage of cases with a complication is 1.35 percentage points lower, and
+the 95% confidence interval does not contain zero (95% CI: -2.14, -0.55). Once we account
+for confounding via weighting, the estimate treatment effect is -0.85 percentage points. We
+estimated 95% confidence intervals using both the cluster-robust sandwich variance estimator
+(95% CI: -1.62, -0.09) and our proposed plug-in variance estimator (95% CI: -1.53, -0.18).
+Because the balance is excellent, further bias reduction via an outcome model does not
+meaningfully change the estimate. In sum, we find that UBR surgeons do indeed appear to
+cause fewer complications, even when accounting for observed confounding. However, the
+magnitude of the treatment effect is smaller once we control for observed covariates.
+7
+Conclusion
+We introduced an approximate balancing weight estimator for designs where treatments
+are administered to entire clusters such as school or hospitals.
+To do so, we considered
+two potential estimands — one adjusting for both unit-level and cluster-level covariates
+and another adjusting for cluster-level covariates alone — that have different identification
+assumptions associated with them. For both of these estimands, we find weights to minimize
+an upper bound on the mean square error of the resulting weighting estimator.
+When
+adjusting for cluster-level covariates alone, the weights are constant within clusters, while
+29
+
+Years Experience
+White
+Urgent Admission
+Surgeon Age
+Self Insurance
+Other Racial Cat.
+No. of Procedures
+No. of Essential Procedures
+No. of Complex Procedures
+No. Comorbidities
+Medicare
+Medicaid
+Male
+Hispanic
+Emergency Admission
+Elective Admission
+Commercial Insur.
+Age
+African−American
+−0.10
+−0.05
+0.00
+0.05
+0.10
+Standardized Difference
+Covariates
+Contrast
+Weighted
+Unweighted
+Figure 4: Balance Plot: UBR vs NUBR surgeons for the set of covariates with the largest
+baseline imbalances.
+30
+
+the weights vary across units within clusters when including unit-level covariates.
+This
+affects the overall variance, as units’ outcomes can be correlated within clusters. We showed
+that when it is sufficient to adjust only for cluster-level covariates in order to estimate
+the ATT, there can be efficiency gains to including unit-level covariates as well, depending
+on the predictive strength of those covariates and the level of correlation between units’
+outcomes within clusters. We also considered two adaptations to deal with cases where it
+is impossible to find weights that achieve good covariate balance: (i) bias-correction via an
+outcome model and (ii) changing the estimand and finding an overlapping weighted subset
+of the data. We then showed how to construct confidence intervals that are asymptotically
+valid under certain conditions. In a simulation study, we demonstrated that our proposed
+weighting estimator outperformed multilevel matching. This was especially true in terms of
+using more of the sample size which resulted in higher efficiency and lower RMSE. In two
+empirical applications, we found balancing weights also had several practical advantages over
+multilevel matching.
+There are several avenues for future work. First, while we propose a heuristic for choosing
+the hyperparameters from the observed data, an important question is how to choose these
+hyperparameters in a rigorous, data-driven way that keeps weights independent of outcomes.
+Second, we can consider resampling approaches to uncertainty quantification such as the
+block weighted bootstrap proposed by Cui et al. (2022). Finally, many COS designs have
+a longitudinal structure, where data is available at multiple granularities over time. For
+example, many policy changes happen at the state-level, but county and municipality-level
+data series exist for the outcome of interest. Exploring these cases and extending our analysis
+to such settings will be important areas for future research.
+31
+
+Bibliography & References Cited
+Abadie, A. and G. W. Imbens (2011). Bias-corrected matching estimators for average treat-
+ment effects. Journal of Business and Economic Statistics 29(1), 1–11.
+Asch, D. A., S. Nicholson, S. Srinivas, J. Herrin, and A. J. Epstein (2009).
+Evaluating
+obstetrical residency programs using patient outcomes. Jama 302(12), 1277–1283.
+Athey, S., G. W. Imbens, and S. Wager (2018). Approximate residual balancing: debiased
+inference of average treatment effects in high dimensions. Journal of the Royal Statistical
+Society: Series B (Statistical Methodology) 80(4), 597–623.
+Bansal, N., K. D. Simmons, A. J. Epstein, J. B. Morris, and R. R. Kelz (2016). Using
+patient outcomes to evaluate general surgery residency program performance.
+JAMA
+surgery 151(2), 111–119.
+Ben-Michael, E., A. Feller, and E. Hartman (2021). Multilevel calibration weighting for
+survey data.
+Ben-Michael, E., A. Feller, D. A. Hirshberg, and J. R. Zubizarreta (2021). The balancing
+act in causal inference. arXiv preprint arXiv:2110.14831.
+Ben-Michael, E., A. Feller, and J. Rothstein (2021). The Augmented Synthetic Control
+Method. Journal of the American Statistical Association 116(536), 1789–1803.
+Chattopadhyay, A., Christopher H. Hase, and J. R. Zubizarreta (2020). Balancing Versus
+Modeling Approaches to Weighting in Practice. Statistics in Medicine in press.
+Cochran, W. G. (1965). The planning of observational studies of human populations. Journal
+of Royal Statistical Society, Series A 128(2), 234–265.
+Cui, C., S. Yang, B. J. Reich, and D. A. Gill (2022). Matching Estimators of Causal Effects
+in Clustered Observational Studies with Application to Quantifying the Impact of Marine
+Protected Areas on Biodiversity.
+Elixhauser, A., C. Steiner, D. R. Harris, and R. M. Coffey (1998). Comorbidity measures
+for use with administrative data. Medical care 36(1), 8–27.
+32
+
+Hainmueller, J. (2011). Entropy Balancing for Causal Effects: A Multivariate Reweighting
+Method to Produce Balanced Samples in Observational Studies. Political Analysis 20,
+25–46.
+Hansen, B. E. and S. Lee (2019).
+Asymptotic theory for clustered samples.
+Journal of
+Econometrics 210(2), 268–290.
+Hazlett, C. (2019). Kernel Balancing : A flexible non-parametric weighting procedure for
+estimating causal effects. Statistica Sincia.
+Hedges, L. V. and E. C. Hedberg (2007). Intraclass correlation values for planning group-
+randomized trials in education. Educational Evaluation and Policy Analysis 29(1), 60–87.
+Hirshberg, D. A., A. Maleki, and J. Zubizarreta (2019). Minimax linear estimation of the
+retargeted mean. arXiv preprint arXiv:1901.10296.
+Hirshberg, D. A. and S. Wager (2021). Augmented minimax linear estimation. The Annals
+of Statistics 49(6), 3206–3227.
+Huber, P. J. (1967). Under nonstandard conditions. In Proceedings of the Fifth Berkeley
+Symposium on Mathematical Statistics and Probability: Weather Modification; University
+of California Press: Berkeley, CA, USA, pp. 221.
+Keele, L., M. Lenard, and L. Page (2021). Matching methods for clustered observational
+studies in education. Journal of Research on Educational Effectiveness 14(3), 696–725.
+Keele, L., M. Lenard, and L. Page (2022). Overlap violations in clustered observational
+studies of educational interventions. Journal of Research on Educational Effectiveness,
+1–18.
+Keele, L. J. and J. Zubizarreta (2017). Optimal multilevel matching in clustered observational
+studies: A case study of the effectiveness of private schools under a large-scale voucher
+system. Journal of the American Statistical Association 112(518), 547–560.
+Li, F., K. L. Morgan, and A. M. Zaslavsky (2018). Balancing covariates via propensity score
+weighting. Journal of the American Statistical Association 113(521), 390–400.
+33
+
+Page, L. C., M. Lenard, and L. Keele (2020, July-Sept). The design of clustered observational
+studies in education. AERA Open 6(3), 1–14.
+Pimentel, S. D., L. C. Page, M. Lenard, and L. J. Keele (2018). Optimal multilevel matching
+using network flows: An application to a summer reading intervention. Annals of Applied
+Statistics 12(3), 1479–1505.
+Raudenbush, S. W. (1997). Statistical analysis and optimal design for cluster randomized
+trials. Psychological Methods 2(2), 173.
+Robins, J. M., A. Rotnitzky, and L. Ping Zhao (1994). Estimation of Regression Coefficients
+When Some Regressors are not Always Observed. Journal of the American Statistical
+Association 89427, 846–866.
+Rubin, D. B. (1974). Estimating causal effects of treatments in randomized and nonrandom-
+ized studies. Journal of Educational Psychology 6(5), 688–701.
+Rubin, D. B. (2007). The design versus the analysis of observational studies for causal effects:
+parallels with the design of randomized trials. Statistics in medicine 26(1), 20–36.
+Rubin, D. B. (2008, September). For objective causal inference, design trumps analysis. The
+Annals of Applied Statistics 2(3), 808–840.
+Rubinstein, M., A. Haviland, and D. Choi (2022). Balancing weights for estimated region-
+level data: the effect of Medicaid Expansion on the uninsurance rate among states that
+did not expand Medicaid.
+Sellers, M. M., L. J. Keele, C. E. Sharoky, C. Wirtalla, E. A. Bailey, and R. R. Kelz (2018).
+Association of surgical practice patterns and clinical outcomes with surgeon training in
+university-or nonuniversity-based residency program. JAMA surgery 153(5), 418–425.
+Small, D. S., T. R. T. Have, and P. R. Rosenbaum (2008, March). Randomization inference
+in a group–randomized trial of treatments for depression: Covariate adjustment, noncom-
+pliance, and quantile effects. Journal of the American Statistical Association 103(481),
+271–279.
+34
+
+Sullivan, M. C., G. Sue, E. Bucholz, H. Yeo, R. H. Bell Jr, S. A. Roman, and J. A. Sosa
+(2012). Effect of program type on the training experiences of 248 university, community,
+and us military-based general surgery residencies. Journal of the American College of
+Surgeons 214(1), 53–60.
+Tarr, A. and K. Imai (2021). Estimating Average Treatment Effects with Support Vector
+Machines.
+Wang, Y. and J. R. Zubizarreta (2019). Minimal dispersion approximately balancing weights:
+asymptotic properties and practical considerations. Biometrika.
+Wang, Y. and J. R. Zubizarreta (2020). Minimal dispersion approximately balancing weights:
+asymptotic properties and practical considerations. Biometrika 107(1), 93–105.
+White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct
+test for heteroskedasticity. Econometrica: Journal of the Econometric Society, 817–838.
+Ye, T., T. Westling, L. Page, and L. Keele (2022). Nonparametric identification of causal ef-
+fects in clustered observational studies with differential selection. Unpublished Manuscript.
+Zaheer, S., S. D. Pimentel, K. D. Simmons, L. E. Kuo, J. Datta, N. Williams, D. L. Fraker,
+and R. R. Kelz (2017). Comparing international and united states undergraduate medical
+education and surgical outcomes using a refined balance matching methodology. Annals
+of surgery 265(5), 916–922.
+Zhao, Q. (2019). Covariate balancing propensity score by tailored loss functions. Annals of
+Statistics 47(2), 965–993.
+Zhao, Q. and D. Percival (2016). Entropy Balancing is Doubly Robust. Journal of Causal
+Inference.
+Zubizarreta, J. R. (2015). Stable weights that balance covariates for estimation with incom-
+plete outcome data. Journal of the American Statistical Association 110(511), 910–922.
+35
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+arXiv:2301.11390v1  [gr-qc]  26 Jan 2023
+Current problems and recent advances in wormhole physics
+(Editorial for a Special issue of Universe journal)
+Kirill A. Bronnikov,a,b,c,1 Sergey V. Sushkovd,2
+a Center of Gravitation and Fundamental Metrology, VNIIMS, Ozyornaya ul. 46, Moscow 119361, Russia
+b Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198,
+Russia
+c National Research Nuclear University “MEPhI”, Kashirskoe sh. 31, Moscow 115409, Russia
+d Institute of Physics, Kazan Federal University, Kremliovskaya str. 16a, Kazan 420008, Russia
+Wormholes are hypothetical space-time tunnels with nontrivial topologies capable of connecting
+either two distant regions of the same universe or two different universes [1–3]. From the theoretical
+point of view, the possibility of their existence is problematic but cannot be ruled out. If wormholes
+do exist, many unusual phenomena can be expected. Among them, probably the most exciting ones
+are shortcuts providing interstellar, intergalactic or inter-universe trips and even time machines.
+At present, wormholes are extremely attractive and popular objects for research, it is sufficient
+to mention that the word “wormhole” is found in the titles of 1614 articles on the resource arXiv.org
+for all years and 175 articles for the last 12 months (as of 19.01.2023). Although many mathematical
+and physical properties of these objects have been discovered and studied in the recent decades,
+there remain multiple unsolved problems and opportunities of interest to be explored.
+The present Special Issue “Recent Advances in Wormhole Physics” is aimed at enlighten-
+ing some recent results in selected areas of wormhole physics. The issue is a collection of fourteen
+papers [4–17] that fairly well characterizes the diversity of subjects and methods of wormhole
+physics.
+Very roughly, wormhole studies may be classified as follows:
+• A search for wormhole solutions in general relativity and other theories of gravity, investiga-
+tions of their properties and conditions of their existence.
+• Studies of mathematical, physical, metaphysical and philosophical consequences of possible
+wormhole existence.
+• Assuming that wormholes do exist in the Universe, studies of their astronomical and as-
+trophysical manifestations, in particular, their possible observational distinctions from black
+holes.
+Needless to say that many studies combine some or all of these trends. Nevertheless, it can be
+more or less conditionally observed that the first trend is represented by the papers [4,13,14,16],
+the second one by [5,7,11,15], and the third one by [6,8–10,12,17].
+Two of the papers are brief reviews [4,9]. Thus, Takafumi Kokubu and Tomohiro Harada [4] con-
+sider different physical aspects of thin-shell wormholes, both in GR and in Einstein–Gauss–Bonnet
+gravity, including spherical, planar and hyperbolic symmetries of space-time and the stability issues.
+Cosimo Bambi and Dejan Stojkovic [9] describe astronomically observable effects of wormholes as
+possible substitutes of black holes, including gravitational lensing and shadows, possible orbiting
+1e-mail: kb20@yandex.ru
+2e-mail: sergey sushkov@mail.ru
+
+2
+star trajectories, accretion disk spectra, and gravitational waves emitted at merges of compact
+objects, of which one or both are wormholes.
+Some of the papers evidently go beyond the traditional subjects of wormhole physics, concerning
+classical traversable wormholes. Thus, Sergey Bondarenko [11] considers their quantum counter-
+parts, called quantum wormholes, and, in particular, their possible role in the origin of a small
+cosmological constant favored by observations. Elias Zafiris and Albrecht von M¨uller [15] discuss
+the possible role of Planck-scale wormholes in the so-called “ER=EPR” conjecture in quantum en-
+tanglement. Alexander Kirillov and Elena Savelova [16] discuss the conditions under which Planck-
+scale virtual wormholes may be converted into macroscopic and observable ones.
+The present
+authors together with Pavel Kashargin [13] discuss wormholes that can emerge in the framework
+of general relativity without any exotic matter in nonstatic space-times.
+The outstanding progress of observational astronomy and cosmology, above all, the discovery
+of gravitational waves and the pictures obtained by the Event Horizon Telescope, are apparently
+reflected in the wealth of studies devoted to possible observable effects due to wormholes. In the
+present issue, we see the discussions of high-energy particle collisions in wormhole space-times
+[6], gravitational lensing by wormholes with unusual topologies [8], peculiar features of accretion
+flows [10] and nearby stellar orbits [12,17] and possible manifestations of a fractal distribution of
+primordial wormholes [11].
+It should be noted that, with the whole diversity of wormhole studies presented in this Special
+issue, some areas turned out to be almost unmentioned. These are axially and cylindrically sym-
+metric wormholes with or without rotation (though, some effects of rotation are discussed in [6])
+— so we would here refer to the reviews [18, 19] and references therein. These are also stability
+studies concerning all kinds of perturbations of static or stationary wormholes (note that Ref. [4]
+only discusses the stability of thin-shell wormholes with respect to shell motion), so please see,
+e.g., [20,21] and references therein for more general studies.
+Concluding, we would like to say that the study of wormholes is far from being complete,
+and one can expect many new interesting physical and mathematical results in this relevant and
+promising area of physics.
+Funding
+S.V.S. is supported by the RSF grant No.
+21-12-00130.
+Partially, this work was done in the
+framework of the Russian Government Program of Competitive Growth of the Kazan Federal
+University. K.B. acknowledges partial support from the Ministry of Science and Higher Education
+of the Russian Federation, Project “Fundamental properties of elementary particles and cosmology”
+No. 0723-2020-0041, and from Project No. FSSF-2023-0003.
+Conflict of interest
+The authors declare that they have no conflicts of interest.
+References
+[1] Morris, M.S.; Thorne, K.S. Wormholes in spacetime and their use for interstellar travel: A tool for
+teaching general relativity, Am. J. Phys. 1988, 56, 395.
+[2] Visser, M. Lorentzian Wormholes: From Einstein to Hawking; American Institute of Physics: Wood-
+bury, NY, USA, 1995; 412p.
+
+3
+[3] Lobo, F.S.N. Exotic solutions in General Relativity: Traversable wormholes and “warp drive” space-
+times. In Classical and Quantum Gravity Research; Nova Science Publishers: New York, NY, USA,
+2008; pp. 1–78; arxiv: 0710.4474.
+[4] Takafumi Kokubu and Tomohiro Harada, Thin-Shell Wormholes in Einstein and Einstein-Gauss-Bonnet
+Theories of Gravity, Universe 2020, 6(11), 197; https://doi.org/10.3390/universe6110197 - 26 Oct 2020
+[5] Sergey Bondarenko, CPTM Discrete Symmetry, Quantum Wormholes and Cosmological Constant Prob-
+lem, Universe 2020, 6(8), 121; https://doi.org/10.3390/universe6080121 - 11 Aug 2020
+[6] Oleg B. Zaslavskii, New Scenarios of High-Energy Particle Collisions Near Wormholes, Universe 2020,
+6(12), 227; https://doi.org/10.3390/universe6120227 - 30 Nov 2020
+[7] Brandon Mattingly, Abinash Kar, Matthew Gorban, William Julius, Cooper K. Watson, MD Ali,
+Andrew Baas, Caleb Elmore, Jeffrey S. Lee, Bahram Shakerin, Eric W. Davis and Gerald B.
+Cleaver, Curvature Invariants for the Alcubierre and Nat´ario Warp Drives, Universe 2021, 7(2), 21;
+https://doi.org/10.3390/universe7020021 - 20 Jan 2021
+[8] Kimet Jusufi, Determining the Topology and Deflection Angle of Ringholes via Gauss-Bonnet Theorem,
+Universe 2021, 7(2), 44; https://doi.org/10.3390/universe7020044 - 16 Feb 2021
+[9] Cosimo
+Bambi
+and
+Dejan
+Stojkovic,
+Astrophysical
+Wormholes,
+Universe
+2021,
+7(5),
+136;
+https://doi.org/10.3390/universe7050136 - 08 May 2021
+[10] Rosaliya M. Yusupova, Ramis Kh. Karimov, Ramil N. Izmailov and Kamal K. Nandi, Accretion Flow
+onto Ellis-Bronnikov Wormhole Universe 2021, 7(6), 177; https://doi.org/10.3390/universe7060177 - 02
+Jun 2021
+[11] Alexander A. Kirillov, Elena P. Savelova and Polina O. Vladykina, Possible Effects of the Fractal
+Distribution of Relic Wormholes Universe 2021, 7(6), 178; https://doi.org/10.3390/universe7060178 -
+03 Jun 2021
+[12] Zdenek Stuchlik and Jaroslav Vrba, Epicyclic Oscillations around Simpson-Visser Regular Black Holes
+and Wormholes, Universe 2021, 7(8), 279; https://doi.org/10.3390/universe7080279 - 01 Aug 2021
+[13] Kirill A. Bronnikov, Pavel E. Kashargin and Sergey V. Sushkov, Magnetized Dusty Black Holes and
+Wormholes, Universe 2021, 7(11), 419; https://doi.org/10.3390/universe7110419 - 03 Nov 2021
+[14] J´ulio
+C.
+Fabris,
+Tales
+Augusto
+Oliveira
+Gomes
+and
+Denis
+Campos
+Rodrigues,
+Black
+Hole and Wormhole Solutions in Einstein-Maxwell-Scalar Theory,
+Universe 2022,
+8(3),
+151;
+https://doi.org/10.3390/universe8030151 - 27 Feb 2022
+[15] Elias Zafiris and Albrecht von M¨uller, The “ER = EPR” Conjecture and Generic Gravitational Prop-
+erties: A Universal Topological Linking Model of the Correspondence between Tripartite Entanglement
+and Planck-Scale Wormholes, Universe 2022, 8(3), 189; https://doi.org/10.3390/universe8030189 - 18
+Mar 2022
+[16] Alexander A. Kirillov and Elena P. Savelova, On Possible Origin of an Artificial Wormhole, Universe
+2022, 8(8), 428; https://doi.org/10.3390/universe8080428 - 19 Aug 2022
+[17] Ramis Kh. Karimov, Ramil N. Izmailov and Kamal K. Nandi, On a Class of Harko-Kovacs-Lobo Worm-
+holes, Universe 2022, 8(10), 540; https://doi.org/10.3390/universe8100540 - 18 Oct 2022
+[18] Kirill A. Bronnikov, Nilton Santos and Anzhong Wang, Cylindrical systems in general relativity (re-
+view). Class. Quantum Grav. 37, 113002 (2020); arXiv: 1901.06561.
+[19] Burkhard Kleihaus, Jutta Kunz, Rotating Wormholes. In: Lobo, F. (ed.), Wormholes, Warp Drives
+and Energy Conditions, Fundamental Theories of Physics, vol 189. Springer, Cham.
+https://doi.org/10.1007/978-3-319-55182-1 3
+[20] Kirill A. Bronnikov, Scalar fields as sources for wormholes and regular black holes. Particles 2018, 1,
+5; arXiv: 1802.00098.
+[21] Francesco Cremona, Livio Pizzocchero, Olivier Sarbach, Gauge-invariant spherical linear perturbations
+of wormholes in Einstein gravity minimally coupled to a self-interacting phantom scalar field. Phys.
+Rev. D 101, 104061 (2020); arXiv: 1911.13103.
+

3NFIT4oBgHgl3EQf5iun/content/tmp_files/load_file.txt

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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='11390v1  [gr-qc]  26 Jan 2023  Current problems and recent advances in wormhole physics  (Editorial for a Special issue of Universe journal)  Kirill A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Bronnikov,a,b,c,1 Sergey V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Sushkovd,2  a Center of Gravitation and Fundamental Metrology, VNIIMS, Ozyornaya ul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' 46, Moscow 119361, Russia  b Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198,  Russia  c National Research Nuclear University “MEPhI”, Kashirskoe sh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' 31, Moscow 115409, Russia  d Institute of Physics, Kazan Federal University, Kremliovskaya str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' 16a, Kazan 420008, Russia  Wormholes are hypothetical space-time tunnels with nontrivial topologies capable of connecting  either two distant regions of the same universe or two different universes [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' From the theoretical  point of view, the possibility of their existence is problematic but cannot be ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' If wormholes  do exist, many unusual phenomena can be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Among them, probably the most exciting ones  are shortcuts providing interstellar, intergalactic or inter-universe trips and even time machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='  At present, wormholes are extremely attractive and popular objects for research, it is sufficient  to mention that the word “wormhole” is found in the titles of 1614 articles on the resource arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='org  for all years and 175 articles for the last 12 months (as of 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Although many mathematical  and physical properties of these objects have been discovered and studied in the recent decades,  there remain multiple unsolved problems and opportunities of interest to be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='  The present Special Issue “Recent Advances in Wormhole Physics” is aimed at enlighten-  ing some recent results in selected areas of wormhole physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' The issue is a collection of fourteen  papers [4–17] that fairly well characterizes the diversity of subjects and methods of wormhole  physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='  Very roughly, wormhole studies may be classified as follows:  A search for wormhole solutions in general relativity and other theories of gravity, investiga-  tions of their properties and conditions of their existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='  Studies of mathematical, physical, metaphysical and philosophical consequences of possible  wormhole existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='  Assuming that wormholes do exist in the Universe, studies of their astronomical and as-  trophysical manifestations, in particular, their possible observational distinctions from black  holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='  Needless to say that many studies combine some or all of these trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Nevertheless, it can be  more or less conditionally observed that the first trend is represented by the papers [4,13,14,16],  the second one by [5,7,11,15], and the third one by [6,8–10,12,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='  Two of the papers are brief reviews [4,9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Thus, Takafumi Kokubu and Tomohiro Harada [4] con-  sider different physical aspects of thin-shell wormholes, both in GR and in Einstein–Gauss–Bonnet  gravity, including spherical, planar and hyperbolic symmetries of space-time and the stability issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='  Cosimo Bambi and Dejan Stojkovic [9] describe astronomically observable effects of wormholes as  possible substitutes of black holes, including gravitational lensing and shadows, possible orbiting  1e-mail: kb20@yandex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='ru  2e-mail: sergey sushkov@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='ru  2  star trajectories, accretion disk spectra, and gravitational waves emitted at merges of compact  objects, of which one or both are wormholes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='  Some of the papers evidently go beyond the traditional subjects of wormhole physics, concerning  classical traversable wormholes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Thus, Sergey Bondarenko [11] considers their quantum counter-  parts, called quantum wormholes, and, in particular, their possible role in the origin of a small  cosmological constant favored by observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Elias Zafiris and Albrecht von M¨uller [15] discuss  the possible role of Planck-scale wormholes in the so-called “ER=EPR” conjecture in quantum en-  tanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Alexander Kirillov and Elena Savelova [16] discuss the conditions under which Planck-  scale virtual wormholes may be converted into macroscopic and observable ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='  The present  authors together with Pavel Kashargin [13] discuss wormholes that can emerge in the framework  of general relativity without any exotic matter in nonstatic space-times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='  The outstanding progress of observational astronomy and cosmology, above all, the discovery  of gravitational waves and the pictures obtained by the Event Horizon Telescope, are apparently  reflected in the wealth of studies devoted to possible observable effects due to wormholes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' In the  present issue, we see the discussions of high-energy particle collisions in wormhole space-times  [6], gravitational lensing by wormholes with unusual topologies [8], peculiar features of accretion  flows [10] and nearby stellar orbits [12,17] and possible manifestations of a fractal distribution of  primordial wormholes [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='  It should be noted that, with the whole diversity of wormhole studies presented in this Special  issue, some areas turned out to be almost unmentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' These are axially and cylindrically sym-  metric wormholes with or without rotation (though, some effects of rotation are discussed in [6])  — so we would here refer to the reviews [18, 19] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' These are also stability  studies concerning all kinds of perturbations of static or stationary wormholes (note that Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' [4]  only discusses the stability of thin-shell wormholes with respect to shell motion), so please see,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=', [20,21] and references therein for more general studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='  Concluding, we would like to say that the study of wormholes is far from being complete,  and one can expect many new interesting physical and mathematical results in this relevant and  promising area of physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='  Funding  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' is supported by the RSF grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='  21-12-00130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='  Partially, this work was done in the  framework of the Russian Government Program of Competitive Growth of the Kazan Federal  University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' acknowledges partial support from the Ministry of Science and Higher Education  of the Russian Federation, Project “Fundamental properties of elementary particles and cosmology”  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' 0723-2020-0041, and from Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' FSSF-2023-0003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='  Conflict of interest  The authors declare that they have no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='  References  [1] Morris, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Thorne, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Wormholes in spacetime and their use for interstellar travel: A tool for  teaching general relativity, Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' 1988, 56, 395.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='  [2] Visser, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Lorentzian Wormholes: From Einstein to Hawking;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' American Institute of Physics: Wood-  bury, NY, USA, 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' 412p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='  3  [3] Lobo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Exotic solutions in General Relativity: Traversable wormholes and “warp drive” space-  times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' In Classical and Quantum Gravity Research;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Nova Science Publishers: New York, NY, USA,  2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' 1–78;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' arxiv: 0710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='4474.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='  [4] Takafumi Kokubu and Tomohiro Harada, Thin-Shell Wormholes in Einstein and Einstein-Gauss-Bonnet  Theories of Gravity, Universe 2020, 6(11), 197;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='3390/universe6110197 - 26 Oct 2020  [5] Sergey Bondarenko, CPTM Discrete Symmetry, Quantum Wormholes and Cosmological Constant Prob-  lem, Universe 2020, 6(8), 121;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='3390/universe6080121 - 11 Aug 2020  [6] Oleg B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Zaslavskii, New Scenarios of High-Energy Particle Collisions Near Wormholes, Universe 2020,  6(12), 227;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='3390/universe6120227 - 30 Nov 2020  [7] Brandon Mattingly, Abinash Kar, Matthew Gorban, William Julius, Cooper K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Watson, MD Ali,  Andrew Baas, Caleb Elmore, Jeffrey S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Lee, Bahram Shakerin, Eric W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Davis and Gerald B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='  Cleaver, Curvature Invariants for the Alcubierre and Nat´ario Warp Drives, Universe 2021, 7(2), 21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
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+page_content='3390/universe7020021 - 20 Jan 2021  [8] Kimet Jusufi, Determining the Topology and Deflection Angle of Ringholes via Gauss-Bonnet Theorem,  Universe 2021, 7(2), 44;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
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+page_content='3390/universe7020044 - 16 Feb 2021  [9] Cosimo  Bambi  and  Dejan  Stojkovic,  Astrophysical  Wormholes,  Universe  2021,  7(5),  136;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
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+page_content='3390/universe7050136 - 08 May 2021  [10] Rosaliya M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Yusupova, Ramis Kh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Karimov, Ramil N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Izmailov and Kamal K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Nandi, Accretion Flow  onto Ellis-Bronnikov Wormhole Universe 2021, 7(6), 177;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
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+page_content='3390/universe7060177 - 02  Jun 2021  [11] Alexander A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Kirillov, Elena P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Savelova and Polina O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Vladykina, Possible Effects of the Fractal  Distribution of Relic Wormholes Universe 2021, 7(6), 178;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
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+page_content='3390/universe7060178 -  03 Jun 2021  [12] Zdenek Stuchlik and Jaroslav Vrba, Epicyclic Oscillations around Simpson-Visser Regular Black Holes  and Wormholes, Universe 2021, 7(8), 279;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
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+page_content='3390/universe7080279 - 01 Aug 2021  [13] Kirill A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Bronnikov, Pavel E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Kashargin and Sergey V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Sushkov, Magnetized Dusty Black Holes and  Wormholes, Universe 2021, 7(11), 419;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
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+page_content='3390/universe7110419 - 03 Nov 2021  [14] J´ulio  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='  Fabris,  Tales  Augusto  Oliveira  Gomes  and  Denis  Campos  Rodrigues,  Black  Hole and Wormhole Solutions in Einstein-Maxwell-Scalar Theory,  Universe 2022,  8(3),  151;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
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+page_content='3390/universe8030151 - 27 Feb 2022  [15] Elias Zafiris and Albrecht von M¨uller, The “ER = EPR” Conjecture and Generic Gravitational Prop-  erties: A Universal Topological Linking Model of the Correspondence between Tripartite Entanglement  and Planck-Scale Wormholes, Universe 2022, 8(3), 189;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='3390/universe8030189 - 18  Mar 2022  [16] Alexander A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Kirillov and Elena P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Savelova, On Possible Origin of an Artificial Wormhole, Universe  2022, 8(8), 428;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='3390/universe8080428 - 19 Aug 2022  [17] Ramis Kh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Karimov, Ramil N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Izmailov and Kamal K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Nandi, On a Class of Harko-Kovacs-Lobo Worm-  holes, Universe 2022, 8(10), 540;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='3390/universe8100540 - 18 Oct 2022  [18] Kirill A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Bronnikov, Nilton Santos and Anzhong Wang, Cylindrical systems in general relativity (re-  view).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Quantum Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' 37, 113002 (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' arXiv: 1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='06561.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='  [19] Burkhard Kleihaus, Jutta Kunz, Rotating Wormholes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' In: Lobo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' (ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' ), Wormholes, Warp Drives  and Energy Conditions, Fundamental Theories of Physics, vol 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Springer, Cham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='1007/978-3-319-55182-1 3  [20] Kirill A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Bronnikov, Scalar fields as sources for wormholes and regular black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Particles 2018, 1,  5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' arXiv: 1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='00098.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='  [21] Francesco Cremona, Livio Pizzocchero, Olivier Sarbach, Gauge-invariant spherical linear perturbations  of wormholes in Einstein gravity minimally coupled to a self-interacting phantom scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' D 101, 104061 (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content=' arXiv: 1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}
+page_content='13103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFIT4oBgHgl3EQf5iun/content/2301.11390v1.pdf'}

3dAzT4oBgHgl3EQf9P73/content/tmp_files/2301.01918v1.pdf.txt

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+Plant Species Richness Prediction from DESIS Hyperspectral Data:
+A Comparison Study on Feature Extraction Procedures
+and Regression Models
+Yiqing Guoa, Karel Mokanya, Cindy Ongb, Peyman Moghadamc, Simon Ferriera, Shaun R.
+Levickd
+aCSIRO Land and Water, Acton, ACT 2601, Australia
+bCSIRO Energy, Kensington, WA 6151, Australia
+cCSIRO Data61, Pullenvale, QLD 4069, Australia
+dCSIRO Land and Water, Winnellie, NT 0822, Australia
+Abstract
+The diversity of terrestrial vascular plants plays a key role in maintaining the stability and pro-
+ductivity of ecosystems. Monitoring species compositional diversity across large spatial scales is
+challenging and time consuming. Airborne hyperspectral imaging has shown promise for mea-
+suring plant diversity remotely, but to operationalise these efforts over large regions we need to
+advance satellite-based alternatives. The advanced spectral and spatial specification of the re-
+cently launched DESIS (the DLR Earth Sensing Imaging Spectrometer) instrument provides a
+unique opportunity to test the potential for monitoring plant species diversity with spaceborne
+hyperspectral data. This study provides a quantitative assessment on the ability of DESIS hy-
+perspectral data for predicting plant species richness in two different habitat types in southeast
+Australia. Spectral features were first extracted from the DESIS spectra, then regressed against
+on-ground estimates of plant species richness, with a two-fold cross validation scheme to assess the
+predictive performance. We tested and compared the effectiveness of Principal Component Anal-
+ysis (PCA), Canonical Correlation Analysis (CCA), and Partial Least Squares analysis (PLS) for
+feature extraction, and Kernel Ridge Regression (KRR), Gaussian Process Regression (GPR), and
+Random Forest Regression (RFR) for species richness prediction. The best prediction results were
+r = 0.76 and RMSE = 5.89 for the Southern Tablelands region, and r = 0.68 and RMSE = 5.95 for
+the Snowy Mountains region. Relative importance analysis for the DESIS spectral bands showed
+Preprint submitted to ISPRS Journal of Photogrammetry and Remote Sensing
+January 6, 2023
+arXiv:2301.01918v1  [cs.LG]  5 Jan 2023
+
+that the red-edge, red, and blue spectral regions were more important for predicting plant species
+richness than the green bands and the near-infrared bands beyond red-edge. We also found that the
+DESIS hyperspectral data performed better than Sentinel-2 multispectral data in the prediction
+of plant species richness. Our results provide a quantitative reference for future studies exploring
+the potential of spaceborne hyperspectral data for plant biodiversity mapping.
+Keywords:
+hyperspectral, remote sensing, vascular plant, biodiversity, species richness, DESIS
+(the DLR Earth Sensing Imaging Spectrometer)
+1. Introduction
+Plant biodiversity is of critical importance to the stability of terrestrial ecosystems (Frankel
+et al., 1995). Anthropogenic activities, such as inappropriate cropping, deforestation, overgrazing,
+and construction, in conjunction with climate change, have been leading to substantial degradation
+and loss of natural habitats, posing imminent threats to vulnerable plant species (Ceballos et al.,
+2015; Tollefson, 2019). Consequently, species extinctions are occurring much faster than the natural
+background rate (Ceballos et al., 2015). Conservation activities have been undertaken in many
+places around the world aiming to reduce the current rate of extinction (Lecl`ere et al., 2020;
+Mokany et al., 2020).
+Spatial mapping of plant biodiversity helps with a better understanding of the distribution
+and temporal trends of plant species richness, facilitating effective policy making in environmental
+conservation and restoration (Stevenson et al., 2021; Myers et al., 2021; De Palma et al., 2021).
+Considerable effort has been made to collect samples of plant species richness through in-situ sur-
+veys (Kattge et al., 2020). Despite the ever increasing amount of data, it has been recognised that
+the completeness and representativeness of biodiversity samples still remain as a major challenge
+for the compilation of up-to-date biodiversity maps with fine resolution and wide coverage (K¨onig
+et al., 2019; Kattge et al., 2020). This data gap is understandable as field expeditions are labour
+2
+
+and time consuming, and sometimes infeasible if the location is remote or hard to access (Wang and
+Gamon, 2019; Guo et al., 2018). Moreover, inconsistencies among collection campaigns in their
+sampling strategies and ground plot sizes, confounded by human subjectivity and bias, further
+hampered the use of ground sampling data in downstream scientific research (Wang and Gamon,
+2019).
+Spaceborne remote sensing has long been deemed as a promising and cost-effective tool for
+mapping plant biodiversity, mainly due to its ability to capture data over large areas and in a
+timely manner (Skidmore et al., 2021; Wang and Gamon, 2019; Bush et al., 2017; Pettorelli et al.,
+2016). Among different types of remote sensing data, hyperspectral data is of particular interest for
+the biodiversity community, as it contains rich features in the spectral domain that can be utilised
+to explore the underlying relationship with plant biodiversity on the ground (Ghiyamat and Shafri,
+2010; Carlson et al., 2007; Guo et al., 2022). Previous studies have shown that plant biodiversity
+is linked to remotely sensed spectral measurements because of a well-founded interrelationship
+between plant species richness and primary productivity (Wang et al., 2016; Grace et al., 2016). It is
+hypothesised that a high diversity of plant species enhances primary production of the community,
+as a result of complementary functions provided by the diversified species composition. These
+interrelationships have the potential to enable researchers to use hyperspectral measurements, and
+their derived spectral indices, as remotely sensed measures of vegetation productivity and estimates
+of species richness.
+Most research into the relationships between biodiversity and hyperspectral data have made use
+of hand-held or airborne hyperspectral sensors, which are more readily available than spaceborne
+hyperspectral data-streams (e.g., Peng et al. (2018), Asner (2008), F´eret and Asner (2014), and
+Hacker et al. (2020)). Spaceborne hyperspectral imaging is relatively rare, with no active satellites
+in orbit since the Hyperion mission (which was active from 2000 to 2017). However, in preparation
+for the upcoming launch of EnMAP (Guanter et al., 2015), the DLR launched an exploratory
+3
+
+system to the International Space Station and embedded into the Multi-User-System for Earth
+Sensing (MUSES) platform in 2018.
+The DLR Earth Sensing Imaging Spectrometer (DESIS)
+(Eckardt et al., 2015; Mafanya et al., 2022) provides a unique opportunity to test the potential for
+monitoring plant species diversity with spaceborne hyperspectral data. It delivers hyperspectral
+images with 235 spectral bands over the visible and near-infrared regions of 400 ∼ 1000 nm, with
+a spectral resolution of 2.55 nm and a spatial resolution of 30 m (Eckardt et al., 2015; Alonso
+et al., 2019; Krutz et al., 2019). The high resolutions in both spectral and spatial domains make
+DESIS a promising data source for estimating biodiversity from space. However, there is so far
+a lack of quantitative studies on assessing the ability of DESIS hyperspectral data for predicting
+plant species richness values on the ground. It is worth noting the added challenge of spaceborne
+hyperspectral imagery such as DESIS having a larger pixel size (30m) than typical airborne (ranging
+from a few centimeters to several meters). As the signal of a large pixel tends to comprise a mix
+of multiple species, many techniques previously used to quantify richness with hyperspectral data,
+which detect individual species and then sum up to get richness (e.g., Asner (2008); F´eret and
+Asner (2014)), cannot be applied.
+The original bands in hyperspectral imagery are not orthogonal but rather highly collinear
+with each other, presenting a high degree of redundancy of information. Such redundancy can
+be removed to some extent by transforming the original spectral bands into an orthogonal space
+of a lower dimensionality.
+Among popular algorithms for dimensionality reduction are Princi-
+pal Component Analysis (PCA) (Xu et al., 2019; Jia and Richards, 1999), Canonical Correlation
+Analysis (CCA) (Zhao et al., 2014b; Richards and Jia, 2006), and Partial Least Squares analysis
+(PLS) (Feilhauer et al., 2015; Hacker et al., 2020; Wang et al., 2019). These dimensionality re-
+duction techniques reduce information redundancy by removing multicollinearity among spectral
+bands, enabling extracting spectral features from the original spectral data of hundreds of bands.
+In contrast to feature selection that chooses a subset of the original bands (or spectral indices
+4
+
+computed from a subset of the original bands), feature extraction techniques such as PCA, CCA,
+and PLS are able to makes use of information in all bands by transforming them into compact
+yet informative features. Compared with pre-defined vegetation indices such as Ratio Vegetation
+Index (RVI) and Normalised Difference Vegetation Index (NDVI), spectral features generated with
+feature extraction have shown better performance in extracting useful information from hyperspec-
+tral measurements (Zhao et al., 2014b). Following the extraction of spectral features, regression
+analysis can then be conducted to explore potential relationships between the extracted features
+and target biological variables. Commonly applied regression algorithms include the Kernel Ridge
+Regression (KRR), Gaussian Process Regression (GPR), and Random Forest Regression (RFR).
+Statistical regression based on extracted spectral features has shown to be effective in addressing
+biological problems with hyperspectral remote sensing data. For example, in a study to detect
+unintended herbicide damage in crops with hyperspectral measurements, dimensionality reduction
+was conducted with CCA in order to extract useful information to discriminate between healthy
+and damaged crops (Zhao et al., 2014b). In another study aiming to retrieve foliar traits from
+hyperspectral data, PLS was used to reduce the spectral dimensionality, with adequate retrieval
+accuracies being achieved for 10 out of the 11 functional traits (Hacker et al., 2020). These studies
+demonstrate that feature extraction and statistical regression can be effective tools in addressing
+biological problems with hyperspectral measurements.
+In this study, we aimed to assess the potential for DESIS hyperspectral data to predict on-
+ground plant species richness in two regions of southeast New South Wales, Australia—the South-
+ern Tablelands and Snowy Mountains. Our approach focused on spectral feature extraction from
+the DESIS spectra, and subsequent regression against field-measured plant species richness. We
+tested the combination of different feature extraction procedures (PCA, CCA, and PLS) and re-
+gression models (GPR, KRR, and RFR) for the predictive performance of species richness with
+DESIS data. Through quantitative analyses, we sought to address primarily the following impor-
+5
+
+tant questions: (1) How much variation in plant species richness can be explained with DESIS
+data? (2) Which parts of the spectrum had the most explanatory power? (3) Could similar results
+be achieved with more readily available multi-spectral imagery such as Sentinel-2?
+2. Materials and Methods
+While DESIS hyperspectral data contain rich spectral information, modelling is needed to link
+such information to field-based measurements of plant species richness. Here we followed a two-step
+approach whereby spectral features were first extracted from DESIS spectra and then correlated to
+species richness through regression. In this section, we start with describing the DESIS spectra and
+in-situ richness samples, followed by introducing the methods for feature extraction, regression,
+and accuracy assessment.
+2.1. Study site and field data
+This study focused on two different habitat types in southeast New South Wales, Australia,
+namely the Southern Tablelands (34°12’26”–34°39’07”S, 150°05’57”–150°40’51”E) and Snowy Moun-
+tains (35°43’58”–36°16’30”S, 148°23’16”–148°39’02”E), as shown in Fig. 1. Both regions are lo-
+cated within the climate zone of Cfb (oceanic climates), according to the K¨oppen–Geiger climate
+classification system.
+The Southern Tablelands region is located to the southwest of Sydney (Fig. 1). It is charac-
+terised by high-altitude plains with a rich biodiversity. There are more than 1200 plant species
+within Southern Tablelands, of which 30 are listed as threatened (Fallding, 2002). A large part
+of the landscape has been transformed into suburbs for residential developments and pastures for
+grazing purposes. Considering the high degree of human interference and habitat alteration, con-
+servation efforts have been undertaken in order to preserve endangered plant species by improving
+habitat connectivity and condition.
+6
+
+Figure 1: Locations of in-situ plant species richness samples collected in field experiments.
+The Snowy Mountains region is located to the southwest of Canberra (Fig. 1). It encompasses
+the highest mountain ranges of the Australian Alps, serving as an important habitat for alpine-
+exclusive species. There are 212 species of vascular plants, of which 21 are endemic (Pickering et al.,
+2008). Due to its unique status in Australia’s ecosystem, the plant biodiversity in Snowy Mountains
+has drawn consistent interest from the research community (e.g. K¨orner (1995), Pickering et al.
+(2008), and Pickering and Green (2009)).
+For on-ground measures of vascular plant species richness, we obtained plant community survey
+data from the NSW BioNet Vegetation Information System database (Government, 2019). Field
+surveys were conducted to collect species richness samples in 2016 and 2017 for the two regions,
+with a sampling plot area of 400 m2 (20 m × 20 m) at each surveying location. A significant
+bushfire event occurred during the 2019–2020 summer, with some of the sampling points situated
+within the affected areas. These bushfire-affected samples were excluded from the data set, based
+7
+
+Southern
+Sydney
+Tablelands
+NT
+Qld
+WA
+SA
+NSW
+Vic
+ACT
+Tas
+Canberra
+Snowy
+Mountains
+N
+100kmon the National Indicative Aggregated Fire Extent Datasets (NIAFED) provided by the Australian
+Government Department of Agriculture, Water and the Environment. After the exclusion, a total
+of 44 and 29 samples were used in this study for analysis for the Southern Tablelands and Snowy
+Mountains regions, respectively. The locations of these samples are shown in Fig. 1, and their
+associated information is summarised in Table 1. For each sampling plot, the number of native
+vascular plant species was calculated and used as the response variable in our analyses.
+Table 1: Information summary of the plant species richness samples collected in field observations.
+Southern Tablelands
+Snowy Mountains
+Number of Samples
+44
+29
+Sampling Time
+Feb 19, 2017 ∼ Dec 07, 2017
+Feb 24, 2016 ∼ Dec 13, 2017
+Plot Area
+400 m2 (20 m × 20 m)
+400 m2 (20 m × 20 m)
+Geo-extent
+34°12’26”–34°39’07”S
+150°05’57”–150°40’51”E
+35°43’58”–36°16’30”S
+148°23’16”–148°39’02”E
+The histograms of species richness distribution for sampling plots in the two regions are shown
+in Fig. 2. Generally, the sampling plots in Southern Tablelands show a higher richness of species
+than Snowy Mountains, with a mean richness value of 44.4 for the former and 23.5 for the latter.
+The difference in species richness can be mainly attributed to the fact that the Southern Tablelands
+is located at lower altitudes with a relatively warmer climate than the mountainous region of Snowy
+Mountains.
+2.2. Satellite Data
+The DESIS spectrometer (Krutz et al., 2019) is embedded in the MUSES platform onboard
+the International Space Station at an altitude of approximately 400 km. It operates in a push-
+broom imaging mode featuring state-of-the-art radiometric and spectral specifications. It delivers
+8
+
+Figure 2: Histograms of species richness distribution for sampling plots in the (a) Southern Table-
+lands, and (b) Snowy Mountains regions.
+hyperspectral images with 235 spectral bands over the visible and near-infrared regions of 400 ∼
+1000 nm, with a spectral resolution of 2.55 nm and a spatial resolution of 30 m. The radiometric
+resolution for each band is 12 bit with 1 bit gain. The signal-to-noise ratio is 195 at the wavelength
+of 550 nm.
+In our study, the DESIS Level-2A product was used. It consisted of surface reflectance images
+9
+
+14
+(a)
+12
+10 
+8 -
+Count of
+6
+4
+2
+0
+25
+30
+35
+40
+45
+50
+55
+60
+Species Richness
+10
+(b)
+8
+6 
+4
+2
+0
+10
+15
+20
+25
+30
+35
+40
+Species Richnesswith atmospheric correction having been applied. The correction was conducted with DLR’s PACO
+(Python Atmospheric COrrection) software (De los Reyes et al., 2018) where the MODTRAN®
+radiative transfer model (Berk, 2016) served as the module for simulating atmospheric effects.
+As inputs for atmospheric simulation, aerosol optical thickness and water vapour content were
+retrieved per pixel using reflectance in the red and NIR bands, and bands around the water
+absorption features of 820 nm, respectively.
+DESIS spectra intersecting with locations of the
+species richness samples were queried within CSIRO’s Earth Analytics and Science Innovation
+(EASI) platform. We selected spectra captured in January 2020 for our analysis. In order to
+moderate the random noise present in the original spectra, the spectral resolution were down-
+sampled from 2.55 nm into 10.2 nm bins with the assumption of a Gaussian-shaped spectral
+response function. The atmospherically affected bands of 759, 769, 933.4, 943.4, and 953.2 nm,
+and the low quality bands of 402.8, 410.3, and 999.5 nm at the left and right ends of the spectrum
+were removed. A total of 52 bands were retained after the removal. Pixels flagged as cloud by the
+DLR Level-1 processing were masked out.
+The Bidirectional Reflectance Distribution Function (BRDF) effect in DESIS data is un-
+neglectable, given the 4.1◦ field of view in conjunction with the ±15◦ along track pointing capability
+of the DESIS sensor, and the ±25◦ along track and −45◦ ∼ +5◦ cross track tilting capability of
+the MUSES platform. Following the approach adopted in Green and Craig (1985) and Ong and
+Cudahy (2014) for correcting BRDF effect in hyperspectral data, in our study each DESIS spectra
+was mean-normalised with each spectral band being divided by the mean value over all bands.
+For comparison purpose, cloud-free Sentinel-2 multispectral data observed closest to the sensing
+time of DESIS spectra were also downloaded. These Sentinel spectra were downloaded as Level-2A
+surface reflectance with atmospheric effects being corrected. Each spectrum consisted of 12 bands
+covering the visible and near-infrared spectral regions. The Sentinel-2 data were re-sampled into
+a spatial resolution of 30 m to be consistent with that of the DESIS data. A comparison between
+10
+
+the DESIS and Sentinel-2 spectra at one of the ground sampling plots is shown in Fig. 3. Both
+spectra cover the visible, near-infrared, and short-wave-infrared (SWIR) regions, including the
+red-edge region that is critical for vegetation mapping. The Sentinel-2 and DESIS spectra show a
+consistent shape in these spectra regions, with DESIS having a much denser band coverage. The
+SWIR region is covered by Sentinel-2 only with its two bands. Though information in SWIR is
+not contained in the DESIS data, it will be provided by the upcoming DLR mission of EnMAP
+(for which DESIS has been served as a preparation mission), as shown in Fig. 3.
+Figure 3: Comparison of the DESIS (before and after pre-processing) and Sentinel-2 spectra at
+one of the ground sampling plots. An EnMAP spectrum simulated for a random location is also
+shown for reference.
+2.3. Dimensionality Reduction for DESIS Hyperspectral Data
+The DESIS hyperspectral data provide rich information in the abundant and spectrally con-
+tinuous bands, but these bands are highly collinear. We performed dimensionality reduction to
+11
+
+0.4
+Visible
+Near-Infrared
+Short-Wave-Infrared
+0.35
+DESIS (After Pre-processing)
+0.3
+○ DESIS (Before Pre-processing
++ Sentinel-2
+0.25
+X EnMAP (Simulated)
+Reflectance
+0.2
+R
+0.15
+0.1
+0.05
+0
+300
+800
+1300
+1800
+2300
+Wavelength (nm)address this problem—testing three different approaches, namely the Principal Component Anal-
+ysis (PCA), Canonical Correlation Analysis (CCA), and Partial Least Squares analysis (PLS).
+These methods aim to find a linear transformation to project the DESIS spectra from the original
+space of n spectral bands to a new space of a reduced dimensionality defined by k uncorrelated
+components, with k being smaller than n.
+Mathematically, the transformation for dimensionality reduction is written as:
+T = XW,
+(1)
+where the input data for the transformation is X, which is a n×m matrix consisting of the original
+spectra, with n being the number of observed spectra and m being the number of spectral bands;
+the output of the transformation is T = [t1, t2, · · · , tk], which is a n × k matrix consisting of k
+components of X; W = [w1, w2, · · · , wk] is a m × k matrix transforming X from the original n-
+dimensional space into a new space of k components. The weights in wi are a measure of relative
+contributions of the original bands to the transformed component ti = Xwi.
+The number of
+components, k, needs to be preset. In this study, k is selected as the one that achieves the highest
+cross-validation accuracy.
+The PCA is an unsupervised algorithm that finds orthogonal components as the ones that
+explain the maximum variance in the spectral data, disregarding the target values of species
+richness. In contrast, CCA and PLS are supervised with both spectral data and species richness
+values being taken into account in the computation of components. The difference between CCA
+and PLS is that CCA seeks to maximise the correlation between computed components and species
+richness values, while PLS aims to maximise the covariance between the two.
+2.4. Estimation of Species Richness with Regression Models
+After reducing the dimensionality of spectral data from the original n bands to k compo-
+nents, regression is conducted to predict species richness from the components, such that the
+12
+
+mismatch between model-predicted and ground-truth species richness is minimised. The Kernel
+Ridge Regression (KRR), Gaussian Process Regression (GPR), and Random Forest Regression
+(RFR) algorithms are employed and compared, covering respectively the deterministic, Bayesian,
+and ensemble approaches to statistical regression. Here we focus on formulating our task of esti-
+mating species richness within the frameworks of these regression approaches.
+The DESIS spectra {xi}n
+i=1 can be represented by their components {ti}n
+i=1 using one of the
+dimensionality reduction methods described in Subsection 2.3. The KRR transforms {ti}n
+i=1 into
+a feature space of high dimensionality (potentially infinite dimensionality) via a function ϕ(ti). A
+linear model in the high-dimensional feature space is non-linear when it projects back to the original
+space, thus enabling capturing non-linear relationships in the data. The number of parameters
+for a linear model increases with space dimensionality, posing a risk of over-fitting. In order to
+constrain the model complexity, a regularisation term is adopted in KRR to penalise the norm of
+the coefficient vector b. The optimisation problem is:
+arg min
+b
+n
+�
+i=1
+(tT
+i b − yi)2 + λ ∥b∥2
+2 ,
+(2)
+where λ is the regularisation parameter controls the relative importance of the regularisation term
+∥b∥2
+2.
+The input data for Eq. 2 are the transformed components {ti}n
+i=1. Solving Eq. 2 does not
+involve calculating ϕ(ti). Instead, it only requires computation of the inner product k(ti, tj) =
+ϕ(ti)ϕ(tj), where ti and tj are pairs from the input data {ti}n
+i=1. In this study, the kernel function
+k(ti, tj) is specified as a combination of a dot-product kernel kd(ti, tj), a radial-basis function
+13
+
+kernel kr(ti, tj), and a white kernel kw(ti, tj):
+k(ti, tj) = kd(ti, tj) + kr(ti, tj) + kw(ti, tj),
+kd(ti, tj) = ti · tj + σ2,
+kr(ti, tj) = exp(−∥ti − tj∥2
+2
+2l2
+),
+kw(ti, tj) = δ
+if ti = tj else 0,
+(3)
+where σ, l, and δ are hyperparameters that need to be selected with grid search. The dot-product
+and radial-basis function kernels account for the linearity and non-linearity of the data, respectively,
+while the white kernel explains the noise in the data. For a new spectrum x with its extracted
+components t, the predicted species richness is ˆy = �n
+i=1 αik(ti, t), where αi = (K + λI)−1yi with
+Ki,j = k(ti, tj).
+In contrast to KRR that is formulated in a deterministic form, GPR is a Bayesian approach for
+regression. The underlying function correlating DESIS components and species richness is assumed
+to be distributed probabilistically as a Gaussian Process (GP):
+f(t) ∼ GP(m(t), k(t, t′))
+(4)
+where m(t) is the mean function which is often set to 0, and k(t, t′) is the covariance function that
+can be specified as a kernel function. The input data for Eq. 4 are the transformed components
+{ti}n
+i=1. The covariance function k(t, t′) for GPR is set in the same way as that for KRR (Eq.
+3), i.e., k(t, t′) = k(ti, tj) where ti and tj are pairs from the input data {ti}n
+i=1.
+In contrast
+to KRR in which the optimal kernel parameters is found by grid search, the parameters of the
+covariance function in GPR can be automatically determined based on gradient-ascent on the
+marginal likelihood function. After the posterior likelihood is determined for the GP, the predicted
+species richness is ˆy = �n
+i=1 αik(ti, t) for a new spectrum, where αi = (K + ϵ2I)−1yi with Ki,j
+being k(ti, tj) and ϵ being a pre-set parameter explaining the noise in the data.
+In addition to KRR and GPR that are respectively belong to the deterministic and Bayesian
+regression categories, we also tested the ensemble method of RFR. A random forest combines a
+14
+
+number of decision tree regressors and takes the average regression result over all trees as the final
+estimate. Due to the ensemble structure, RFR tends to produce robust regression results with
+high resistance to overfitting and data noise. The training of a random forest regressor minimises
+the following optimization function:
+arg min
+θj
+1
+d
+d
+�
+j=1
+n
+�
+i=1
+[h(ti; θj) − yi]2 ,
+(5)
+where h(·) is the decision tree regressor with θj being the parameters of the jth tree; d is the
+number of decision trees, which was set to 100 in this study.
+2.5. Determination of Hyperparameters
+Hyperparameters that need to be pre-set included the number of components k in the dimen-
+sionality reduction methods PCA, CCA, and PLS, and the kernel parameters σ, l, and δ in the
+non-linear regression method KRR. The number of components k determines how much informa-
+tion to retain after dimensionality reduction. An optimal selection of k is able to reduce data
+redundancy without excessively discarding useful information in the original DESIS spectra. In
+this study, we tested different k values ranging from 1 to 10. The optimal k value was selected
+based on the accuracy of species richness prediction, and the amount of variance in the spectral
+data that the retained components could explain. The tunable hyperparameters σ, l, and δ for
+the kernel function define the non-linearity structure of the regression model KRR. These kernel
+parameters were selected based on grid search. A grid of values was tested with each parameter
+varying from 10−5 to 105 on a logarithmic scale. The combination of σ, l, and δ that produced
+the best performance in species richness prediction was selected.
+2.6. Accuracy Assessment and Analyses
+A two-fold validation scheme was used in this study for assessing the modelling accuracy. For
+each of the Southern Tablelands and Snowy Mountains regions, the whole data set was randomly
+partitioned into two subsets (Subsets I and II), with Subset I dedicated to training and Subset II
+15
+
+for validation (Round I), followed by Subset II for training and Subset I for validation (Round II).
+This procedure was repeated 100 times with the data set being partitioned differently each time.
+Correlation diagrams were plotted between the ground-truth species richness and the predicted
+values from the DESIS spectra. The coefficient of correlation (r) and Root-Mean-Square Error
+(RMSE) were calculated to evaluate the performance of the models. Results with different feature
+extraction procedures (PCA, CCA, and PLS) and regression models (KRR, GPR, and RFR) were
+computed and compared.
+2.7. Band Importance Analysis
+The DESIS data consist of spectral measurements over the visible and near-infrared bands
+from 400 nm to 1000 nm. Importance analysis was conducted in order to analyse which spectral
+bands provide more explanatory power than others in predicting plant species richness. We used
+the vector length of contribution values of each band to all components used in the regression
+weighted by the partial correlation coefficients as the importance index of the band:
+Ii =
+�
+�
+�
+�
+k
+�
+j=1
+(wi,j · pj)2,
+(6)
+where Ii is the importance index for the ith band; wi,j is the (i, j)th element of the weight matrix
+W in Eq. 1, representing the contribution of the ith band to the jth component; pj is the partial
+correlation coefficient of the jth component with the target variable of species richness; k is the
+number of components used in regression. In our study, the importance indices Ii are normalised
+to relative values ˜Ii with sum over all bands equal to one:
+˜Ii =
+Ii
+�m
+i=1 Ii
+,
+(7)
+where m is the number of spectral bands.
+2.8. Comparison with Multispectral Data
+Spaceborne multispectral imagery such as Sentinel-2 is more readily available than hyperspec-
+tral.
+Though Sentinel-2 images have less bands than the DESIS hyperspectral data, they are
+16
+
+delivered on a more stable and systematic basis. In this study, an analysis is conducted to see if
+Sentinel-2 multispectral data are able to achieve comparable results. Through this analysis, we
+hoped to examine how well Sentinel-2 could serve as a substitute for plant biodiversity mapping
+in instances where hyperspectral data are unavailable.
+The Sentinel-2 data set described in Section 2.2 was used for the comparison.
+In order to
+conduct a fair comparison between hyperspectral and multispectral data with minimised differences
+in instrumental specifications and acquisition conditions, a Sentinel-2-like synthetic data set was
+simulated from the DESIS data. The simulation involved resampling the DESIS spectra using
+the spectral response functions of Sentinel-2’s visible and near-infrared bands. Both the real and
+synthetic Sentinel-2 sets were used for species richness prediction, with results being compared to
+those achieved with DESIS data.
+3. Results
+3.1. Spectral Reflectance Differences Between Species Richness Classes
+The blue, green, and red curves in Fig. 4 show the DESIS spectra averaged over ground sam-
+pling plots with species richness falling into the low, intermediate, and high tertiles, respectively.
+Results for the Southern Tablelands and Snowy Mountains regions are displayed in Figs 4a and
+b, respectively.
+For each region, it was observed that plots of higher richness showed a lower
+reflectance in the visible range of 400 ∼ 680 nm. Considering that major absorption features of
+chlorophyll are located within the visible region (Zhao et al., 2014b), the lower reflectance in this
+spectral portion might indicate a higher concentration of chlorophyll. It was also observed that
+plots of higher richness showed a higher reflectance in the near-infrared plateau of 780 ∼ 1000 nm
+and a steeper red edge between 680 ∼ 780 nm, which might suggest a larger Leaf Area Index (LAI)
+(Delegido et al., 2013) and a greater vegetation vigor (Boochs et al., 1990) for those plots. On the
+basis of these observations, the spectral shape of high richness plots, as compared with spectra
+17
+
+of intermediate and low richness plots, may imply a generally richer vegetation. This is consis-
+tent with findings reported in literature that high species richness enhances primary productivity
+(Wang et al., 2016; Grace et al., 2016) and biomass (Malhi et al., 2020; Tilman et al., 1997).
+3.2. Hyperparameter Selection Results
+Fig. 5 shows the r and RMSE values achieved with different numbers of components (ranging
+from 1 to 10) being selected as features. The results were averaged over the two study regions.
+It was observed that two components achieved the best performance for all the three feature
+extraction methods of PCA (Fig.
+5a), CCA (Fig.
+5b), and PLS (Fig.
+5c).
+When only one
+component was used with more information in the original spectral data being discarded, lower
+r values and higher RMSE values were also observed, indicating a poorer performance compared
+with that achieved by two components. When more than two components were retained with a
+higher degree of data redundancy presenting, weaker results were also observed. The eigenvalue,
+percentage of explained variance, and cumulative percentage of explained variance with different
+components are shown in Table 2. It was seen that with two components, the PCA, CCA, and PLS
+could explain 93.81%, 89.34%, and 87.38% of variance in the DESIS data, respectively. Based on
+these results, the number of components, k, was set to two in our experiments for dimensionality
+reduction with PCA, CCA, and PLS. The first and second components are plotted in Fig. 6. The
+first component depicted the general shape of the spectral brightness, while the second component
+highlighted more on local spectral features of the spectrum.
+18
+
+Figure 4: Average DESIS reflectance spectra calculated from field sample plots with low, interme-
+diate, and high species richness for the (a) Southern Tablelands and (b) Snowy Mountains regions.
+19
+
+0.30
+(a)
+0.25
+Low richness plots
+Intermediate richness plots
+0.20
+High richness plots
+Reflectance
+0.15
+0.10
+0.05
+0.00
+400
+500
+600
+700
+800
+900
+1000
+Wavelength (nm)
+0.30
+(b)
+0.25
+Low richness plots
+Intermediate richness plots
+0.20
+High richness plots
+Reflectance
+0.15
+0.10
+0.05
+0.00
+400
+500
+600
+700
+800
+900
+1000
+Wavelength (nm)Figure 5: Impact of number of components on the estimation accuracy of plant species richness
+with (a) Principal Component Analysis (PCA), (b) Canonical Correlation Analysis (CCA), and
+(c) Partial Least Squares analysis (PLS).
+20
+
+0.7
+(a)
+Root Mean Square Error (RMSE)
+r
+Coefficient of Regression (r)
+RMSE
+￥
+8.5
+0.6
+8
+0.5
+7.5
+0.4
+- 6.5 
+0.3
+6
+2
+3
+4
+5
+8
+9
+10
+Number of Components
+0.7
+(b)
+Root Mean Square Error (RMSE)
+Coefficient of Regression (r)
+RMSE
+8.5
+0.6
+8
+0.5
+7.5
+7
+0.4
+ 6.5
+0.3
+1
+2
+3
+4
+6
+7
+8
+9
+10
+5
+Number of Components
+0.7
+(c)
+Root Mean Square Error (RMSE)
+r
+Coefficient of Regression (r)
+RMSE
+8.5
+0.6
+8
+0.5
+7.5
+/
+0.4
+6.5
+0.3
+1
+2
+3
+4
+5
+6
+9
+10
+Number of ComponentsTable 2: The eigenvalue, percentage of explained variance, and cumulative percentage of explained variance for components produced with
+Principal Component Analysis (PCA), Canonical Correlation Analysis (CCA), and Partial Least Squares analysis (PLS).
+Component
+PCA
+CCA
+PLS
+Eigenvalue
+% of
+Variance
+Cumulative %
+of Variance
+Eigenvalue
+% of
+Variance
+Cumulative %
+of Variance
+Eigenvalue
+% of
+Variance
+Cumulative %
+of Variance
+1
+44.35
+84.48
+84.48
+40.43
+74.61
+74.61
+40.96
+75.58
+75.58
+2
+4.90
+9.34
+93.81
+7.98
+14.73
+89.34
+6.39
+11.80
+87.38
+3
+2.22
+4.23
+98.04
+4.16
+7.67
+97.01
+4.94
+9.12
+96.50
+4
+0.44
+0.84
+98.87
+0.61
+1.13
+98.14
+0.70
+1.29
+97.79
+5
+0.22
+0.41
+99.29
+0.48
+0.89
+99.03
+0.41
+0.75
+98.55
+6
+0.10
+0.18
+99.47
+0.12
+0.23
+99.26
+0.23
+0.42
+98.97
+7
+0.04
+0.08
+99.55
+0.10
+0.19
+99.45
+0.14
+0.26
+99.22
+8
+0.04
+0.08
+99.63
+0.08
+0.14
+99.59
+0.11
+0.19
+99.42
+9
+0.03
+0.06
+99.69
+0.06
+0.11
+99.69
+0.08
+0.14
+99.56
+10
+0.02
+0.04
+99.73
+0.04
+0.08
+99.77
+0.06
+0.11
+99.67
+21
+
+Figure 6: The first and second components for the transformed DESIS spectra.
+Left column:
+Result for the Southern Tablelands with (a) Principal Component Analysis (PCA), (b) Canonical
+Correlation Analysis (CCA), and (c) Partial Least Squares analysis (PLS); Right column: Result
+for the Snowy Mountains region with (d) PCA, (e) CCA, and (f) PLS.
+22
+
+0.6
+0.6
+(a)
+1st Component
+(d)
+1st Component
+0.4
+2nd Component
+0.4
+2nd Component
+0.2
+0.2
+Component
+Component
+0.0
+0.0
+-0.2
+-0.2
+-0.4
+-0.4 
+-0.6
+-0.6
+400
+600
+800
+1000
+400
+600
+800
+1000
+Wavelength (nm)
+Wavelength (nm)
+0.6
+0.6
+(b)
+1st Component
+(e)
+1st Component
+0.4 -
+2nd Component
+0.4 -
+2nd Component
+0.2
+0.2
+Component
+0.0
+0.0
+-0.2
+-0.2
+-0.4 
+-0.4 
+-0.6
+-0.6 
+400
+600
+800
+1000
+400
+600
+800
+1000
+Wavelength (nm)
+Wavelength (nm)
+0.6
+0.6
+(c)
+1st Component
+(f)
+1st Component
+0.4 -
+2nd Component
+0.4
+2nd Component
+0.2
+0.2
+Component
+Component
+0.0
+0.0
+-0.2
+-0.2
+-0.4 
+-0.4
+-0.6
+-0.6
+400
+600
+800
+1000
+400
+600
+800
+1000
+Wavelength (nm)
+Wavelength (nm)The grid searching result for the optimal KRR hyperparameters σ, l, and δ is shown in Fig. 7.
+The best combination of kernel parameters was σ = 103, l = 103, and δ = 10. This combination
+of kernel parameter values was used as default values for KRR in our experiments.
+Figure 7: Selection of kernel parameters σ, l, and δ based on grid search for Kernel Ridge Regression
+(KRR). The best performing combinations of kernel parameters are circled in black.
+3.3. Plant Species Richness Prediction
+The assessment of different dimensionality reduction and regression methods (Table 3) showed
+that, for the Southern Tablelands region, the best result was achieved with a combination of
+PLS for dimensionality reduction and GPR for regression, with r being 0.76 and RMSE being
+23
+
+= 10-3
+= 10-5
+10-1
+105
+105
+105
+103
+103
+103
+101
+101
+101
+10-1
+10-1,
+10-1
+10-3
+10-3
+10-3
+10-5
+10-5.
+10-5
+10-5
+10-3
+10-1
+101
+103
+105
+10-5
+10-3
+10-1
+101
+103
+105
+10-5
+10-3
+10-1
+101
+103
+105
+0
+0
+0
+S= 101
+= 103
+= 105
+105
+105
+105
+103
+103
+103
+101
+101
+101
+10-1
+10-1
+10-1
+10-3
+10-3
+10-3
+10-5
+10-5
+10-5
+10-5
+10-3
+10-1
+101
+103
+105
+10-5
+10-3
+10-1
+101
+103
+105
+10-5
+10-3
+10-
+101
+103
+105
+0
+0
+0
+RMSE
+10
+>105.89. Results with PLS as the dimensionality reduction method performed better than those with
+PCA and CCA. The PLS-based models achieved 0.75 ∼ 0.76 for r and 5.89 ∼ 5.92 for RMSE,
+respectively, better than the PCA-based models with 0.70 ∼ 0.71 for r and 5.99 ∼ 6.02 for RMSE,
+and the CCA-based models with 0.69 ∼ 0.73 for r and 5.98 ∼ 6.02 for RMSE.
+Table 3: The coefficient of correlation (r) and Root-Mean-Square Error (RMSE) between model
+predicted and ground truth plant species richness for the Southern Tablelands region.
+Dimensionality
+Reduction
+Regression
+r
+RMSE
+PCA
+KRR
+0.70
+6.01
+PCA
+GPR
+0.71
+5.99
+PCA
+RFR
+0.70
+6.02
+CCA
+KRR
+0.69
+6.02
+CCA
+GPR
+0.72
+5.98
+CCA
+RFR
+0.73
+6.00
+PLS
+KRR
+0.75
+5.92
+PLS
+GPR
+0.76
+5.89
+PLS
+RFR
+0.75
+5.91
+The assessment results for the Snowy Mountains region in shown in Table 4.
+The Snowy
+Mountains region is located at high altitudes with less human interference and a generally lower
+species richness than the Southern Tablelands (Fig. 2). Compared with the results for the Southern
+Tablelands in Table 3, it can be seen that generally the r values were lower and RMSE were higher
+for the Snowy Mountains region (Table 4). The best r and RMSE were 0.68 and 5.95, respectively,
+achieved with a combination of PLS for dimensionality reduction and RFR for regression.
+The correlation diagrams in Fig. 8 show the relationship between the ground-truth species
+24
+
+Table 4: The coefficient of correlation (r) and Root-Mean-Square Error (RMSE) between model
+predicted and ground truth plant species richness for the Snowy Mountains region.
+Dimensionality
+Reduction
+Regression
+r
+RMSE
+PCA
+KRR
+0.51
+6.17
+PCA
+GPR
+0.54
+6.03
+PCA
+RFR
+0.56
+6.01
+CCA
+KRR
+0.52
+6.14
+CCA
+GPR
+0.53
+6.08
+CCA
+RFR
+0.54
+5.99
+PLS
+KRR
+0.64
+6.08
+PLS
+GPR
+0.66
+5.97
+PLS
+RFR
+0.68
+5.95
+richness and the predicted values from the DESIS spectra, with data samples from the validation
+set. These results were produced with the best performing models in Tables 3 and 4.
+3.4. Generalised Modelling Results
+Figs 8a and 8b show results with modelling being conducted for the Southern Tablelands and
+Snowy Mountains regions separately. When we modelled the pooled data from both regions, we
+found that accuracy decreased (Fig. 9). The prediction result was r = 0.61 and RMSE = 10.1,
+which was lower than modelling with data from only one region (Figs 8a and 8b). This indicates
+that location-specific modelling performs better than using one model to describe multiple regions.
+The difficulty of modelling relationships between hyperspectral data and plant species richness
+for multiple regions, compared modelling for each region separately, is worth further investigation.
+25
+
+Figure 8: Correlation diagrams between the ground-truth species richness and the predicted values
+from the DESIS spectra for the (a) Southern Tablelands and (b) Snowy Mountains regions. The
+r and RMSE stand for the coefficient of correlation and root-mean-square error.
+Different regions may differ in the assemblages of plant species, and their compositional and struc-
+tural properties, resulting in extra variations in hyperspectral data in addition to those induced by
+the richness of species. These additional variations may add complexity in exploring useful infor-
+mation in hyperspectral data to predict plant species richness. The location-specific relationship
+26
+
+(a)
+55
+Regression Line
+50
+1:1 Line
+45
+Confidence Band
+40
+35
+r = 0.76
+RMSE = 5.89
+30
+25
+25
+30
+35
+40
+45
+50
+55
+Ground Truth Species Richness
+(b)
+35
+Regression Line
+30
+1:1 Line
+Confidence Band
+25
+20
+r = 0.66
+RMSE = 5.97
+15
+10
+10
+15
+20
+25
+30
+35
+Ground Truth Species Richnesshyperspectral data and plant species richness calls for location-dependant modelling or encoding
+location information into the input spectra in future studies when mapping plant species richness
+at continental or global scales is attempted.
+Figure 9: Correlation diagrams between the ground-truth species richness and values predicted
+from DESIS spectra with data congregating both of the Southern Tablelands and Snowy Mountains
+regions. The r and RMSE stand for the coefficient of correlation and root-mean-square error.
+3.5. The Relative Importance of Spectral Bands
+The relative importance of DESIS bands in predicting plant species richness is shown in Fig.
+10, with the aim to analyse which parts of the spectrum had the most explanatory power. Subplots
+10a, b, and c display the results with PCA, CCA, and PLS being used as the feature extraction
+procedure, respectively. From these subplots, it was observed that bands in the red-edge spectral
+region of approx. 700 ∼ 720 nm showed the highest importance. This may suggest that the slope
+of the red-edge is an important spectral feature for plant species richness prediction, considering
+that the low, intermediate, and high species richness plots showed varied red-edge slopes in Fig.
+4.
+This observation might be supported by literature that the red-edge is a critical spectral
+27
+
+60
+Regression Line
+Model Predicted Species Richness
+50
+1:1 Line
+Confidence Band
+40
+30
+r = 0.61
+RMSE = 10.1
+20
+10
+10
+20
+30
+40
+50
+60
+Ground Truth Species Richnessregion for vegetation mapping as it is closely related to biological variables such as leaf area index
+(Delegido et al., 2013), plant vigour (Boochs et al., 1990), and biochemical contents (Mutanga and
+Skidmore, 2007). Followed by the red-edge, the visible range of approx. 400 ∼ 700 nm also showed
+high importance. The importance of visible bands in plant species richness prediction might be
+justified by the fact that, this portion of the spectrum, especially bands in red and blue, is the major
+leaf pigment absorption range. It is sensitive to mainly chlorophyll a and b contents, according
+to the sensitivity analysis result reported in Zhao et al. (2014b,a). Though less important than
+the red-edge and visible regions, the near-infrared region with wavelengths longer than 720 nm
+also showed some explanatory power. This indicated that the near-infrared region also provided
+contributory information in predicting plant species richness, as this portion of the spectrum, often
+characterised by high reflectance for vegetation, is sensitive to leaf thickness (Zhao et al., 2014a)
+and the amount, arrangement, and inclination of leaves in the canopy (Knipling, 1970).
+3.6. Comparison with Multispectral Data
+Figs 11a and 11b show correlation diagrams between the ground-truth species richness and
+values predicted from the real Sentinel-2 multispectral data for the Southern Tablelands and Snowy
+Mountains regions, respectively. It is seen that, a prediction result of r = 0.66 and RMSE = 6.27
+was achieved for the Southern Tablelands region (Fig. 11a), and r = 0.57 and RMSE = 6.31 for
+the Snowy Mountains region (Fig. 11b). The prediction results with the Sentinel-2-like synthetic
+data set are shown in Fig. 12. For the Southern Tablelands, the r and RMSE values are 0.65 and
+6.19, while for the Snowy Mountains, the r and RMSE values are 0.57 and 6.39.
+These multispectral results were slightly worse than the hyperspectral results shown in Fig. 8.
+This could be explained by the fact that hyperspectral data contain richer information in the spec-
+tral domain. However, when taking data availability and stability into consideration, spaceborne
+multispectral imagery could serve as a reliable alternative for plant biodiversity mapping. Also,
+data such as Sentinel has much more consistent and regular sampling over time, and harnessing the
+28
+
+Figure 10: Relative importance analysis for the DESIS bands in predicting plant species richness
+with spectral features being extracted by (a) Principal Component Analysis (PCA), (b) Canonical
+Correlation Analysis (CCA), and (c) Partial Least Squares analysis (PLS).
+29
+
+0.08
+(a)
+: Importance
+0.06
+0.04
+Relative I
+0.02
+0.00
+400
+500
+600
+700
+800
+900
+1000
+Wavelength (nm)
+0.08
+(b)
+: Importance
+0.06
+0.04
+Relative I
+0.02
+0.00
+400
+500
+600
+700
+800
+900
+1000
+Wavelength (nm)
+0.08
+(c)
+Relative Importance
+0.06
+0.04
+0.02
+0.00
+400
+500
+600
+700
+800
+900
+1000
+Wavelength (nm)Figure 11: Correlation diagrams between the ground-truth species richness and values predicted
+from the real Sentinel-2 multispectral data set for the (a) Southern Tablelands and (b) Snowy
+Mountains regions. The r and RMSE stand for the coefficient of correlation and root-mean-square
+error.
+added information in temporal changes in multispectral signal could provide further explanatory
+power for biodiversity analyses.
+30
+
+(a)
+55
+Regression Line
+50
+1:1 Line
+Confidence Band
+45
+40
+35
+r = 0.66
+30
+RMSE = 6.27
+25
+25
+30
+35
+40
+45
+50
+55
+Ground Truth Species Richness
+(b)
+35
+Model Predicted Species Richness
+Regression Line
+1:1 Line
+30
+Confidence Band
+25
+20
+r = 0.57
+15 
+RMSE = 6.31
+10
+10
+15
+20
+25
+30
+35
+Ground Truth Species RichnessFigure 12: Correlation diagrams between the ground-truth species richness and values predicted
+from the Sentinel-2-like synthetic multispectral data set for the (a) Southern Tablelands and (b)
+Snowy Mountains regions. The r and RMSE stand for the coefficient of correlation and root-mean-
+square error.
+31
+
+(a)
+55
+Regression Line
+50
+l:1 Line
+Confidence Band
+45
+40
+35
+r = 0.65
+30
+RMSE = 6.19
+25
+25
+30
+35
+40
+45
+50
+55
+Ground Truth Species Richness
+(b)
+35
+Regression Line
+1:1 Line
+30
+Confidence Band
+25
+20
+r= 0.57
+15
+RMSE = 6.39
+10
+10
+15
+20
+25
+30
+35
+Ground Truth Species Richness4. Discussions
+4.1. Comparison to Previous Studies
+Predicting plant biodiversity from remotely sensed measurements helps with making evidence-
+informed environmental policies and conducting effective conservation activities. Hyperspectral
+imagery delivered by the recently launched DESIS instrument opens the potential for plant bio-
+diversity monitoring at a finer spatial scale and with a higher accuracy. In this study, we take
+the the Southern Tablelands and Snowy Mountains regions in southeast Australia as experimental
+sites to test the relationship between DESIS data and plant species richness. A two step approach
+is proposed, where feature extraction techniques are first used to reduce the dimensionality of hy-
+perspectral data, followed by regression models with kernel functions to account for the linearity,
+non-linearity, and noise in data.
+Obtaining informative features from hyperspectral data is important to the success of sub-
+sequent interference of plant species richness. In previous studies, features have been primarily
+selected as a subset of the original bands, or as spectral indices computed from a subset of the
+original bands (e.g., Malhi et al. (2020), Peng et al. (2018), and Wang et al. (2016)). The bands
+or indices are often selected based on our a priori knowledge of the spectral properties, such as
+absorption features of biochemical contents. Though selected features usually offer good explain-
+ability, it means we have to discard information in unselected bands. Considering the large number
+of bands in hyperspectral data, many of them would be discarded as the high collinearity of hy-
+perspectral data often requires a considerable reduction of dimensionality. In contrast, the feature
+extraction approach, as adopted in this work, makes use of the information in all original bands
+by transforming them into a new feature space of lower and non-collinear dimensionality.
+Linear models have been primarily employed in previous studies to relate features to species
+richness. For example, multiple linear regression has been adopted by Wang et al. (2016) and
+Malhi et al. (2020), and stepwise linear regression by Peng et al. (2018). However, the relationship
+32
+
+between features and richness might not necessarily follows simply a linear pattern. In our study,
+a novel kernel function is proposed (Eq. 3), with the dot-product and radial-basis function kernels
+account for the linearity and non-linearity of the data, respectively, and the white kernel explains
+the noise in the data. The ability of our model to explore both linear and non-linear patterns
+distinguishes our work from aforementioned studies.
+4.2. Mechanism of Plant Species Richness Prediction
+Though detailed field surveys have been deemed as the most accurate and reliable way for
+assessing plant biodiversity, remote sensing data can serve as a proxy for large-scale and cost-
+effective biodiversity mapping.
+The mechanism justifying the use of remote sensing has been
+widely discussed in literature. It is worth noting the exact mechanism is dependant on which
+method is used and the sensor specifications (e.g., pixel size and spectral range). The richness of
+species can be estimated either directly from raw spectra, or via the extracted plant functional
+traits or types (Wang and Gamon, 2019). Methods based on the spectral variation hypothesis is
+also intriguing, whereby the spectral variation across spatially adjacent pixels is employed as the
+proxy (Palmer et al., 2002; Fassnacht et al., 2022). Here we focus on discussing the underlying
+mechanism that underpins our study where the DESIS spectra with a pixel size of 30 m and a
+spectral range of 400–1000 nm are linked to on-ground species richness via feature extraction and
+statistical regression.
+It has been reported in literature that plant communities of high diversity tend to have an
+enhanced primary productivity (Wang et al., 2016; Grace et al., 2016) and a higher above ground
+biomass (Malhi et al., 2020; Tilman et al., 1997). Though the reason to explain the richness–
+productivity/biomass relationship is a subject of debate, a common theory is that the comple-
+mentary roles played by different species lead to lower nutrient losses and more sustainable soils
+(Tilman et al., 1996). Species complementarity allows plants to capture resources in ways that are
+complementary in space or/and time, leading to increased biomass production (Cardinale et al.,
+33
+
+2007). For example, a high number of species allows plants to reside in various partitions of niches,
+resulting in a denser occupation of space and a higher efficiency of water, nutrition, and sunlight
+usage.
+Based on this relationship, many studies have successfully estimated plant species richness from
+hyperspectral measurements (e.g., Wang et al. (2016), Malhi et al. (2020), and Peng et al. (2018)),
+given that remotely sensed hyperspectral data is a good proxy of vegetative biomass and primary
+productivity. A positive and dynamic productivity–diversity relationship is observed in a prairie
+grassland experiment at Cedar Creek, Minnesota, USA, with NDVI being employed as a proxy of
+vegetation productivity to estimate species richness (Wang et al., 2016). The correlation between
+spectral indices and plant species diversity is also reported in Peng et al. (2018) for a semi-arid
+sandland ecosystem in Inner Mongolia, China.
+It is worth noting that previous studies are primarily focused on ground-measured (e.g., (Peng
+et al., 2018) and Wang et al. (2016)) or airborne (e.g., Asner (2008)) hyperspectral data, with a
+limited spatial range. The recently launched DESIS and PRISMA (Pignatti et al., 2013; Verrelst
+et al., 2021), and the upcoming EnMAP (Environmental Mapping and Analysis Program) (Guanter
+et al., 2015) missions, enable us to test the potential of spaceborne hyperspectral measurements
+in plant species richness mapping. Our study shows a promising correlation between the two, and
+finds that the correlation is location-dependent.
+4.3. Limitations
+Over the past decades, in-situ samples of plant species richness have been collected via various
+survey campaigns. In total, more than 188 thousands of samples have been gathered in Australia
+as of the year of 2018 (Gellie et al., 2018).
+However, most of the samples are not able to be
+matched with a DESIS observation that is temporally close enough to them, as DESIS has not
+been in operation until 2018. In this study, in order to avoid large temporal discrepancies, we have
+limited our on-ground samples to those that are less than three years apart with their associated
+34
+
+DESIS spectra. The limited spatial coverage of DESIS images, and bushfires during the 2019–2020
+summer, have further reduced the number of available samples. As a result, analyses in this study
+are conducted on a relatively small number of samples. Nevertheless, the ever increasing amount
+of both satellite images and on-ground samples will enable more comprehensive assessment of the
+potential of spaceborne hyperspectral remote sensing for plant biodiversity mapping. Moreover,
+though the data set used in this work does not have information on which strata the observed species
+come from, it is worth splitting out richness for different strata in future field surveys. With strata
+information on record, we might be able to model tree richness and understory richness separately,
+and then combine them once predicted to get total richness. The results reported in this study
+may serve as a basis for future studies.
+It is important to note that the DESIS pixel size (30×30 m) does not match exactly with the plot
+area of ground species richness samples (400 m2 = 20×20 m). Though richness measurements could
+be scaled to a larger or smaller plot area using an assumed power relationship (S = cAz) between
+species richness (S) and plot area (A) (Rosenzweig, 1995), the location-specific power parameter
+(z) is often hard to be accurately determined without adequate knowledge about the experiment
+site. It is suggested that, in future field campaigns, it is worthwhile to gather information on
+the richness-area relationship, in order to facilitate accurate up- and down-scaling of plot areas.
+Potential inaccuracies in geo-registration of DESIS images and in geo-positioning measurements
+during field surveys may also result in geo-mismatch between pixels and ground plots. Better geo-
+registration and geo-positioning accuracies would help reduce uncertainties in predicting species
+richness in future works.
+The DESIS sensor covers the visible and near-infrared (VNIR) portions of the spectrum. As
+shown in this study, this spectral range is informative in plant biodiversity mapping. However, it is
+also worthwhile to leverage the potential of the short-wave-infrared (SWIR) bands. The upcoming
+EnMAP imaging spectroscopy mission (Guanter et al., 2015) is scheduled to be launched in 2022.
+35
+
+The EnMAP dual-spectrometer instrument, covering both VNIR and SWIR from 420 nm to 2450
+nm (as shown in Fig. 3), will provide an opportunity to integrate information from SWIR for plant
+biodiversity mapping. In addition to hyperspectral optical imagery, the combination of data from
+other sensor types, such synthetic-aperture radar (SAR) or LiDAR data, could also be explored
+to improve our ability in remote mapping of plant biodiversity.
+5. Conclusion
+Spaceborne hyperspectral remote sensing is a promising and cost-effective data source to en-
+able plant biodiversity mapping. Thanks to its advanced spectral and spatial specifications, the
+recently launched hyperspectral instrument DESIS (the DLR Earth Sensing Imaging Spectrom-
+eter) opens up an opportunity to monitor plant biodiversity at a finer spatial scale and with a
+higher accuracy. In this study, we assessed the ability of DESIS hyperspectral data in predicting
+plant species richness in the Southern Tablelands and Snowy Mountains regions in southeast New
+South Wales, Australia. The spectral features were firstly extracted, and then correlated to plant
+species richness via statistical regression. We evaluated the performance of several combinations
+of feature extraction procedures (PCA, CCA, and PLS) and regression models (KRR, GPR, and
+RFR). The main findings of this study are summarised as follows:
+(1) Plant species richness values were predicted from DESIS data in the two study regions.
+Prediction accuracies fell within a comparable range for different combinations of feature extraction
+techniques and regression models (Tables 3 and 4). The best prediction results were r = 0.76 and
+RMSE = 5.89 for Southern Tablelands region, and r = 0.68 and RMSE = 5.95 for the Snowy
+Mountains region.
+(2) The correlation between DESIS hyperspectral data and plant species richness was region-
+specific. Modelling the correlation separately for each region produced better results than building
+a single model for all regions.
+36
+
+(3) The relative importance analysis conducted among DESIS bands showed that the red-edge,
+red, and blue spectral regions are more important in predicting plant species richness than the
+green bands and the near-infrared bands beyond red-edge (noting that the SWIR region is not
+sampled by DESIS).
+(4) The DESIS hyperspectral data performed better than multispectral data in predicting
+plant species richness, indicating that the provision of richer information in the spectral domain is
+important for diversity mapping.
+Results shown in this study provided a quantitative reference on the potential for spaceborne
+hyperspectral data to be used in the mapping of on-ground plant species richness. Future studies
+should focus on extending the current approach to larger areas, investigating the potential of
+upcoming hyperspectral missions that extend into the SWIR region, and exploring the combination
+of data from other sensor types.
+6. Acknowledgement
+The authors are grateful to the anonymous reviewers for their important and insightful com-
+ments for improving this manuscript.
+References
+Alonso, K., Bachmann, M., Burch, K., Carmona, E., Cerra, D., De los Reyes, R., Dietrich, D.,
+Heiden, U., H¨olderlin, A., Ickes, J., et al., 2019. Data products, quality and validation of the
+DLR Earth sensing imaging spectrometer (DESIS). Sensors 19, 4471.
+Asner, G.P., 2008. Hyperspectral remote sensing of canopy chemistry, physiology, and biodiversity
+in tropical rainforests. Hyperspectral remote sensing of tropical and sub-tropical forests , 261–
+296.
+Berk, A., 2016. MODTRAN 5.4. 0 User’s Manual. Spectral Sciences Inc., Burlingon, MA .
+37
+
+Boochs, F., Kupfer, G., Dockter, K., K¨uhbauch, W., 1990.
+Shape of the red edge as vitality
+indicator for plants. International Journal of Remote Sensing 11, 1741–1753.
+Bush, A., Sollmann, R., Wilting, A., Bohmann, K., Cole, B., Balzter, H., Martius, C., Zlinszky,
+A., Calvignac-Spencer, S., Cobbold, C.A., et al., 2017. Connecting Earth observation to high-
+throughput biodiversity data. Nature Ecology & Evolution 1, 1–9.
+Cardinale, B.J., Wright, J.P., Cadotte, M.W., Carroll, I.T., Hector, A., Srivastava, D.S., Loreau,
+M., Weis, J.J., 2007. Impacts of plant diversity on biomass production increase through time
+because of species complementarity.
+Proceedings of the National Academy of Sciences 104,
+18123–18128.
+Carlson, K.M., Asner, G.P., Hughes, R.F., Ostertag, R., Martin, R.E., 2007. Hyperspectral remote
+sensing of canopy biodiversity in Hawaiian lowland rainforests. Ecosystems 10, 536–549.
+Ceballos, G., Ehrlich, P.R., Barnosky, A.D., Garc´ıa, A., Pringle, R.M., Palmer, T.M., 2015. Ac-
+celerated modern human–induced species losses: Entering the sixth mass extinction. Science
+Advances 1, e1400253.
+De Palma, A., Hoskins, A., Gonzalez, R.E., B¨orger, L., Newbold, T., Sanchez-Ortiz, K., Ferrier, S.,
+Purvis, A., 2021. Annual changes in the Biodiversity Intactness Index in tropical and subtropical
+forest biomes, 2001–2012. Scientific Reports 11, 1–13.
+Delegido, J., Verrelst, J., Meza, C., Rivera, J., Alonso, L., Moreno, J., 2013. A red-edge spectral
+index for remote sensing estimation of green LAI over agroecosystems. European Journal of
+Agronomy 46, 42–52.
+Eckardt, A., Horack, J., Lehmann, F., Krutz, D., Drescher, J., Whorton, M., Soutullo, M., 2015.
+DESIS (DLR Earth sensing imaging spectrometer for the ISS-MUSES platform), in: 2015 IEEE
+International Geoscience and Remote Sensing Symposium (IGARSS), IEEE. pp. 1457–1459.
+38
+
+Fallding, M., 2002. A planning framework for natural ecosystems of the ACT and NSW Southern
+Tablelands. Natural Heritage Trust, NSW National Parks and Wildlife Service.
+Fassnacht, F.E., M¨ullerova, J., Conti, L., Malavasi, M., Schmidtlein, S., 2022. About the link
+between biodiversity and spectral variation. Applied Vegetation Science , e12643.
+Feilhauer, H., Asner, G.P., Martin, R.E., 2015. Multi-method ensemble selection of spectral bands
+related to leaf biochemistry. Remote Sensing of Environment 164, 57–65.
+Frankel, O.H., Brown, A.H., Burdon, J.J., 1995. The Conservation of Plant Biodiversity. Cam-
+bridge University Press.
+F´eret,
+J.B.,
+Asner,
+G.P.,
+2014.
+Mapping
+tropical
+forest
+canopy
+di-
+versity
+using
+high-fidelity
+imaging
+spectroscopy.
+Ecological
+Applica-
+tions
+24,
+1289–1296.
+URL:
+https://esajournals.onlinelibrary.wiley.
+com/doi/abs/10.1890/13-1824.1,
+doi:https://doi.org/10.1890/13-1824.1,
+arXiv:https://esajournals.onlinelibrary.wiley.com/doi/pdf/10.1890/13-1824.1.
+Gellie, N.J., Hunter, J.T., Benson, J.S., Kirkpatrick, J.B., Cheal, D.C., McCreery, K., Brockle-
+hurst, P., 2018. Overview of plot-based vegetation classification approaches within Australia.
+Phytocoenologia 48, 251–272.
+Ghiyamat, A., Shafri, H.Z., 2010. A review on hyperspectral remote sensing for homogeneous
+and heterogeneous forest biodiversity assessment. International Journal of Remote Sensing 31,
+1837–1856.
+Government, N., 2019. BioNet Systematic Flora Survey. https://www.environment.nsw.gov.
+au/research/VISplot.htm. [Online; accessed 11-February-2022].
+Grace, J.B., Anderson, T.M., Seabloom, E.W., et al., 2016. Integrative modelling reveals mecha-
+nisms linking productivity and plant species richness. Nature 529, 390–393.
+39
+
+Green, A., Craig, M., 1985. Analysis of aircraft spectrometer data with logarithmic residuals, in:
+JPL Proc. of the Airborne Imaging Spectrometer Data Anal. Workshop, pp. 111–119.
+Guanter, L., Kaufmann, H., Segl, K., Foerster, S., Rogass, C., Chabrillat, S., Kuester, T., Hollstein,
+A., Rossner, G., Chlebek, C., et al., 2015. The EnMAP spaceborne imaging spectroscopy mission
+for Earth observation. Remote Sensing 7, 8830–8857.
+Guo, Y., Jia, X., Paull, D., 2018. Effective sequential classifier training for svm-based multitempo-
+ral remote sensing image classification. IEEE Transactions on Image Processing 27, 3036–3048.
+Guo, Y., Mokany, K., Ong, C., Moghadam, P., Ferrier, S., Levick, S., 2022. Quantitative assessment
+of DESIS hyperspectral data for plant biodiversity estimation in Australia, in: 2022 IEEE
+International Geoscience and Remote Sensing Symposium, IEEE. pp. 1744–1747.
+Hacker, P.W., Coops, N.C., Townsend, P.A., Wang, Z., 2020. Retrieving foliar traits of quercus
+garryana var. garryana across a modified landscape using leaf spectroscopy and LiDAR. Remote
+Sensing 12, 26.
+Jia, X., Richards, J.A., 1999. Segmented principal components transformation for efficient hyper-
+spectral remote-sensing image display and classification. IEEE Transactions on Geoscience and
+Remote Sensing 37, 538–542.
+Kattge, J., B¨onisch, G., D´ıaz, S., Lavorel, S., Prentice, I.C., Leadley, P., Tautenhahn, S., Werner,
+G.D., Aakala, T., Abedi, M., et al., 2020. TRY plant trait database–enhanced coverage and
+open access. Global Change Biology 26, 119–188.
+Knipling, E.B., 1970. Physical and physiological basis for the reflectance of visible and near-infrared
+radiation from vegetation. Remote Sensing of Environment 1, 155–159.
+K¨onig, C., Weigelt, P., Schrader, J., Taylor, A., Kattge, J., Kreft, H., 2019. Biodiversity data
+integration—the significance of data resolution and domain. PLoS Biology 17, e3000183.
+40
+
+K¨orner, C., 1995. Alpine plant diversity: a global survey and functional interpretations, in: Arctic
+and alpine biodiversity: Patterns, causes and ecosystem consequences. Springer, pp. 45–62.
+Krutz, D., M¨uller, R., Knodt, U., G¨unther, B., Walter, I., Sebastian, I., S¨auberlich, T., Reulke,
+R., Carmona, E., Eckardt, A., et al., 2019. The instrument design of the DLR Earth sensing
+imaging spectrometer (DESIS). Sensors 19, 1622.
+Lecl`ere, D., Obersteiner, M., Barrett, M., Butchart, S.H., Chaudhary, A., De Palma, A., DeClerck,
+F.A., Di Marco, M., Doelman, J.C., D¨urauer, M., et al., 2020. Bending the curve of terrestrial
+biodiversity needs an integrated strategy. Nature 585, 551–556.
+Mafanya, M., Tsele, P., Zengeya, T., Ramoelo, A., 2022. An assessment of image classifiers for
+generating machine-learning training samples for mapping the invasive Campuloclinium macro-
+cephalum (Less.) DC (pompom weed) using DESIS hyperspectral imagery. ISPRS Journal of
+Photogrammetry and Remote Sensing 185, 188–200.
+Malhi, R.K.M., Anand, A., Mudaliar, A.N., Pandey, P.C., Srivastava, P.K., Sandhya Kiran, G.,
+2020. Synergetic use of in situ and hyperspectral data for mapping species diversity and above
+ground biomass in Shoolpaneshwar Wildlife sanctuary, Gujarat. Tropical Ecology 61, 106–115.
+Mokany, K., Ferrier, S., Harwood, T.D., Ware, C., Di Marco, M., Grantham, H.S., Venter, O.,
+Hoskins, A.J., Watson, J.E., 2020.
+Reconciling global priorities for conserving biodiversity
+habitat. Proceedings of the National Academy of Sciences 117, 9906–9911.
+Mutanga, O., Skidmore, A.K., 2007. Red edge shift and biochemical content in grass canopies.
+ISPRS Journal of Photogrammetry and Remote Sensing 62, 34–42.
+Myers, B.J., Weiskopf, S.R., Shiklomanov, A.N., Ferrier, S., Weng, E., Casey, K.A., Harfoot, M.,
+Jackson, S.T., Leidner, A.K., Lenton, T.M., et al., 2021.
+A new approach to evaluate and
+reduce uncertainty of model-based biodiversity projections for conservation policy formulation.
+BioScience 71, 1261–1273.
+41
+
+Ong, C., Cudahy, T., 2014. Mapping contaminated soils: using remotely-sensed hyperspectral data
+to predict pH. European journal of soil science 65, 897–906.
+Palmer, M.W., Earls, P.G., Hoagland, B.W., White, P.S., Wohlgemuth, T., 2002. Quantitative
+tools for perfecting species lists. Environmetrics 13, 121–137.
+Peng, Y., Fan, M., Song, J., Cui, T., Li, R., 2018. Assessment of plant species diversity based on
+hyperspectral indices at a fine scale. Scientific Reports 8, 1–11.
+Pettorelli, N., Wegmann, M., Skidmore, A., M¨ucher, S., Dawson, T.P., Fernandez, M., Lucas, R.,
+Schaepman, M.E., Wang, T., O’Connor, B., et al., 2016. Framing the concept of satellite remote
+sensing essential biodiversity variables: challenges and future directions. Remote Sensing in
+Ecology and Conservation 2, 122–131.
+Pickering, C., Green, K., 2009. Vascular plant distribution in relation to topography, soils and
+micro-climate at five GLORIA sites in the Snowy Mountains, Australia. Australian Journal of
+Botany 57, 189–199.
+Pickering, C., Hill, W., Green, K., 2008. Vascular plant diversity and climate change in the alpine
+zone of the Snowy Mountains, Australia. Biodiversity and Conservation 17, 1627–1644.
+Pignatti, S., Palombo, A., Pascucci, S., Romano, F., Santini, F., Simoniello, T., Umberto, A.,
+Vincenzo, C., Acito, N., Diani, M., et al., 2013. The PRISMA hyperspectral mission: Science
+activities and opportunities for agriculture and land monitoring, in: 2013 IEEE International
+Geoscience and Remote Sensing Symposium-IGARSS, IEEE. pp. 4558–4561.
+De los Reyes, R., Richter, R., Langheinrich, M., Pflug, B., Schwind, P., 2018.
+Validation of
+a new atmospheric correction software using AERONET reference data PACO: Python-based
+Atmospheric COrrection, in: Workshop on Land Product Validation and Evolution (LPVE2018),
+pp. 1–1.
+42
+
+Richards, J.A., Jia, X., 2006. Remote Sensing Digital Image Analysis [4th Edition]. Springer.
+Rosenzweig, M.L., 1995. Species diversity in space and time. Cambridge University Press.
+Skidmore, A.K., Coops, N.C., Neinavaz, E., Ali, A., Schaepman, M.E., Paganini, M., Kissling,
+W.D., Vihervaara, P., Darvishzadeh, R., Feilhauer, H., et al., 2021. Priority list of biodiversity
+metrics to observe from space. Nature Ecology & Evolution 5, 896–906.
+Stevenson, S.L., Watermeyer, K., Caggiano, G., Fulton, E.A., Ferrier, S., Nicholson, E., 2021.
+Matching biodiversity indicators to policy needs. Conservation Biology 35, 522–532.
+Tilman, D., Knops, J., Wedin, D., Reich, P., Ritchie, M., Siemann, E., 1997. The influence of
+functional diversity and composition on ecosystem processes. Science 277, 1300–1302.
+Tilman, D., Wedin, D., Knops, J., 1996. Productivity and sustainability influenced by biodiversity
+in grassland ecosystems. Nature 379, 718–720.
+Tollefson, J., 2019. Humans are driving one million species to extinction. Nature 569, 171–172.
+Verrelst, J., Rivera-Caicedo, J.P., Reyes-Mu˜noz, P., Morata, M., Amin, E., Tagliabue, G., Pani-
+gada, C., Hank, T., Berger, K., 2021. Mapping landscape canopy nitrogen content from space
+using PRISMA data. ISPRS Journal of Photogrammetry and Remote Sensing 178, 382–395.
+Wang, R., Gamon, J.A., 2019. Remote sensing of terrestrial plant biodiversity. Remote Sensing of
+Environment 231, 111218.
+Wang, R., Gamon, J.A., Montgomery, R.A., et al., 2016. Seasonal variation in the NDVI–species
+richness relationship in a prairie grassland experiment (Cedar Creek). Remote Sensing 8, 128.
+Wang, Z., Townsend, P.A., Schweiger, A.K., Couture, J.J., Singh, A., Hobbie, S.E., Cavender-
+Bares, J., 2019. Mapping foliar functional traits and their uncertainties across three years in a
+grassland experiment. Remote Sensing of Environment 221, 405–416.
+43
+
+Xu, M., Jia, X., Pickering, M., Jia, S., 2019. Thin cloud removal from optical remote sensing images
+using the noise-adjusted principal components transform. ISPRS Journal of Photogrammetry
+and Remote Sensing 149, 215–225.
+Zhao, F., Guo, Y., Huang, Y., Reddy, K.N., Lee, M.A., Fletcher, R.S., Thomson, S.J., 2014a.
+Early detection of crop injury from herbicide glyphosate by leaf biochemical parameter inversion.
+International Journal of Applied Earth Observation and Geoinformation 31, 78–85.
+Zhao, F., Huang, Y., Guo, Y., Reddy, K.N., Lee, M.A., Fletcher, R.S., Thomson, S.J., 2014b. Early
+detection of crop injury from glyphosate on soybean and cotton using plant leaf hyperspectral
+data. Remote Sensing 6, 1538–1563.
+44
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+Phase separation of passive particles in active liquids
+Pragya Kushwaha1, Vivek Semwal2, Sayan Maity1, Shradha Mishra2, Vijayakumar Chikkadi1
+1 Indian Institute of Science Education and Research Pune, India 411008
+2 Indian Institute of Technology (BHU) Varanasi, India 221005
+(Dated: January 30, 2023)
+The transport properties of colloidal particles in active liquids have been studied extensively. It
+has led to a deeper understanding of the interactions between passive and active particles. However,
+the phase behavior of colloidal particles in active media has received little attention. Here, we present
+a combined experimental and numerical investigation of passive colloids dispersed in suspensions of
+active particles. Our study reveals dynamic clustering of colloids in active media due to an interplay
+of active noise and an attractive effective potential between the colloids. The size-ratio of colloidal
+particles to the bacteria sets the strength of the interaction. As the relative size of the colloids
+increases, the effective potential becomes stronger and the average size of the clusters grows. The
+simulations reveal a macroscopic phase separation of passive colloids at sufficiently large size-ratios.
+We will present the role of density fluctuations and hydrodynamic interactions in the emergence of
+effective interactions.
+The Brownian colloids self-assemble to display a wide
+variety of phases depending on their shapes and inter-
+actions [1–3]. Their equilibrium phase behavior is gov-
+erned by the principles of equilibrium statistical mechan-
+ics [4, 5]. However, our understanding of the collective
+behavior of colloids far from equilibrium remains a chal-
+lenge [6, 7]. In recent years, active matter has emerged
+as a new paradigm for understanding nonequilibrium sys-
+tems [8–11]. They are known to display many interest-
+ing phenomena such as flocking [12, 13], motility induced
+phase separation [14–16], active turbulence [17], superflu-
+idity [18], that are absent in equilibrium systems. There-
+fore, active matter offers novel approaches to colloidal
+assembly in systems far from equilibrium.
+In this let-
+ter, we have investigated the phase behavior of colloidal
+particles dispersed in active liquids.
+Wu and Libchaber [19] did seminal experiments on
+the active transport of colloidal particles in suspensions
+of bacteria. They discovered anomalous diffusion and a
+large effective diffusion constant, when compared to dif-
+fusion at equilibrium, which inspired a slew of theoreti-
+cal investigations and detailed experiments [20–27]. The
+subsequent efforts have elucidated how enhanced diffu-
+sion arises due to an interplay of entrainment of colloids
+by bacteria, far-field hydrodynamic interactions, direct
+collisions, and the relative size of bacteria and colloid
+[23–25]. Further, the effective interaction between a pair
+of passive particles in active media has been the focus
+of several investigations. It has been predicted to be at-
+tractive, repulsive, and long-ranged, depending on the ge-
+ometry of passive particles, the activity of active species,
+and their density [28–36]. This understanding has opened
+new routes to colloidal assembly mediated by active flu-
+ids [7, 39]. The phase behavior of active-passive mixtures
+is a topic of recent interest [11, 37–43, 45, 47], where ex-
+perimental investigations are scarce [6, 7]. On the one
+hand, theory and simulations at high Peclet numbers
+have shown that homogeneous mixtures of active and
+passive particles are unstable.
+The underlying physics
+is similar to motility induced phase separation (MIPS)
+[40, 42]. On the other hand, in the diffusive limit, theory
+and simulations of nonequilibrium binary mixtures with
+different diffusivities and temperatures reveal phase sep-
+aration [41, 45, 46] due to spinodal-like instability. Sur-
+prisingly, there is little known about mixtures at moder-
+ate Peclet numbers. This is the range where most of the
+active matter experiments involving living matter or syn-
+thetic systems, such as diffusio-phoretic colloids, fall. A
+recent study of colloids in active suspensions of bacteria
+reports dynamical clustering and absence of phase sepa-
+ration at moderate Peclet numbers [6]. The conclusions
+were based on the phase diagram obtained from varia-
+tions of Peclet number and rotation rate of active par-
+ticles. In contrast, earlier numerical studies have shown
+a macroscopic phase separation [43]. Therefore, it is not
+clear whether active-passive mixtures show macroscopic
+phase separation at moderate Peclet numbers.
+This letter presents a combined experimental and nu-
+merical study of the phase behavior of colloidal parti-
+cles in active liquids. The experiments were performed
+using colloids in bacteria suspensions, and simulations
+of active-passive mixtures were realized using Brownian
+dynamics [48–50]. Earlier simulations of active-passive
+mixtures, by one of the authors of this letter, had shown
+a significant influence of the size-ratio of passive to ac-
+tive particles on their phase diagram [43].
+Motivated
+by this study, our experiments were performed over a
+range of densities and sizes of passive colloidal particles
+in active suspensions of bacteria.
+The colloids display
+dynamic clustering due to an interplay of activity and an
+attractive effective potential. However, the average size
+of the clusters increases with the size of colloidal par-
+ticles, suggesting an enhanced interaction between the
+particles.
+Using simulations, we confirm an attractive
+effective potential between passive particles in an active
+medium.
+The strength of the interaction is shown to
+grow with an increasing size-ratio. When the size-ratio
+is sufficiently large, the interactions are strong enough to
+arXiv:2301.11771v1  [cond-mat.soft]  27 Jan 2023
+
+2
+drive the phase separation of passive colloids. The ori-
+gin of the effective potential in simulations appears to be
+related to long-ranged density fluctuations of active par-
+ticles. In contrast, the correlations of density fluctuations
+of bacteria decay rapidly in experiments. These results
+indicate a hydrodynamic origin of effective interactions
+between the colloids in our experiments. Thus, shedding
+new light on the phase behavior of passive particles in
+active media.
+The active suspensions were prepared using E.coli cells
+(U5/41 type strain). The cells were cultured using well
+established protocols in the literature [18, 44]. They are
+suspended in a motility media to get desired concentra-
+tions. Details of the method are given in the supplemen-
+tary section. The density of bacteria in our experiments
+was well below the density threshold for the onset of col-
+lective motion. The average speed and average size of
+bacteria cells were estimated to be v = 33.84±9.98 µm/s
+[supplementary Fig. S1(a)] and l = 2.68 ± 0.86 µm [sup-
+plementary Fig. S1(b)], respectively. Their rotational dif-
+fusion time scale was estimated to be τr = 1.67 s [supple-
+mentary Fig. S1(c)]. The Peclet number, which is defined
+as Pe = τrv/l, turns out to be Pe ∼ 21 for our system.
+The phase behavior of colloidal particles in suspensions
+of bacteria was investigated by varying the size and den-
+sity of the beads at a constant density of bacteria. The
+diameters of the particles used in the experiments were
+σ = 7µm, 10µm and 15µm, and their density is varied
+from φ ∼ 0.1 − 0.4, where φ = N ∗ πσ2/(4A) is the area
+fraction, N is the number of colloidal particles in the field
+of view of area A. The size-ratio S = σ/l is defined as
+the ratio of the diameter of colloids to the length of the
+bacteria.
+The simulations were performed using a binary mix-
+ture of active Brownian particles (ABP) with Na small
+active particles of radius aa and Np big passive particles
+of radius ap (ap > aa) moving on a two dimensional fric-
+tional substrate. The active particles are associated with
+a self propulsion speed v and an orientation unit vector
+ˆvi. The equations of motion and other simulation details
+are given in the supplementary section. We simulate the
+system in a square box of size lbox × lbox, with periodic
+boundary conditions. The system is defined by the area
+fractions φa = Naπa2
+a/l2
+box and φp = Npπa2
+p/l2
+box of the
+active and passive particles respectively, the activity v of
+active particles and the size-ratio (S = ap/aa) defined as
+the ratio of the radius of a passive particle to the radius
+of an active particle. We start with a random homoge-
+neous distribution of active and passive particles in the
+box and with random directions for the velocity of active
+particles. The Eqs. (S1-S3) are updated for all particles
+and one simulation step is counted after a single update
+for all the particles. The simulations does not include
+the hydrodynamic interactions that are present in exper-
+iments. The effect of hydrodynamic interaction can be
+included using coarse-grained studies similar to [51].
+The colloids used in our experiments are bigger than
+5 µm, so they are non-Brownian particles. However, they
+FIG. 1. The structure of passive particles in active suspen-
+sions.
+Main panel : The pair correlation function g(r) for
+φ ∼ 0.10 and S ∼ 2.5, 3.5, and 5.5.
+The g(r) curves are
+shifted along the y − axis for clarity. Insets: The bright field
+images of particles at φ ∼ 0.10 and size ratios S ∼ 2.5 (left)
+and S ∼ 5.5 (right), respectively. The scale bar in the images
+is 50µm.
+diffuse in suspensions of bacteria due to active fluctua-
+tions with a characteristic super-diffusive motion on short
+time scales and a diffusive motion on long time scales.
+To investigate their collective behavior in active suspen-
+sions, we first analyze their pair correlation function g(r),
+which is shown in the main panel of Fig. 1 at an area
+fraction of φ ∼ 0.1 and size ratios S ∼ 2.5 − 5.5. The
+normalized g(r) for different size-ratios is shifted along
+the y-axis for clarity.
+What is prominent is the pres-
+ence of a sharp peak at r = σ, and additional peaks
+develop at r = 1.7σ and r = 2σ with increasing size ra-
+tio. The peak at 2σ indicates a second shell of neighbors,
+and the one at 1.7σ is a signature of hexagonal order-
+ing in the cluster. These observations are evident in the
+bright field images presented in the insets of Fig.1. The
+larger size ratios lead to larger clusters with enhanced
+order. These images are reminiscent of clustering in sys-
+tems of purely active particles [15]. However, the clusters
+of passive particles in our experiments break and form
+much more rapidly. A real-time video of dynamic clus-
+ter formation is presented in the supplementary section
+Video. SV1 for φ ∼ 0.10 and S ∼ 2.5. Recent simulations
+have reported similar dynamic clustering and traveling
+interfaces of active-passive particles that are not observed
+in our current study [40, 42]. One of the main difference
+between our experiments and these simulations are the
+large Peclet numbers used in simulations. Further, as re-
+ported by earlier investigations, self-propulsion of parti-
+cles is a manifestation of an attractive effective potential
+between the passive particles due to active fluctuations
+[28].
+We next turn our attention to cluster size distribution
+(CSD), p(n), which is a count of clusters of n particles
+[43, 52]. The clusters in our experiments were determined
+by setting a distance criterion of rc ≤ 1.1σ to identify
+pairs of particles as neighbors. This was set based on the
+
+60
+*S=2.5
+S=3.5
+8
+50
+88
+☆-S=5.5
+%
+8
+8
+40
+8
+888
+88
+30
+b
+20
+10
+0.5
+1
+1.5
+2
+2.5
+3
+0
+Y3
+FIG. 2. Cluster statistics of passive particles. (a) Cluster size
+distribution in the main panel is shown for different size-ratios
+S ∼ 2.5, 3.5, and 5.5 at a density of φ ∼ 0.10. The symbols
+distinguish different size ratios.
+The inset shows the CSD
+plot for same size ratios at φ ∼ 0.3. (b) The average cluster
+size < n > for varying S. The curves with different symbols
+correspond to different particle densities, ranging from φ ∼
+0.1 − 0.4.
+position of the first peak of g(r) in Fig.1, and to account
+for small polydispersity (< 5%) in the size distribution
+of colloidal particles. The results of our analysis are pre-
+sented in Figs. 2a & 2b. The main panel in Fig.2a shows
+CSD for varying size ratios of S ∼ 2.5, 3.5, and 5.5 at
+a density of φ ∼ 0.1. For small size ratios S < 5, p(n)
+has an exponential form exp(−n/n0) as observed in the
+equilibrium case [50]. The clustering is weak at these size
+ratios, however, for S > 5 the p(n) displays a power-law
+decay with an exponential cut-off at large n, i.e., it is best
+described by p(n)/p(1) ∼ 1/nα exp(−n/n0). The fits of
+this form to our data are shown in the figure using dashed
+lines. These results indicate that the characteristic size
+of clusters grows with increasing size ratio. The growth
+of clusters is dramatic at larger area fractions, the inset
+of Fig.2a shows cluster distribution at φ ∼ 0.3.
+We further elucidate the clustering of colloids by com-
+puting the average cluster size using the expression <
+n >= � n p(n), which is presented in Fig. 2b where
+the curves with different symbols correspond to different
+area fractions ranging from φ ∼ 0.1 − 0.4. These mea-
+surements were made in the steady state where the mean
+cluster sizes fluctuates around a mean value. This data is
+provided in Fig. S2(a-c) for various size ratios and area
+fractions for over 5000 frames or 500 s. What is clear
+from Fig. 2b is that increasing the size-ratio or the rel-
+ative size of colloids leads to larger cluster sizes. This
+suggests that the effective potential between the colloids
+becomes stronger with an increasing size ratio. One can
+intuitively understand the underlying physics by consid-
+ering the interaction between an isolated colloidal parti-
+cle and a swimmer. When the size of a particle is small,
+a bacterium entrains the particle to larger distances be-
+fore changing its direction of motion. However, when the
+particle is large, the entrainment distance is small, and
+the scattering angle of the swimmer is large [25]. It in-
+dicates that the bacteria can suppress cluster formation
+when the colloidal particles are smaller.
+What is not
+clear from our experiments is whether larger size-ratios
+lead to a macroscopic phase separation in our system. To
+understand this aspect, we turn to numerical simulations
+that allow a detailed exploration of parameter space.
+The first quantity we have calculated in the simulations
+is the effective potential between two passive particles in
+the medium of ABPs with torque. In order to calculate
+the effective potential between two passive particles, we
+choose Np = 2 at positions r1 and r2, respectively, in a
+system of ABPs with Na = 1800. We keep r1 fixed and
+slowly vary r2 in small steps of ∆x = 0.5aa starting from
+the zero surface to surface distance between two passive
+particles. The cartoon of the system simulated for the
+force calculation for a fixed r is shown in Fig. S3 (SM).
+In the figure, ABPs are shown in red and passive particles
+in blue for S = 8. For resolution, only a part of the sys-
+tem near the two passive particles is shown. The active
+particles’ positions and orientations are updated accord-
+ing to the Eqns. (S1 and S2). For each configuration at a
+given distance between two passive particles, the system
+is allowed to reach the steady state. Further, we use the
+steady state configuration to calculate the force FS(r)
+between two-passive particles at a surface to surface sep-
+aration r, such that FS(r) = F12(r)+�Na
+i=1 F1i(r). Here
+F12(r) is the force due to passive particle 2nd on 1st, and
+�Na
+i=1 F1i(r) represents the sum of all the forces due to
+active particles on 1st passive particle for a given con-
+figuration of two passive particles at separation r. The
+potential is then calculated by integrating the force over
+the distance U(r) =
+� r
+−∞ FS(r)dr [53–55]. Here we set
+the lower limit as one-fourth of the box-length. The re-
+sults are averaged over 30 independent realizations.
+We calculated the effective potentials U(r) for Pe = 25
+(which is comparable to the experimental system) and
+four size-ratios S = 3, 5, 8 and 10.
+The comparable
+size-ratio in experimental system is S ∼ (2.5 to 5.5).
+We first plot the effective potential U(r). In the main
+panel of Fig. 3(a) we show the plot of U(r) vs.
+r for
+the system for S = 3, 5, 8 and 10. The distance is nor-
+malised by the radius of active particles, which is kept
+fixed to 0.1.
+The negative side of the potential shows
+attraction and the positive nature is repulsion. For all
+the parameters the potential approaches zero at large
+distances, and it is negative at intermediate distances.
+The depth of the potential becomes deeper with increas-
+ing S. The inset shows the effective potential with the
+distance r scaled by the size of passive particles. Sur-
+prisingly, the minima of the potentials for the size ratios
+S = 5, 8 and 10 fall at r/ap = 1, which implies that the
+length scale characterizing the range of the interaction
+potential is set by size passive particles. We investigate
+further the origin of long-range interactions by consider-
+ing a single passive particle in the center of our system,
+as shown in Fig. S4. It is evident that the passive par-
+ticle disturbs the density field of active particles, lead-
+ing to clustering around the passive particle. The main
+panel of Fig. 3(b) shows the normalised density correla-
+tion C(r) = (⟨ρ(0)ρ(r)⟩ − ⟨ρ(r)⟩2)/(
+�
+ρ(r)2�
+− ⟨ρ(r)⟩2) of
+
+(a)
+10
+100
+P(n)/P(1)
+米
+10-5
+米
+10-2
+100
+104
+/(u)
+n
+P
+10
+V
+S=2.5
+米
+S=3.5
+S=5.5
+10-6
+★
+100
+102
+m(b)
+20
+1Φ=0.10
+fΦ=0.20
+Φ=0.30
+15
+IΦ=0.40
+V
+V
+10
+5
+2
+3
+4
+5
+6
+S4
+FIG. 3.
+Top left figure: The effective potential between a
+pair ,of colloidal particles. The main panel shows the plot of
+the effective potential U(r) vs. distance r for Pe = 25 and
+size-ratios S = 3, 5, 8, and 10. The inset shows the effec-
+tive potential with the scaled distance r/ap. Top right figure:
+The main panel shows normalized correlations of density fluc-
+tuations C(r) due to a passive particle. The inset show the
+length scale extracted from C(r) as function of the size-ratio.
+The length scale is expressed in terms of active particle size.
+Bottom panel: Snapshots of the system obtained from the
+microscopic simulation: two types of particles for different
+size ratio S = 3, 5 and 8 (left, central and right columns) at
+Pe = 25. Red particles are ABPs and blue particles are pas-
+sive particles, for fixed packing fraction φ = 0.60 in a system
+of size lbox = 140aa.
+active particles calculated from the surface of the big pas-
+sive particle for four different size ratios S = 3, 5, 8 and
+10. The inset of Fig. 3(b) shows the typical size of clus-
+ters L(S) around a single passive particle in the center of
+the box for different size ratios S. The L(S) is measured
+in terms of size of ABP. Clearly, L(S) increases with in-
+creasing S. The number fluctuations of active particles
+around an isolated passive particle yield similar conclu-
+sions. The Fig. S5 (SM) shows number fluctuations for
+three different sizes of the passive particle or for three
+different size ratios S = 3, 5 and 8. The details of the cal-
+culations are give in the supplementary material. For all
+the cases the graph is a power law with ∆N ≃ N α, where
+α ≃ 0.7 for moderate N for all S and starts to deviate
+for large N. The deviation appears at relatively larger N
+on increasing size ratio. Hence increasing the size of pas-
+sive particle increases the stretch of density fluctuation of
+ABP’s. These results establishes that the density fluctu-
+ations play a central role in the emergence of long-range
+effective attractive interactions between passive particles
+in our simulations.
+We elucidate the effect of such effective potential, full
+microscopic simulations of mixtures of active and passive
+were performed using the Eqs. (S1-S3). We simulated the
+system for Pe = 25 and size ratio S = 3, 5 and 8, which
+are close to experimental values. In the bottom panel of
+Fig. 3 the steady state snapshots of passive (blue, big-
+ger) and active (red, smaller) are shown for different size
+ratios S = 3, 5 and 8 respectively. Clusters with moder-
+ate to strong ordering is found on increasing S. For small
+S = 3 clusters are present but without strong local hexag-
+onal ordering, whereas as we increase S the ordering and
+clustering is enhanced. We also calculated the percent of
+passive particles participating in the largest cluster for
+different size ratios and it increases from 35% to 67% as
+we increase size from 3 to 8 (data not shown). Hence for
+large size ratio passive particles show the macroscopic
+phase separation.
+A similar examination of correlations of density fluc-
+tuations of bacteria in experiments reveals that they are
+suppressed, which is evident from C(r) in Fig. S6. The
+clustering of colloidal particles appears to arise from their
+hydrodynamic interactions. An earlier numerical study
+of active-passive matter with pusher-type swimmers at
+dilute concentrations had shown hydrodynamic interac-
+tions to stabilize colloidal clusters [59]. In addition, a re-
+cent theoretical model of active gels shows a long-ranged
+attractive effective potential between colloids due to hy-
+drodynamic effects [58]. Considering these studies, hy-
+drodynamics is likely to promote the formation of col-
+loidal clusters.
+Our investigations conclude that the interplay of ef-
+fective potential and active noise determines the phase
+behavior of colloidal particles in active liquids.
+The
+strength of the effective potential is set by the size ra-
+tio of passive particles to active ones; larger size ra-
+tios lead to stronger interactions.
+The simulations re-
+veal a long-ranged effective potential extending to sev-
+eral active particle diameters. It appears to emerge from
+the long-ranged density fluctuations of active particles
+in the system. When the size-ratio is small, the passive
+particles display dynamic clusters that form and break
+rapidly. However, the effective potential is strong enough
+to lead to phase separation of passive particles at suffi-
+ciently large size-ratios. These are the novel features of
+active-passive mixtures absent in the equilibrium ana-
+log of colloid-polymer mixtures where the range of effec-
+tive potential is short-ranged. The density fluctuations of
+Bacteria are suppresses in experiments. Further investi-
+gation is needed to understand the role of hydrodynamic
+interactions on the effective potential of colloids in our
+experiments with active suspensions.
+We thank Chaitanya Athale,
+Apratim Chatterji,
+Thomas Pucadyil, Sunish Radhakrishnan, Rajesh Singh,
+and Ganesh Subramanian for helpful discussions and sup-
+port. We thank Madan Rao for drawing our attention
+to [58], and Kumar Gourav for assistance in the initial
+stages of experiments. V.C. acknowledges financial sup-
+port from IISER Pune and DST/SERB under the project
+grant CRG/2021/007824.
+P.K. is supported by CSIR-
+UGC fellowship 1353. V.S. and S. M. thank I.I.T. (BHU)
+Varanasi computational facility.
+V.S. thanks DST IN-
+SPIRE (INDIA) for the research fellowship. S.M. thanks
+DST, SERB (INDIA), Project No.
+ECR/2017/000659
+for partial financial support.
+
+S=3
+S=5
+0.5
+S=8
+S=10
+0
+-0.5
+0.5
+0
+-1
+-0.5
+-1
+-1.5
+-1.5
+-2
+0
+2
+4
+r/a
+p
+-2
+10
+20
+30
+40
+r/a
+a24
+0.8
+16
+0.6
+C
+0.4
+4
+6
+8
+10
+S
+S=3
+S=5
+0.2
+S=8
+S=10
+0
+10
+20
+30
+40
+r/a
+aS=3
+S=5
+S=85
+[1] S. Sacanna, M. Korpics, K. Rodriguez et al., Shaping
+colloids for self-assembly, Nat. Commun. 4, 1688 (2013).
+[2] T. Hueckel, G.M. Hocky, and S. Sacanna, Total synthesis
+of colloidal matter, Nat Rev Mater 6, 1053–1069 (2021).
+[3] S. Glotzer and M. Solomon, Anisotropy of building
+blocks and their assembly into complex structures, Na-
+ture Mater. 6, 557–562 (2007).
+[4] P. N. Pusey, in Liquids, Freezing and Glass Transition
+edited by J. P. Hansen, D. Levesque, and J. Zinn-Justin
+(North-Holland, 1991), pp. 765–942.
+[5] W. C. K. Poon, “Colloidal suspensions,” in Oxford Hand-
+book of Soft Condensed Matter, edited by E. Terentjev
+and D. A. Weitz (Oxford University Press, Oxford, 2015),
+pp. 1–49.
+[6] S. Gokhale, J. Li, A. Solon, J. Gore, and N. Fakhri, Dy-
+namic clustering of passive colloids in dense suspensions
+of motile bacteria, Phys. Rev. E, 105 (5), 054605 (2022).
+[7] F. Kummel, P. Shabestari, C. Lozano, G. Volpe, and C.
+Bechinger, Formation, compression and surface melting
+of colloidal clusters by active particles, Soft Matter 11,
+6187 (2015).
+[8] P. Romanczuk, M. Bar, W. Ebeling, and B Lindner, Ac-
+tive Brownian particles , Eur. Phys. J. Special Topics
+202,1-162(2012)
+[9] S. Ramaswamy, Active matter, Journal of Statistical Me-
+chanics: Theory and Experiment, 2017,054002,(2017)
+[10] M. C. Marchetti, J.-F. Joanny, S. Ramaswamy, T. B.
+Liverpool, J. Prost, M. Rao, and R. A. Simha, Hydrody-
+namics of soft active matter, Rev. Mod. Phys. 85, 1143
+(2013).
+[11] C. Bechinger, R.D. Leonardo, H. Lowen, C. Reichhardt,
+G. Volpe, G. Volpe, Active particles in complex and
+crowded environments, Rev. Mod. Phys. 88, 045006
+(2016).
+[12] T. Vicsek, A. Czir´ok, E. Ben-Jacob, I. Cohen, and O.
+Shochet, Novel type of phase transition in a system of
+self-driven particles, Phys. Rev. Lett. 75, 1226 (1995).
+[13] A. Bricard, J.-B. Caussin, N. Desreumaux, O. Dauchot,
+and D. Bartolo, Emergence of macroscopic directed mo-
+tion in populations of motile colloids, Nature 503, 95
+(2013).
+[14] M. E. Cates and J. Tailleur, Motility-Induced Phase
+Separation, Annu. Rev. Condens. Matter Phys. 6, 219
+(2015).
+[15] I. Buttinoni, J. Bialk´e, F. K¨ummel, H. L¨owen, C.
+Bechinger, T. Speck, Dynamical clustering and phase
+separation in suspensions of self-propelled colloidal par-
+ticles, Phys. Rev. Lett. 110, 238301 (2013).
+[16] J. Palacci, S. Sacanna, A. P. Steinberg, D. J. Pine and P.
+M. Chaikin, Living Crystals of Light-Activated Colloidal
+Surfers, Science, 339, 936 (2013).
+[17] H. Wensink et al., Meso-scale turbulence in living fluids,
+Proc. Natl. Acad. Sci. USA 109, 14308 (2012).
+[18] H.M. Lopez, J. Gachelin, C. Douarche, H. Auradou, and
+E. Clement, Turning bacteria suspensions into superflu-
+ids, Phys. Rev. Lett. 115, 028301 (2015).
+[19] X.-L. Wu and A. Libchaber, Particle Diffusion in a Quasi-
+Two-Dimensional Bacterial Bath, Phys. Rev. Lett. 84,
+3017 (2000).
+[20] K.C. Leptos, J.S. Guasto, J.P. Gollub, A.I. Pesci, and
+R.E. Goldstein, Dynamics of enhanced tracer diffusion
+in suspensions of swimming eukaryotic microorganisms,
+Phys. Rev. Lett. 103, 198103 (2009).
+[21] C. Valeriani, M. Li, J. Novosel, J. Arlt, and D. Maren-
+duzzo, Colloids in a bacterial bath: simulations and ex-
+periments, Soft Matter 7, 5228 (2011).
+[22] J.-L. Thiffeault, Distribution of particle displacements
+due to swimming microorganisms, Phys. Rev. E 92,
+023023 (2015).
+[23] R. Jeanneret, D. O. Pushkin, V. Kantsler, and M.
+Polin, Entrainment dominates the interaction of microal-
+gae with micron-sized objects, Nat. Commun. 7, 12518
+(2016).
+[24] A. J. T. M. Mathijssen, R. Jeanneret, and M. Polin,
+Universal entrainment mechanism controls contact times
+with motile cells, Phys. Rev. Fluids 3, 033103 (2018).
+[25] H. Shum and J. M. Yeomans, Entrainment and scattering
+in microswimmer-colloid interactions, Phys. Rev. Fluids
+2, 113101 (2017).
+[26] L. Ortlieb, S. Rafai, P. Peyla, C. Wagner, and T. John,
+Statistics of colloidal suspensions stirred by microswim-
+mers, Phys. Rev. Lett. 122, 148101 (2019).
+[27] A. Lagarde, N. Dages, T. Nemoto, V. Demery, D. Bar-
+tolo, and T. Gibaud, Colloidal transport in bacteria sus-
+pensions: from bacteria collision to anomalous and en-
+hanced diffusion, Soft Matter 16, 7503 (2020).
+[28] L. Angelani, C. Maggi, M. L. Bernardini, A. Rizzo, and
+R. Di Leonardo, Effective interactions between colloidal
+particles suspended in a bath of swimming cells, Phys.
+Rev. Lett. 107, 138302 (2011).
+[29] D. Ray, C. Reichhardt, and C. J. Olson Reichhardt,
+Casimir effect in active matter systems, Phys. Rev. E
+90, 013019 (2014).
+[30] R. Ni, M.A. Cohen Stuart, and P.G. Bolhuis, Tunable
+long range forces mediated by self-propelled colloidal
+hard spheres, Phys. Rev. Lett. 114, 018302 (2015).
+[31] M. Z. Yamchi and A. Naji, Effective interactions between
+inclusions in an active bath, J. Chem. Phys. 147, 194901
+(2017).
+[32] F. Feng, T. Lei, and N. Zhao, Tunable depletion force
+in active and crowded environments, Phys. Rev. E 103,
+022604 (2021).
+[33] F. Smallenburg and H. L¨owen, Swim pressure on walls
+with curves and corners, Phys. Rev. E 92, 032304 (2015).
+[34] J. Harder, S. A. Mallorya, C. Tung, C. Valeriani, and A.
+Cacciuto, The role of particle shape in active depletion,
+J. Chem. Phys. 141, 194901 (2014).
+[35] P. Liu, S. Ye, F. Ye, K. Chen, and M. Yang, Constraint
+dependence of active depletion forces on passive particles,
+Phys. Rev. Lett. 124, 158001 (2020).
+[36] Y. Baek, A. P. Solon, X. Xu, N. Nikola, and Y. Kafri,
+Generic long-range interactions between passive bodies
+in an active liquid, Phys. Rev. Lett. 120, 058002 (2018).
+[37] S. R. McCandlish, A. Bhaskaran, and M. Hagan, Sponta-
+neous segregation of self-propelled particles with different
+motilities, Soft Matter 8, 2527 (2012).
+[38] S. C. Takatori and J. F. Brady, A theory for the phase
+behavior of mixtures of active particles, Soft Matter 11,
+7920 (2015).
+[39] A. K. Omar, Y. Wu, Z. G. Wang, and J. F. Brady, Swim-
+ming to stability: Structural and dynamical control via
+active doping, ACS Nano 13, 560 (2019).
+
+6
+[40] J. Stenhammar, R. Wittkowski, D. Marenduzzo, and M.
+E. Cates, Activity-induced phase separation and self-
+assembly in mixtures of active and passive particles,Phys.
+Rev. Lett. 114, 018301 (2015).
+[41] S. N. Weber, C. A. Weber and E. Frey, Binary mixtures
+of particles with different diffusivities demix, Phys. Rev.
+Lett. 116, 058301 (2016).
+[42] A. Wysocki, R. G. Winkler, and G. Gompper, Propagat-
+ing interfaces in mixtures of active and passive Brownian
+particles, New J. Phys. 18, 123030 (2016).
+[43] P. Dolai, A. Simha, and S. Mishra, Phase separation in
+binary mixtures of active and passive particles, Soft Mat-
+ter 14, 6137 (2018).
+[44] J. Adler, and B. Templeton, The effect of environmental
+conditions on the motility of Escherichia coli. J. Gen.
+Microbiol. 46, 175–184 (1967).
+[45] E. Ilker and J.-F. Joanny, Phase separation and nucle-
+ation in mixtures of particles with different temperatures,
+Phys. Rev. Res. 2, 023200 (2020).
+[46] A. Y. Grosberg and J.-F. Joanny, Nonequilibrium statis-
+tical mechanics of mixtures of particles in contact with
+different thermostats, Phys. Rev. E 92, 032118 (2015).
+[47] F. Hauke, H. Lowen, and B. Liebchen, Clustering-
+induced velocity-reversals of active colloids mixed with
+passive particles, J. Chem. Phys. 152, 014903 (2020).
+[48] R. L. Schilling and L. Partzsch, in Brownian Motion:An
+Introduction to Stochastic Processes, contributed by B.
+B¨ottcher (De Gruyter Textbook, 2012).
+[49] Z. Schuss, Brownian Dynamics at Boundaries and Inter-
+faces: In Physics, Chemistry, and Biology (Springer, New
+York, 2015)
+[50] Y. Fily and M. C. Marchetti, Athermal Phase Separa-
+tion of Self-Propelled Particles with No Alignment, Phys.
+Rev. Lett. 108, 235702 (2012).
+[51] A. Tiribocchi, R. Wittkowski, D. Marenduzzo, and M.
+E. Cates, Active Model H: Scalar Active Matter in
+a Momentum-Conserving Fluid, Phys. Rev. Lett., 115,
+188302 (2015).
+[52] F. Peruani and M. B¨ar, A kinetic model and scaling prop-
+erties of non-equilibrium clustering of self-propelled par-
+ticles, New J. Phys., 15, 065009(2013).
+[53] J. P. Singh, S. Pattanayak, S. Mishra and J. Chakrabarti,
+Effective single component description of steady state
+structures of passive particles in an active bath, The
+Journal of Chemical Physics, 156,(2022).
+[54] J. Dzubiella, J. Chakrabarti and H. Lowen, Tuning col-
+loidal interactions in subcritical solvents by solvophobic-
+ity: Explicit versus implicit modeling, J. Chem. Phys.
+131,044513(2009).
+[55] J Chakrabarti, S Chakrabarti and H Lown, Short ranged
+attraction and long ranged repulsion between two solute
+particles in a subcritical liquid solvent, J. Phys. Condense
+Matter 18,81-87(2006).
+[56] S. Asakura and F. Oosawa, On Interaction between Two
+Bodies Immersed in a Solution of Macromolecules, J.
+Chem. Phys. 22, 1255(1954).
+[57] H.N.W. Lekkerkerker, W. C. K. Poon, P. N. Pusey, A.
+Stroobants, and P. B. Warren, Phase Behaviour of Col-
+loid + Polymer Mixtures, Europhys. Lett. 20, 559 (1992).
+[58] A.S. Vishen, J. Prost, M. Rao, Breakdown of effective
+temperature, power law interactions, and self-propulsion
+in a momentum-conserving active fluid, Phys. Rev. E.
+100 (6), 062602 (2019).
+[59] R.C. Krafnick and A.E. Garcia, Impact of hydrodynam-
+ics on effective interactions in suspensions of active and
+passive matter, Phys. Rev. E 91, 022308 (2015).
+
+Supplementary information
+Phase separation of passive particles in active liquids
+Pragya Kushwaha1, Vivek Semwal2, Sayan Maity1, Shradha Mishra2, Vijayakumar Chikkadi11
+11 Indian Institute of Science Education and Research Pune, India 411008
+2 Indian Institute of Technology (BHU) Varanasi, India 221005
+I.
+METHODS
+A.
+Experiments
+The active suspensions were prepared using E.coli cells (U5/41 type strain). The bacteria were grown overnight at
+37◦C in LB agar plate containing 1% tryptone, 1% NaCl, 0.5% yeast extract and 1.5% agar. A single colony of E.
+coli was added to 10 ml of LB broth and kept at 37◦C until OD600 (optical density at 600 nm wavelength) reached
+a value of 1.3. Bacterial cells were then harvested and washed three times with motility media (10 mM potassium
+phosphate (pH 7.0), 0.1 mM EDTA , 0.002% Tween-20 and 50 mM L-Serine) by centrifugation at 3000 rpm for 5
+minutes at room temperature to remove the traces of LB broth. The pellet was later resuspended in motility media
+to get desired concentrations. The observation chamber was created using a circular cavity of size 1 cm and 100 µm
+deep, which was glued to a PEG coated coverslip using double-sided tape. The bacteria density of the sample with
+OD600 = 1 was estimated to contain 6 × 109 cells/ml (b0). The density of bacteria in our experiments was fixed at
+10b0, which was well below the density threshold for the onset of collective motion.
+B.
+Simulations
+The simulations were performed using a binary mixture with Na small active particles of radius aa and Np big
+passive particles of radius ap (ap > aa) moving on a two dimensional frictional substrate. The active particles are
+associated with a self propulsion speed v and an orientation unit vector ˆvi = (cosθi; sinθi), where θi is the angle
+between the velocity vector and a reference direction. The motion of active Brownian particles (ABP) is governed by
+the following Langevin equations:
+dri
+dt = vˆvi − µ1
+�
+j̸=i
+Fij +
+�
+2DT ηT i
+(S1)
+dθi
+dt = Γ
+�
+i
+sin(θi − θij) +
+�
+2Drηri.
+(S2)
+Here µ1 is the mobility and Fij is the force acting on particle i due to particle j. The noise term is defined as
+< ηr,T i(t)ηr,T j(t′) >= 2Dr,T δijδ(t − t′), DT and Dr are the translational and rotational diffusion constants of active
+particles and Γ is the magnitude of alignment and θij = arg (ri − rj). The persistence length lp = v/Dr of active
+particles is constant in our simulations, it is set at lp = 20aa. The other constants in our simulations are DT = 0.005
+and Γ = 1
+The equation of motion for passive particles is
+dri
+dt = µ2
+�
+j̸=i
+Fij,
+(S3)
+where µ2 is the mobility of passive particles. There is no translational noise in Eq. 3, so the dynamics of passive
+particles is only due to interaction force. We choose the mobility of both species to be the same i.e., µ1 = µ2. Particles
+interact through short ranged soft repulsive forces Fij = Fijˆrij, where Fij = k(ai + aj − rij) when rij ≤ (ai + aj)
+and Fij = 0 otherwise; rij = |ri − rj| and k is a constant.
+The elastic time scale in our system is defined by
+(µk)−1 = (150)−1.
+arXiv:2301.11771v1  [cond-mat.soft]  27 Jan 2023
+
+2
+II.
+SUPPLEMENTARY FIGURES
+A.
+Peclet number estimation
+FIG. S1.
+(a) Histogram of bacteria velocities. The average velocity is < v >= 33.84 ± 9.98 µm/s. (b) Histogram of the size of
+the bacteria. The size of the bacteria is its length along the longer axis. The average length of the cells is < l >= 2.68±0.86 µm.
+(c) The rotational diffusion time of the bacteria was estimated from their normalised velocity auto-correlation function. The
+dashed line is an exponential fit to the data, which gives a rotational diffusion time of τr ∼ 1.67 s.
+III.
+SUPPLEMENTARY FIGURES
+A.
+Steady state in experiments
+FIG. S2.
+The average cluster size in our system as a function of frame number for a range of area fractions from φ ∼ 0.1 − 0.4
+and at S ∼ 2.5 (a), S ∼ 3.5 (b) and S ∼ 5.5 (c). The steady state of the system is evident from these figures. The total
+duration of the measurement was 500 s
+
+350
+(a)
+300
+250
+200
+Count
+150
+100
+50
+0
+20
+40
+60
+um/sec
+225
+(b)
+20
+15
+Count
+10
+5
+0
+2
+3
+1
+4
+5
+um(c)
+100
+é(t).é(0)
+10-1
+0
+1
+2
+3
+4
+t(s)5.5
+(a)
+Φ=0.10
+5
+- Φ=0.20
+-Φ=0.30
+cluster size
+4.5
+-Φ=0.39
+4
+3.5
+3
+2.5
+2
+ha
+1.5
+1
+1000
+2000
+3000
+4000
+5000
+Frames(b)
+—Φ=0.11
+7
+ - Φ=0.22
+--Φ=0.30
+-Φ=0.40
+6
+5
+4
+3
+2
+1
+1000
+2000
+3000
+4000
+5000
+Frames(c)
+一Φ=0.11
+30
+ - Φ=0.21
+---Φ=0.30
+size
+-Φ=0.40
+25
+cluster s
+20
+Average
+15
+10
+5
+1000
+2000
+3000
+4000
+5000
+Frames3
+B.
+Calculation of effective potential
+FIG. S3. Snapshot of the part of the system to calculate the potential. The two bigger particles are passive particles with the
+left one marked as particle 1 and the right one marked 2 with positions r1 and r2, respectively. The red circles are ABPs. The
+line shows the surface to surface distance r between two passive particles
+C.
+Density of active particles in the vicinity of an isolated passive particle in simulations
+FIG. S4.
+Density fluctuations of active particles due to a single passive particle in the center of the system. Instantaneous
+snapshots of the system for four size ratios S = 3, 5, 8 and 10, from left to right. Many such configurations are used to calculate
+the density correlations C(r). Red particles are ABP’s and blue particle at the center is the bigger passive particle.
+D.
+Number fluctuations of active particles in the vicinity of an isolated passive particle in simulations
+FIG. S5. Number fluctuation ∆N vs. N for different size ratios. The slopes of dotted line is 0.5 and dashed line is 0.7
+For calculating number fluctuation, we start with an annular disc with an inner radius the radius of the passive
+particle and outer radius is varied. The mean and variance of number of ABP’s are calculated for different outer
+
+S=3
+S=5
+S=8
+slope=0.5
+slope=0.7
+10
+10
+100
+1000
+N4
+radius of the disc. The same is repeated for three different sizes of the passive particle or for three different size ratios
+S = 3, 5 and 8. In the Fig. S6 we show the plot of ∆N vs. N for three sizes S = 3, 5 and 8. For all the cases the
+graph is power law with ∆N ≃ N α, where α ≃ 0.7 for moderate N for all S and starts to deviate for large N. The
+deviation appears at relatively larger N on increasing size ratio. Hence increasing the size of passive particle increases
+the stretch of density fluctuation of ABP’s and it might be one of the factor to introduce a long ranged attraction
+among passive particles for larger size ratio.
+E.
+Correlations of density fluctuations of bacteria in the vicinity of an isolated colloidal particle in
+experiments
+FIG. S6. The correlations of density fluctuations C(r), as defined in the main text, is shown for two different size ratios S ∼ 2.5
+and 5.5. The x-axis is scaled by the size of the bacteria l. It is clear from these results that the density fluctuations are
+suppressed in experiments.
+
+ε-S = 5.5
+e-S = 2.5
+0.8
+0.6
+0.4
+r
+0.2
+0
+-0.2
+-0.4
+5
+10
+15
+0
+20
+r/l

69E0T4oBgHgl3EQfwAF2/content/tmp_files/2301.02626v1.pdf.txt

@@ -0,0 +1,1615 @@
+A hierarchical equations of motion (HEOM) analog for systems with delay: illustrated
+on inter-cavity photon propagation
+Robert Fuchs1 and Marten Richter1, ∗
+1Institut f¨ur Theoretische Physik, Nichtlineare Optik und Quantenelektronik,
+Technische Universit¨at Berlin, Hardenbergstr. 36, EW 7-1, 10623 Berlin, Germany
+(Dated: January 9, 2023)
+Over the last two decades, the hierarchical equations of motion (HEOM) of Tanimura and Kubo
+have become the equation of motion-based tool for numerically exact calculations of system-bath
+problems.
+The HEOM is today generalized to many cases of dissipation and transfer processes
+through an external bath.
+In spatially extended photonic systems, the propagation of photons
+through the bath leads to retardation/delays in the coupling of quantum emitters. Here, the idea
+behind the HEOM derivation is generalized to the case of photon retardation and applied to the
+simple example of two dielectric slabs. The derived equations provide a simple reliable framework
+for describing retardation and may provide an alternative to path integral treatments.
+After the hierarchical equations of motion (HEOM)
+were initially invented by Tanimura and Kubo [1, 2] to
+solve numerically exactly the open quantum system prob-
+lem with a Debye spectral density, the HEOM did not
+immediately take off, since the limited numeric capabil-
+ities did not allow for a versatile implementation at the
+time. However, the idea to use the time constant deriva-
+tive of the Debye spectral density time correlation func-
+tion stuck.
+Recently, various implementations [2–7] of
+HEOM followed after sufficient computing power became
+available. Soon after its invention, many generalizations
+using arbitrary spectral densities by decomposition into
+summed Debye form spectral densities were also devel-
+oped.
+For most system-bath approaches it provides a
+well-established path to a numerically exact solution.
+A different type of system-bath problem is the prop-
+agation of quantum states e.g. through a bath of pho-
+tons or phonons [8–17]. A typical problem is describing
+quantum interconnects for quantum computing and cryp-
+tography applications. Recently, various applications of
+these systems with a delay caused by the propagation
+through the bath were investigated [8–17] including the
+development of different methods. However, the number
+of propagating photons is still limited, as it was for the
+open quantum systems approaches until HEOM imple-
+mentations became widespread, along with other meth-
+ods such as tensor networks [14, 15, 18–33]. In this pa-
+per, an analysis of the HEOM derivation in the context
+of delay is carried out and HEOM analog equations for
+systems with delay are derived. We demonstrate that the
+approach leads to a systematic set of equations ordered
+by the number of photons propagating through the bath.
+In the future, combinations with, e.g., tensor networks
+or automatic derivation may lead to an additional route
+to solve problems involving delays.
+The paper starts with a derivation of the HEOM ana-
+log for open quantum systems with delay and illustrates
+its potential with a simple photon propagation example.
+HEOM derivation: An HEOM analog with delay is
+derived for an open quantum system with: H = Hs +
+Hb + Hsb. Here, Hs is the Hamiltonian of the system,
+which consists of quantum emitters in different spatially
+separated cavities. Hb is the bath Hamiltonian contain-
+ing the propagating photon modes. Finally, Hsb is the
+system-bath coupling Hamiltonian.
+In open quantum
+systems, only the observables of the system are of in-
+terest, which can be calculated from the relevant density
+matrix ρs(t) = trB(ρ(t)).
+Its calculation is the main
+objective of HEOM, where we transfer the steps by Tan-
+imura and Kubo [1] to systems with delay. We assume a
+factorized initial state ρ(t0) = ρs(t0) ⊗ ρB, where ρB is a
+harmonic bath state. The system dynamics obey:
+ρs(t) = trB (T←U(t, t0)
+exp
+�
+− i
+ℏ
+� t
+t0
+dτU(t0, τ)Hsb,−(τ)U(τ, t0)
+�
+ρs(t0) ⊗ ρB) ,
+(1)
+where ALρ = Aρ, ARρ = ρA, and A− = AL − AR
+define the Liouville space operators acting on Liouville
+operator ρ for any Hilbert space operator A [34] and
+U(t, t0) = T←exp
+�
+− i
+ℏ
+� t
+t0 dτ(Hs,−(τ) + Hb,−(τ))
+�
+with
+time ordering operator T←. Hs,−(τ) may also contain
+Lindblad operators for describing external processes act-
+ing on the joint system-bath state. Following the HEOM
+derivation [1] and the path integral derivation from [17],
+we convert Eq. (1) to path integral form:
+ρs(t) = trB (
+T←
+M
+�
+i=0
+Ui,i−1exp
+�� t0+εi
+t0+ε(i−1)
+dτU †
+i−1(τ)Hsb,−(τ)Ui−1(τ)
+�
+ρs(t0) ⊗ ρB)
+(2)
+with ε
+=
+(t − t0)/M and M
+→
+∞ (in the fol-
+lowing equations the limit is always assumed).
+Fur-
+thermore, Ui,j
+=
+U(t0 + εi, t0 + εj) and Ui(τ)
+=
+U(τ, t0 + ε(i)).
+For
+small
+ε,
+the
+approxima-
+tion Ui,i−1 exp
+�� t0+εi
+t0+ε(i−1) dτU †
+i−1(τ)Hsb(τ)Ui−1(τ)
+�
+≈
+arXiv:2301.02626v1  [quant-ph]  6 Jan 2023
+
+2
+Ui,i−1 + ε · Ui,i−1/2Hsb(t0 + ε(i − 1/2))Ui−1/2,i−1 =:
+Ui,i−1 + ε · U (1)
+sb (i) holds, yielding:
+ρs(t) = trB
+�
+T←
+M
+�
+i=0
+(Ui,i−1 + ε · U (1)
+sb (i))ρs(t0) ⊗ ρB
+�
+.
+We assume linear system-bath coupling:
+Hsb
+=
+�
+ijµ CijµAijBµ with system Aij and linear bath opera-
+tor Bµ. For a system A and bath B Liouville operator
+the relation (AB)− = A+B− + A−B+ holds, so U (1)
+sb (i)
+can be written as a sum over products of the system
+and bath operators U (1)
+sb (i) = �
+l A(1)
+l
+(i)B(1)
+l
+(i), and we
+define A(0)
+l
+= U s
+i,i−1δl,0 and B(0)
+l
+= U b
+i,i−1δl,0 with the
+system and bath parts of Ui,i−1. With these relations,
+we write ρS in terms of a system part S and an influence
+functional (similar form as in [17]),
+ρs(t) =
+1
+�
+k1... kM=0
+�
+l1... lM
+� M
+�
+i=1
+εki
+�
+S(k1l1, ... , kMlM)
+× I(k1l1, ... , kMlM).
+(3)
+The system part is still an operator S(k1l1, ... , kMlM) =
+T←
+�M
+i=1 A(ki)
+li
+(i)ρs(t0), while the influence functional
+I(k1l1, ... , kMlM) = trB(T←
+�M
+i=1 B(ki)
+li
+(i)ρB) is just a
+number. Since ρB is assumed to be a harmonic bath equi-
+librium state, Wick’s theorem allows us to factorize the
+influence functional I into expectation values of two bath
+operators B(1)
+l
+(·). Furthermore, for small ε, the system
+propagator is roughly U s
+i,i−1 ≈ Ids − i
+ℏεHs,−(t0 + ε(i −
+1/2)). Using the approximations of the time propagators
+Figure 1. (a) Model of two open QNM cavities with dissi-
+pation rates γµ and effective inter-cavity coupling strength
+Vµη. (b) 1D model with two slabs of width L = 21 µm with
+constant permittivity ϵR = π2 serving as QNM cavities, sit-
+ting against a background ϵB = 1. (c) Scheme of the HEOM
+depicting a process including inter-cavity transfer and dissi-
+pation.
+and using Wick’s theorem we obtain,
+ρs(t + ε) ≈ ρs(t) − ε i
+ℏHs,−(t0 + ε(M + 1/2))ρs(t)
++
+�
+lM+1
+T←εA(1)
+lM+1
+�
+k1... kM
+�
+l1... lM
+� M
+�
+i=1
+εki
+�
+Aki
+li (i)ρs(t0)
+×
+M
+�
+m=1
+trB(B(1)
+lM+1(M + 1)U B
+M,m+1B(1)
+lm (m)ρB)δkm,1
+I(k1l1, ... , km−1lm−1, 00, km+1lm+1... , kMlM),
+including only the terms at most linear in ε. Collecting
+the terms linear in ε yields the derivative of ρs[1]:
+∂tρs(t) = − i
+ℏHs,−(t)ρs(t)
+(4)
++
+�
+l˜l
+A(1)
+l
+(t)
+� t
+t0
+dt1⟨B(1)
+l
+(t)B(1)
+˜l
+(t1)⟩Bρ(1)
+s˜l (t, t1),
+where ⟨A⟩B = trB(AρB) and the bath correlation func-
+tion is in the interaction picture, and
+ρ(1)
+sl (t, ˜t) = δkm1δlml
+�
+T←
+M
+�
+i=1.i̸=m
+B(ki)
+li
+(i)
+�
+B
+�
+k1...kM
+�
+l1...lM
+T←A(1)
+l
+(m)
+�
+�
+M
+�
+i=1,i̸=m
+εkiA(ki)
+li
+(i)
+�
+� ρs(t0),
+with ˜t = mε + t0.
+Here, the derivation deviates from
+the original recipe of Kubo and Tanimura, since the as-
+sumption of a spectral density in Debye form (simple
+exponential e−γt in time) is not compatible with systems
+including delay. Generalizations of HEOM usually rely
+on a decomposition of the spectral density into a sum of
+exponential functions to recover the Debye form. How-
+ever, an expansion of the correlation function for the de-
+lay case using e−γ|t−tdelay| does not yield the advantages
+of Kubo’s and Tanimura’s approach, since the original
+relies on a time constant derivative of the Debye spec-
+tral density time correlation function. Instead, a delayed
+correlation of the above form introduces a sign change at
+t = tdelay, so that a dependence of ρ(n) on earlier inte-
+gration times is unavoidable in the case with delay. Thus
+the integration over t1 is not included in the definition
+of ρ(1) in contrast to the original HEOM [1]. Keeping
+the general form of the bath correlation function is more
+flexible than using a special form, which would simplify
+the equations of motion in the following. ρ(1)
+sl (·, t1) de-
+scribes bath disturbances to the system density matrix,
+which are initially caused by an interaction with A(1)
+˜l
+at
+time t1 (similar to the auxiliary dimensions in extended
+TCL [35]). Of course, the additional time argument pre-
+vents direct numerical implementations for increasing n.
+But specific bath correlation functions together with an-
+alytic calculation or tensor network methods [14, 15, 29–
+33, 36–40] will allow solutions nevertheless.
+Using the
+
+ER
+ER3
+same technique as for ∂tρs(t) yields:
+∂tρ(1)
+sl1(t, t1) = − i
+ℏHs,−(t)ρ(1)
+sl1(t, t1)
++
+�
+l2˜l2
+A(1)
+l2 (t)
+� t
+t0
+dt2⟨B(1)
+l2 (t)B(1)
+˜l2 (t2)⟩Bρ(2)
+sl1˜l2(t, t2, t1)
++ δ(t − t1)A(1)
+l1 (t1)ρs(t1 − 0+).
+(5)
+where we use the interaction picture for the bath correla-
+tion function. Instead of an initial condition ρ(1)
+sl1(t1, t1) =
+Al1(t1)ρs(t1 − 0+), the δ term at the time of the initial
+condition is included. I.e., ρ(1)
+sl1(·, t1) is equal to zero (in
+the delta case) or not defined (in the initial condition
+case) before time t1. Note that t1, t2 of ρ(2)(t, t2, t1) are
+not time ordered since different delay/retardation times
+can occur in open quantum systems.
+The form of ρ(2) points to a general definition of ρ(n)
+starting with ρ(0)(t) = ρs(t):
+ρ(n)
+s˜l1... ˜ln(t, ˜tn, ... , ˜t1) =
+�
+k1...kM
+�
+l1...lM
+T←
+�
+�
+n
+�
+j=1
+A(1)
+˜lj (mj)δ˜lj,lmj δ˜kmj 1
+�
+�
+�
+�
+M
+�
+j=1,∧n
+i=1j̸=mi
+εkjA(kj)
+lj
+(j)
+�
+� ρs(t0)
+�
+T←
+M
+�
+j=1,∧n
+i=1j̸=mi
+B(kj)
+lj
+(j)
+�
+B
+,
+(6)
+with ˜ti = miε + t0. Analogous to ρ(1), this yields:
+∂tρ(n)
+sl1... ln(t, t1, ... , tn) = − i
+ℏHs,−(t)ρ(n)
+sl1... ln(t, t1, ... , tn)
++
+�
+ln+1˜ln+1
+A(1)
+ln+1(t)
+� t
+t0
+dtn+1⟨B(1)
+ln+1(t)B(1)
+˜ln+1(tn+1)⟩B
+ρ(n+1)
+sl1...ln˜ln+1(t, t1, . . . , tn+1)
++
+n
+�
+p=1
+δ(t − tp)A(1)
+lp (tp)
+(7)
+× ρ(n−1)
+sl1... lp−1lp+1... ln(tp − 0+, t1, ... , tp−1, tp+1, ... tn+1).
+The last term is again a replacement to an initial condi-
+tion: ρ(n)
+sl1... ln(tp, t1, ... , tn) = Alp(tp)ρ(n−1)
+sl1... lp−1lp+1ln(tp −
+0+, t1, ... , tp−1, tp+1, ... , tn) with tp = maxi(ti), and it is
+clear that ρ(n)
+sl1... ln(t, t1, ... , tn) = 0 for t < tp. So for the
+last term only p with the largest time tp contributes. Fur-
+thermore ρ(n)
+s
+is invariant under permutations of t1, ... , tn
+including their corresponding l1, ... , ln.
+The physics behind Eq. (7) is very accessible: n corre-
+sponds to the maximum number of photons traveling in
+the bath at a given time t, so an exact truncation of
+Figure 2. (a) Single-photon occupations of the two 1D dielec-
+tric slabs from Fig. 1. The dotted lines show the full reference
+wave function solution. (b) Absolute values of the two-photon
+coherences. For both cases, the QNM frequencies of the slabs
+are identical ˜ω1 = (0.06 − 0.0124i) eV, with coupling strength
+VBA = VAB = 0.0062 eV.
+the equations based on the traveling photons is possible.
+Note, the photons on the left and right side states of
+the density matrix count accumulating, so a transfer of a
+single photon density requires two traveling photons (left
+and right side of density matrix), as opposed to one trav-
+eling photon for a single photon coherence.
+For other
+open quantum system equations of motion techniques
+such as Nakajima-Zwanzig [41] or time convolution less
+(TCL) equations [41], the generators K in the equations
+of motion contain the system-bath coupling in any order.
+A calculation of higher-order contributions from K is gen-
+erally cumbersome involving higher products of system-
+bath correlation functions as well as a truncation at a
+given photon number. For the HEOM analog, only one
+system-bath correlation function appears in the second
+term of Eq. (7) cleanly separating on photon number.
+The first term of Eq. (7) describes the system dynam-
+ics. The second term represents the absorption of a bath
+photon, which entered the bath at time tn+1. The last
+term describes photon emission into the bath.
+Two photon propagation: As a benchmark for the new
+approach, we consider two spatially separated quasinor-
+mal mode (QNM) cavities, coupled to a common pho-
+tonic bath (Fig. 1(a)). The QNMs ˜fµ are an open sys-
+tem analog to normal modes, which solve the Helmholtz
+equation under an outgoing radiation condition [42–48].
+QNMs have complex eigenfrequencies ˜ωµ = ωµ−iγµ with
+photon decay rate γµ > 0. Here, two dielectric slabs serve
+as QNM cavities as in Fig. 1(b). We assume an effective
+1D problem with homogenous continuation in the y, z
+direction . The model allows the analytical calculation
+of the modes (assuming a constant real permittivity ϵR)
+and coupling elements (cf. [49]).
+We include only the
+lowest energy QNM, assuming that all other modes are
+off-resonance.
+Since the slabs are identical, both have
+the same frequency ˜ωA = ˜ωB = ˜ω1. However, we keep
+the indices for generality. The slabs are separated by the
+distance R, which is large enough for a separate quanti-
+zation of the modes without inter-cavity coupling.
+As a first step, we consider a population with initially
+
+1.00
+1.00
+(a)
+slab A
+(b)
+(20ps/00)
+0.75
+slab B
+Occupation
+(02/ps/00)
+0.75
+Amplitude 
+(11/ps/00)
+0.50
+0.50
+0.25
+0.25
+0.00
+0.00
+0
+100
+200
+300
+0
+100
+200
+300
+t [ps]
+t [ps]4
+one excitation (one photon on each side of the density
+matrix) in slab A and a vacuum bath. Therefore, the
+hierarchy truncates at n = 2, i.e. ρ(n) = 0, n > 2, and:
+ρ(2)
+s,l1,l2(t, t1, t2) =
+Θ(t1 − t2)U s(t, t1)A(1)
+l1 (t1)ρ(1)
+s,l2(t1 − 0+, t2)
++ Θ(t2 − t1)U s(t, t2)A(1)
+l2 (t2)ρ(1)
+s,l1(t2 − 0+, t1),
+(8)
+using the initial conditions for ρ(2). The truncation is
+exact since the maximal number of propagating photons
+at any time is set by the initial conditions.
+Inserting Eq. (8) into Eq. (5), we obtain:
+∂tρ(1)
+sl1(t, t1) = − i
+ℏHs,−(t)ρ(1)
+sl1(t, t1)
++
+�
+l2˜l2
+A(1)
+˜l2 (t)
+� t1
+t0
+dt2⟨B(1)
+˜l2 (t)B(1)
+l2 (t2)⟩B
+× U s(t, t1)A(1)
+l1 (t1)ρ(1)
+sl2(t1 − 0+, t2)
++
+�
+l2˜l2
+A(1)
+˜l2 (t)
+� t
+t1
+dt2⟨B(1)
+˜l2 (t)B(1)
+l2 (t2)⟩B
+× U s(t, t2)A(1)
+l2 (t2)ρ(1)
+sl1(t2 − 0+, t1)
++ δ(t − t1)A(1)
+l1 (t1)ρs(t1 − 0+).
+(9)
+Eqs. (9) and (4) form a closed set of equations of motion
+for the system density matrix that are exactly solvable
+(cf. [49]) for at most two traveling photons.
+Fig. 1(c)
+illustrates connections between the equations with one
+photon traveling from time t1 = t − τ until t through the
+bath, requiring the calculation of ρ(1). Intermittently a
+second photon is emitted into the bath at t2.
+The
+dynamics
+of
+a
+specific
+system
+are
+deter-
+mined
+by
+the
+system-bath
+correlation
+function
+trB(B(1)
+˜l2 (t)B(1)
+l2 (t1)ρB), which describes the emission
+of a photon into the bath at time t1 and reabsorption
+at time t.
+The correlation function results from the
+system-bath coupling and reads (cf. [49]):
+Cµη(t − t′) ≈ 2Vµηℏ2 (Θ(t − t′)δ(t − t′ − τ)
++ Θ(t′ − t)δ(t − t′ + τ)) ,
+(10)
+where µ, η are cavity indices (A or B).
+The coupling
+strength is given by Vµη = (1 + δµη)γ1/2 with the cavity
+decay rate γ1. Due to topology, the inter-cavity coupling
+is exactly half the dissipation rate.
+For the 1D case,
+a photon emitted away from the other cavity will not
+return, while a photon emitted towards the other cavity
+can be transferred into that cavity. In higher dimensions,
+the inter-cavity coupling will be much smaller than the
+dissipation rate. The delay time τ in Eq. (10) depends
+implicitly on the involved cavities, with τ = R/c, µ ̸= η,
+and 0 otherwise. For one initial excitation, three system
+states |A⟩ = |10⟩, |B⟩ = |01⟩, |0⟩ = |00⟩ contribute, with
+the excitation in slab A or B, or both slabs in the ground
+state, respectively. Inserting Eq. (10) into Eq. (4) yields
+the equations of motion. As an example, the occupation
+in slab A ⟨A|ρs(t)|A⟩ evolves as (cf. [49]):
+∂t⟨A|ρs(t)|A⟩ = −2γA⟨A|ρs(t)|A⟩
+− 2V ∗
+BAe−iωBτ⟨0|ρ(1)L
+s,0B(t, t − τ)|A⟩ + c.c. (11)
+For the auxiliary density matrix ρ(1), starting from
+Eq. (9) results in (cf. [49]):
+∂t⟨0|ρ(1)L
+s,0B(t, t1)|A⟩ = δ(t − t1)⟨B|ρs(t1)|A⟩
+= −γA⟨0|ρ(1)L
+s,0B(t, t1)|A⟩
+− 2VBAeiωBτ⟨B|ρ(1)R
+s,B0(t1, t − τ)|0⟩
+− 2VBAeiωAτ⟨0|ρ(1)L
+s,0B(t − τ, t1)|A⟩.
+(12)
+The remaining equations for the occupation in B, the
+coherences, and matrix elements for ρ(1) are calculated
+analogously (cf. [49]).
+For time-non-local interactions,
+the system density matrix in Eq. (11) only couples to the
+first auxiliary density matrix ρ(1). Time-local processes
+such as the cavity photon dissipation are included in the
+zeroth step of the hierarchy.
+Fig. 2(a) shows the time dynamics of the single-photon
+occupations in slabs A and B. The model system allows a
+calculation using the wave function (cf. [49]) as a bench-
+mark.
+The HEOM (solid lines) and exact wave func-
+tion (dotted lines) results agree perfectly. Note, that the
+HEOM allows in principle the inclusion of Lindblad terms
+(e.g. for pumping), which the wave function does not.
+Over time the single excitation in slab A will dissipate
+into the bath. However, some photons are transferred to
+the QNM of slab B with delay τ ≈ 44 ps. For the used
+parameters, the occupation in B is even larger than the
+occupation in A after some time. Eventually, the system
+arrives at a trapped state [14, 50–55] due to constructive
+interference from the inter-cavity transfer.
+Note, that the HEOM allows in principle the inclu-
+sion of Lindblad terms (e.g.
+for pumping), which the
+wave function does not. Also an extension to two pho-
+ton processes is feasible for the HEOM. Fig. 2(b) shows
+the two-photon coherences (two photons on one side
+of the density matrix, none on the other) for the two
+slabs from Fig. 1(b), which includes at most two trav-
+eling photons, resulting in a calculation analogous to
+Fig. 2(a). The amplitudes of the intra-cavity coherences
+⟨20|ρs|00⟩/⟨02|ρs|00⟩ resemble the dynamics of the den-
+sities in Fig. 2(a), since in principle the same indepen-
+dent processes are involved. The inter-cavity coherence
+⟨11|ρs|00⟩ requires the transfer of just one photon and
+thus shows a rapid increase after t = τ.
+In the final
+equilibrium state the probability (coherence squared) of
+an inter-cavity contribution matches the sum of the two
+intra-cavity probabilities.
+
+5
+A feasible calculation of the exact solution as shown
+here is limited to a small number of photons by the ex-
+ponential scaling of the numerical complexity with the
+number of excitations.
+For systems requiring a higher
+number of traveling photons, a calculation of the higher
+steps in the hierarchy via matrix product states or other
+tensor networks [14, 15, 29–33, 36–40] may be possible as
+well as analytic calculations in special setups. Further-
+more, the HEOM allows a perturbative truncation of the
+hierarchy for systems with a small system-bath coupling.
+Thus, at least an approximate solution may be possible
+for higher excitation numbers.
+In conclusion, we analyzed the derivation of hierar-
+chical equations of motion and transferred the idea to
+open quantum systems with delay. The resulting equa-
+tions allow a natural, easy truncation on the number of
+excitations in the bath, which is otherwise cumbersome
+for Nakajima-Zwanzig or time convolution-less equations.
+The first implementation for single- and multi-photon
+transfer between two cavities demonstrated the feasibility
+of the approach. We expect that in the future more de-
+manding implementations including tensor network ap-
+proaches may allow the simulation of several photons
+traveling through complex quantum networks.
+∗ marten.richter@tu-berlin.de
+[1] Y. Tanimura and R. Kubo, Time evolution of a quantum
+system in contact with a nearly gaussian-markoffian noise
+bath, Journal of the Physical Society of Japan 58, 101
+(1989).
+[2] Y. Tanimura, Numerically “exact” approach to open
+quantum dynamics: The hierarchical equations of mo-
+tion (heom), The Journal of chemical physics 153, 020901
+(2020).
+[3] L. Ye, X. Wang, D. Hou, R.-X. Xu, X. Zheng, and Y. Yan,
+Heom-quick: a program for accurate, efficient, and uni-
+versal characterization of strongly correlated quantum
+impurity systems, WIREs Computational Molecular Sci-
+ence 6, 608 (2016).
+[4] N. Lambert, T. Raheja, S. Ahmed, A. Pitchford, and
+F. Nori, Bofin-heom: A bosonic and fermionic numer-
+ical hierarchical-equations-of-motion library with appli-
+cations in light-harvesting, quantum control, and single-
+molecule electronics, arXiv preprint arXiv:2010.10806
+(2020).
+[5] T. Kramer, M. Noack, A. Reinefeld, M. Rodr´ıguez, and
+Y. Zelinskyy, Efficient calculation of open quantum sys-
+tem dynamics and time-resolved spectroscopy with dis-
+tributed memory heom (dm-heom), Journal of Compu-
+tational Chemistry 39, 1779 (2018).
+[6] J. Seibt and O. K¨uhn, Strong exciton-vibrational cou-
+pling in molecular assemblies. dynamics using the po-
+laron transformation in heom space, The Journal of Phys-
+ical Chemistry A 125, 7052 (2021).
+[7] T. Kramer, M. Noack, J. R. Reimers, A. Reinefeld,
+M. Rodr´ıguez, and S. Yin, Energy flow in the photosys-
+tem i supercomplex: Comparison of approximative theo-
+ries with dm-heom, Chemical Physics 515, 262 (2018).
+[8] R. F. Oulton, V. J. Sorger, D. Genov, D. Pile, and
+X. Zhang, A hybrid plasmonic waveguide for subwave-
+length confinement and long-range propagation, Nature
+Photonics 2, 496 (2008).
+[9] M. I. Stockman, Nanofocusing of optical energy in ta-
+pered plasmonic waveguides, Physical review letters 93,
+137404 (2004).
+[10] A. Orieux, M. A. Versteegh, K. D. J¨ons, and S. Ducci,
+Semiconductor devices for entangled photon pair genera-
+tion: a review, Reports on Progress in Physics 80, 076001
+(2017).
+[11] M. Weiß and H. J. Krenner, Interfacing quantum emit-
+ters with propagating surface acoustic waves, Journal of
+Physics D: Applied Physics 51, 373001 (2018).
+[12] H. Jayakumar, A. Predojevi´c, T. Kauten, T. Huber,
+G. S. Solomon, and G. Weihs, Time-bin entangled pho-
+tons from a quantum dot, Nature communications 5, 1
+(2014).
+[13] A. Carmele and S. Reitzenstein, Non-markovian features
+in semiconductor quantum optics: quantifying the role
+of phonons in experiment and theory, Nanophotonics 8,
+655 (2019).
+[14] H. Pichler and P. Zoller, Photonic circuits with time de-
+lays and quantum feedback, Phys. Rev. Lett. 116, 093601
+(2016).
+[15] O. Kaestle, R. Finsterhoelzl, A. Knorr, and A. Carmele,
+Continuous and time-discrete non-markovian system-
+reservoir interactions:
+Dissipative coherent quantum
+feedback in liouville space, Phys. Rev. Research 3, 023168
+(2021).
+[16] S. Arranz Regidor, G. Crowder, H. Carmichael, and
+S. Hughes, Modeling quantum light-matter interactions
+in waveguide qed with retardation, nonlinear interac-
+tions, and a time-delayed feedback:
+Matrix product
+states versus a space-discretized waveguide model, Phys.
+Rev. Research 3, 023030 (2021).
+[17] M. Richter and S. Hughes, Enhanced tempo algorithm
+for quantum path integrals with off-diagonal system-bath
+coupling: Applications to photonic quantum networks,
+Phys. Rev. Lett. 128, 167403 (2022).
+[18] A. Caldeira and A. Leggett, Path integral approach to
+quantum brownian motion, Physica A 121, 587 (1983).
+[19] Y. Tanimura and S. Mukamel, Real-time path-integral
+approach to quantum coherence and dephasing in nona-
+diabatic transitions and nonlinear optical response, Phys.
+Rev. E 47, 118 (1993).
+[20] N. Makri and D. E. Makarov, Tensor propagator for itera-
+tive quantum time evolution of reduced density matrices.
+i. theory, J. Chem. Phys. 102, 4600 (1995).
+[21] N. Makri and D. E. Makarov, Tensor propagator for it-
+erative quantum time evolution of reduced density ma-
+trices. ii. numerical methodology, J. Chem. Phys. 102,
+4611 (1995).
+[22] A. Vagov, M. D. Croitoru, M. Gl¨assl, V. M. Axt, and
+T. Kuhn, Real-time path integrals for quantum dots:
+Quantum dissipative dynamics with superohmic environ-
+ment coupling, Phys. Rev. B 83, 094303 (2011).
+[23] A. Strathearn, B. W. Lovett, and P. Kirton, Efficient real-
+time path integrals for non-markovian spin-boson mod-
+els, New Journal of Physics 19, 093009 (2017).
+[24] A. Strathearn, P. Kirton, D. Kilda, J. Keeling, and B. W.
+Lovett, Efficient non-markovian quantum dynamics using
+time-evolving matrix product operators, Nature commu-
+
+6
+nications 9, 3322 (2018).
+[25] D. Gribben, D. M. Rouse, J. Iles-Smith, A. Strathearn,
+H. Maguire, P. Kirton, A. Nazir, E. M. Gauger, and
+B. W. Lovett, Exact dynamics of nonadditive environ-
+ments in non-markovian open quantum systems, PRX
+Quantum 3, 010321 (2022).
+[26] M. Cygorek, M. Cosacchi, A. Vagov, V. M. Axt, B. W.
+Lovett, J. Keeling, and E. M. Gauger, Simulation of open
+quantum systems by automated compression of arbitrary
+environments, Nature Physics , 1 (2022).
+[27] J. Prior, I. de Vega, A. W. Chin, S. F. Huelga, and M. B.
+Plenio, Quantum dynamics in photonic crystals, Phys.
+Rev. A 87, 013428 (2013).
+[28] F. Caycedo-Soler, A. Mattioni, J. Lim, T. Renger,
+S. Huelga, and M. Plenio, Exact simulation of pigment-
+protein complexes unveils vibronic renormalization of
+electronic parameters in ultrafast spectroscopy, Nature
+Communications 13, 1 (2022).
+[29] S. R. Clark, J. Prior, M. J. Hartmann, D. Jaksch, and
+M. B. Plenio, Exact matrix product solutions in the
+Heisenberg picture of an open quantum spin chain, New
+Journal of Physics 12, 025005 (2010).
+[30] A.
+H.
+Werner,
+D.
+Jaschke,
+P.
+Silvi,
+M.
+Kliesch,
+T. Calarco, J. Eisert, and S. Montangero, Positive tensor
+network approach for simulating open quantum many-
+body systems, Phys. Rev. Lett. 116, 237201 (2016).
+[31] R. Rosenbach, J. Cerrillo, S. F. Huelga, J. Cao, and M. B.
+Plenio, Efficient simulation of non-markovian system-
+environment interaction, New Journal of Physics 18,
+023035 (2016).
+[32] F. A. Schr¨oder, D. H. Turban, A. J. Musser, N. D.
+Hine, and A. W. Chin, Tensor network simulation of
+multi-environmental open quantum dynamics via ma-
+chine learning and entanglement renormalisation, Nature
+communications 10, 1 (2019).
+[33] A. D. Somoza, O. Marty, J. Lim, S. F. Huelga, and M. B.
+Plenio, Dissipation-assisted matrix product factorization,
+Phys. Rev. Lett. 123, 100502 (2019).
+[34] V.
+Chernyak
+and
+S.
+Mukamel,
+Collective
+coor-
+dinates
+for
+nuclear
+spectral
+densities
+in
+energy
+transfer
+and
+femtosecond
+spectroscopy
+of
+molecu-
+lar
+aggregates,
+J.
+Chem.
+Phys.
+105,
+4565
+(1996),
+https://doi.org/10.1063/1.472302.
+[35] M. Richter and A. Knorr, A time convolution less density
+matrix approach to the nonlinear optical response of a
+coupled system–bath complex, Annals of Physics 325,
+711 (2010).
+[36] R. Or´us, A practical introduction to tensor networks:
+Matrix product states and projected entangled pair
+states, Annals of Physics 349, 117 (2014).
+[37] U. Schollw¨ock, The density-matrix renormalization group
+in the age of matrix product states, Annals of physics
+326, 96 (2011).
+[38] J. Cirac, D. P´erez-Garc´ıa, N. Schuch, and F. Verstraete,
+Matrix product density operators: Renormalization fixed
+points and boundary theories, Annals of Physics 378, 100
+(2017).
+[39] F. Verstraete and J. I. Cirac, Matrix product states rep-
+resent ground states faithfully, Physical Review B 73,
+094423 (2006).
+[40] G. Vidal, Classical simulation of infinite-size quantum
+lattice systems in one spatial dimension, Physical review
+letters 98, 070201 (2007).
+[41] H.-P. Breuer, F. Petruccione, et al., The theory of open
+quantum systems (Oxford University Press on Demand,
+2002).
+[42] G. Garc´ıa-Calder´on and R. Peierls, Resonant states and
+their uses, Nuclear Physics A 265, 443 (1976).
+[43] K. Lee, P. Leung, and K. Pang, Dyadic formulation of
+morphology-dependent resonances. i. completeness rela-
+tion, JOSA B 16, 1409 (1999).
+[44] E. A. Muljarov, W. Langbein, and R. Zimmermann,
+Brillouin-wigner perturbation theory in open electro-
+magnetic systems, EPL (Europhysics Letters) 92, 50010
+(2011).
+[45] P. T. Kristensen, C. Van Vlack, and S. Hughes, Gener-
+alized effective mode volume for leaky optical cavities,
+Optics letters 37, 1649 (2012).
+[46] C.
+Sauvan,
+J.-P.
+Hugonin,
+I.
+S.
+Maksymov,
+and
+P. Lalanne, Theory of the spontaneous optical emission
+of nanosize photonic and plasmon resonators, Physical
+Review Letters 110, 237401 (2013).
+[47] S. Franke, S. Hughes, M. K. Dezfouli, P. T. Kristensen,
+K. Busch, A. Knorr, and M. Richter, Quantization of
+quasinormal modes for open cavities and plasmonic cav-
+ity quantum electrodynamics, Physical review letters
+122, 213901 (2019).
+[48] P. T. Kristensen,
+K. Herrmann,
+F. Intravaia, and
+K. Busch, Modeling electromagnetic resonators using
+quasinormal modes, Advances in Optics and Photonics
+12, 612 (2020).
+[49] See supplemental material at [url will be inserted by pub-
+lisher] for (i) analytic coupling elements,(ii) equations of
+motions and (iii) approach using the wave function.
+[50] M. Bello, G. Platero, J. I. Cirac, and A. Gonz´alez-Tudela,
+Unconventional quantum optics in topological waveguide
+qed, Science advances 5, eaaw0297 (2019).
+[51] S. Hughes and G. S. Agarwal, Anisotropy-induced quan-
+tum interference and population trapping between or-
+thogonal quantum dot exciton states in semiconduc-
+tor cavity systems, Physical review letters 118, 063601
+(2017).
+[52] A. L. Grimsmo, Time-delayed quantum feedback control,
+Physical review letters 115, 060402 (2015).
+[53] N. N´emet, A. Carmele, S. Parkins, and A. Knorr, Com-
+parison between continuous-and discrete-mode coherent
+feedback for the jaynes-cummings model, Physical Re-
+view A 100, 023805 (2019).
+[54] K.
+Barkemeyer,
+R.
+Finsterh¨olzl,
+A.
+Knorr,
+and
+A. Carmele, Revisiting quantum feedback control: dis-
+entangling the feedback-induced phase from the corre-
+sponding amplitude, Advanced Quantum Technologies 3,
+1900078 (2020).
+[55] R. Finsterh¨olzl, M. Katzer, and A. Carmele, Nonequi-
+librium non-markovian steady states in open quantum
+many-body systems: Persistent oscillations in heisenberg
+quantum spin chains, Physical Review B 102, 174309
+(2020).
+[56] P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J.-P.
+Hugonin, Light interaction with photonic and plasmonic
+resonances, Laser & Photonics Reviews 12, 1700113
+(2018).
+[57] R.-C. Ge, P. T. Kristensen, J. F. Young, and S. Hughes,
+Quasinormal mode approach to modelling light-emission
+and propagation in nanoplasmonics, New Journal of
+Physics 16, 113048 (2014).
+[58] S. Franke, M. Richter, J. Ren, A. Knorr, and S. Hughes,
+Quantized quasinormal-mode description of nonlinear
+
+7
+cavity-qed effects from coupled resonators with a fano-
+like resonance, Physical Review Research 2, 033456
+(2020).
+[59] T. Gruner and D.-G. Welsch, Green-function approach
+to the radiation-field quantization for homogeneous and
+inhomogeneous kramers-kronig dielectrics, Phys. Rev. A
+53, 1818 (1996).
+[60] S. Mukamel, Principles of nonlinear optical spectroscopy,
+6 (Oxford University Press on Demand, 1999).
+
+S1
+Analytic coupling elements
+We use analytic expressions of the mode frequencies,
+decay constants, and coupling elements for numeric eval-
+uation. For linearly polarized waves and assuming a ho-
+mogeneous continuation in y, z-direction, the problem re-
+duces to the 1D model from Fig. 1(b). The QNM within
+each slab is given by [48, 56]
+˜fµ(x)
+���
+|x|<L/2 = einRkµx + e−inRkµx+iµπ,
+(S1)
+where nR = √ϵR is the refractive index of the slab and
+kµ = ˜ωµ/c is the QNM wavenumber.
+The QNM fre-
+quency ˜ωµ is [48, 56]
+˜ωµL/c = 2πµ + iln
+�
+(nR − nB)2/(nr + nB)2�
+2nR
+.
+(S2)
+Thus,
+the
+frequency
+of
+the
+first
+QNM
+˜f1(x)
+is
+˜ω1 = ω1 − iγ1 = (1 − i0.21)L/c. The second QNM ˜f2(x)
+has a resonance frequency that is twice as large. Hence,
+as a first approximation, we take only the first QNM in
+our calculations.
+Outside
+of
+the
+cavity
+(|x|
+>
+L/2),
+we
+re-
+place
+the
+QNMs
+with
+regularized
+modes
+[57]
+˜Fµ(x, ω)
+=
+� L/2
+−L/2 dx′GB(x, x′, ω)∆ϵ(x′) ˜fµ(x′)
+=
+(x/|x|)Mµ(ω)eiω|x|/c, where ∆ϵ(x) = ϵR − ϵB, |x| < L/2,
+and 0 otherwise, and
+Mµ(ω) = i
+2L(π2 − 1)
+�
+si
+�(ω + π˜ωµ)L
+2c
+�
+−si
+�(ω − π˜ωµ)L
+2c
+��
+(S3)
+is an analytical factor that vanishes for ω
+→
+∞.
+si(x)
+=
+sin(x)/x is the unnormalized sinc-function.
+GB(x, x′, ω) = ie−iω|x−x′|/c/2 is the vacuum Green’s
+function for the case of linearly polarized waves, solving
+the Helmholtz equation
+�
+∂2
+x + ω2
+c2
+�
+GB(x, x′, ω) = ω2
+c2 δ(x − x′).
+(S4)
+We locate the slab A at x = 0 and slab B at x = R (cf.
+Fig. 1(b)), so that ˜f1(x) = ˜fA(x) and ˜fB(x) = ˜fA(x−R).
+We quantize the QNMs following the procedure laid out
+in [47], with minor adjustments due to the 1D nature of
+the problem, e.g. taking the 1D analog of the electric
+field quantization and QNM Green function instead of
+the 3D expressions that were used in [47]. Since the QNM
+quantization relies on a complex dissipative permittivity,
+we add a constant imaginary part to the permittivities
+of the slabs and background medium: ϵα = ϵR/B + iακ
+(cf. [58]) so that the original values are retained in the
+limit α → 0. Taking the 1D analog of the quantization
+in dissipative media from [59], we find the electric field
+operator to be
+Eα(x) =
+� ∞
+0
+dω
+�
+dx′ i
+ωϵ0
+Gα(x, x′, ω)ˆjα(x′, ω) + H.a.,
+(S5)
+where G(x, x′, ω) is the Greens function of the dissipa-
+tive medium and ˆjα(x, ω) = ω
+�
+(ℏϵ0/π)ϵα
+I (x, ω)ˆb(x, ω)
+is the noise-current density operator, with ϵI denot-
+ing the imaginary part of the permittivity and ˆb(x, ω)
+a Bosonic photon annihilation operator.
+We use the
+Green’s function expansion in terms of QNMs [43, 56, 57]
+G(x, x′, ω) = �
+µ=A,B Aµ(ω) ˜fµ(x) ˜fµ(x′), where Aµ(ω) =
+ω/(2(˜ωµ − ω)), and the QNM functions ˜fµ are replaced
+with regularized modes ˜Fµ outside their respective cav-
+ity volumes. Inserting the QNM Green’s function into
+Eq. (S5), we find QNM operators analogous to [47]:
+˜aA =
+�
+2
+πωA
+� ∞
+0
+dωAA(ω)
+�� L/2
+−L/2
+dx
+�
+ϵα
+I (x, ω) ˜f α
+A(x)ˆb(x, ω)
++ lim
+λ→∞
+� λ
+L/2
+dx
+�
+ϵα
+I (x, ω) ˜F α
+A(x, ω)ˆb(x, ω)
++ lim
+λ→∞
+� −L/2
+−λ
+dx
+�
+ϵα
+I (x, ω) ˜F α
+A(x, ω)ˆb(x, ω)
+�
+,
+(S6)
+which depend implicitly on α → 0. In the first integral,
+the limit α → 0 can be carried out immediately, so that
+this contribution vanishes, because limα→0 ϵα
+I = 0.
+In
+the other two integrals, the order of the limits cannot be
+exchanged, as pointed out in [58], so the limit λ → ∞ has
+to be taken first. The operators for the QNMs of cavity
+B are defined analogously, just spatially shifted by R.
+The QNM operators defined in Eq. (S6) are non-Bosonic,
+with
+�
+˜aA, ˜a†
+A
+�
+= SAA, and
+SAA =
+2
+πωA
+� ∞
+0
+dωAA(ω)A∗
+A(ω)
+×
+�
+lim
+λ→∞
+� λ
+L/2
+dxϵα
+I (x, ω) ˜F α
+A(x, ω) ˜F ∗,α
+A (x, ω)
++ lim
+λ→∞
+� −L/2
+−λ
+dxϵα
+I (x, ω) ˜F α
+A(x, ω) ˜F ∗,α
+A (x, ω)
+�
+.
+(S7)
+Analogous to [58], we employ the Helmholtz equation of
+the background Green’s function (Eq. (S4)) to reduce the
+integral over x to the value of the modes at the limits of
+the integration volume. Taking the limit λ → ∞ first
+and then α → 0, we find
+SAA = 2c
+γ1
+|M1(˜ω1)|2 ,
+(S8)
+
+S2
+where we used ˜ωA = ˜ω1.
+The overlap integral [˜aA, ˜a†
+B] = SAB is calculated ac-
+cordingly. We make use of the fact that the two slabs are
+identical except for their spatial separation and hence
+˜ωA = ˜ωB = ˜ω1, to obtain
+SAB = 2c
+γ1
+|M1(˜ω1)|2 Re
+� ˜ω1
+2ω1
+e−iω1R/c
+�
+e−γ1R/c.
+(S9)
+Since
+��Re
+�
+˜ω1e−iω1R/c/(2ω1)
+���
+<
+1, it follows that
+|SAB/SAA| < e−γ1R/c, due to the retarded interaction
+between the slabs.
+The QNMs penetrate through the
+boundary of the slab so that there is a non-zero overlap
+even without time delay. However, the mode is concen-
+trated at the cavity so that the overlap is small if the
+slabs are well enough separated. Below, the correlation
+functions are discussed for the case with finite time delay.
+The QNMs’ wavelength is λ1 = 2L, so a separation of a
+few dozen wavelengths, as used in the main text, leads
+to negligible contributions of the overlap.
+Thus, the QNM operators are symmetrized indepen-
+dently within their respective cavities similar to the
+single-cavity case in [47]:
+ˆaµ =
+�
+dx
+� ∞
+0
+dωLµ(x, ω)ˆb(x, ω),
+(S10)
+with
+Lµ(x, ω) = S−1/2
+µµ
+�
+γµϵI(x, ω)
+πcωµ|M1(˜ωµ)|Aµ(ω) ˜fµ(x),
+(S11)
+and the mode function ˜fµ is replaced by the regularized
+mode ˜Fµ outside the slab volume. The imaginary part
+of the permittivity and the bounds of the spatial integral
+include implicit limits, as discussed above.
+We now define continuum operators ˆc(x, ω) = ˆb(x, ω) −
+�
+µ=A,B L∗
+µ(x, ω)ˆaµ [58], which commute with the sym-
+metrized Bosonic QNM operators and serve as the bath.
+While they are generally non-Bosonic, as a first ap-
+proximation, we neglect the non-Bosonic contributions.
+This allows us to decompose the full Hamiltonian H =
+ℏ
+�
+dx
+� ∞
+0
+dωωˆb†(x, ω)ˆb(x, ω) into system and bath parts
+[58]:
+HS = ℏ
+�
+µ=A,B
+ωµˆa†
+µˆaµ,
+HB = ℏ
+�
+dx
+� ∞
+0
+dωωˆc†(x, ω)ˆc(x, ω),
+HSB = ℏ
+�
+µ=A,B
+�
+dx
+� ∞
+0
+dωgµ(x, ω)ˆc(x, ω)ˆa†
+µ + H.a.
+(S12)
+The
+coupling
+elements
+gµ(x, ω)
+=
+−S−1/2
+µµ
+×
+�
+ϵI(x, ω)/(2πωµ)ω ˜fµ(x),
+are derived from the pro-
+jectors Lµ(x, ω), with the pole at ω = ˜ωµ removed
+during the derivation, as shown in [58].
+To derive the coupling strength of the interaction
+between the slabs mediated via the bath, we calculate
+the correlation function [34, 60] that characterizes the
+system-bath interaction in the HEOM formalism:
+Cµη(t − t′) = ℏ2
+� ∞
+0
+dω
+� ∞
+0
+dω′
+�
+dx
+�
+dx′e−iω(t−t′)
+×gµ(x, ω)g∗
+η(x′, ω′)
+�
+ˆc(x, ω)ˆc†(x′, ω′)
+�
+B .
+(S13)
+For ρB = |0⟩⟨0| (no initial photons), the trace results in
+a delta function, so only an integral over the coupling
+elements remains. This is calculated similarly to [58], i.e.
+by assuming that the coupling is sharply peaked at the
+QNM frequency, so that
+�
+dxgµ(x, ω)g∗
+η(x, ω) ≈ S−1
+11
+2c
+γ1
+|M1(˜ω1)|2 γ1
+2π
+×
+�
+eiωRµη/c + e−iωRµη/c�
+, (S14)
+where |Rµη| is R if µ ̸= η and 0 otherwise. Using S11 =
+2c|M1(˜ω1)|2/γ1, and defining the retardation time τ =
+Rµη/c as an implicit function of µ and η, the correlation
+function becomes
+Cµη(t − t′) = γ1ℏ2
+2π
+� ∞
+0
+dω
+�
+eiωτ + e−iωτ�
+e−iω(t−t′),
+(S15)
+As a final approximation, we extend the lower limit to
+−∞ [58], to obtain the correlation function in Eq. (10).
+Calculation of the equations of motion
+For equations of motion of the density-matrix ele-
+ments, we convert Eq. (4) to a more explicit form by
+replacing the indices l → (α, ν1ν2, x, ω) of the system
+and bath operators
+Al(t) → ˆAα
+ν1ν2(t),
+Bl(t) →
+(S16)
+�
+µ
+⟨ν1|ˆaµ|ν2⟩
+�
+dx
+� ∞
+0
+dωg∗
+µ(x, ω)ˆc†,α(x, ω) + H.a.,
+with ˆAν1ν2 = |ν1⟩⟨ν2|, so that
+∂tρs(t) = − i
+ℏHs,−(t)ρs(t)
++
+�
+α,β=L,R
+�
+ν1...ν4
+(−1)α+β ˆAα
+ν1ν2(t)
+×
+� t
+t0
+dt1Cβ
+ν1ν2ν3ν4(t, t1)ρ(1)β
+s,ν3ν4(t, t1),
+(S17)
+
+S3
+where |νi⟩ is a system state, and α, β determines whether
+the operator acts on the left or right side of the den-
+sity matrix.
+The sign is negative if α ̸= β.
+The
+correlation function is defined as CL
+ν1ν2ν3ν4(t, t1)
+=
+�
+µη⟨ν1|ˆa†
+µ|ν2⟩⟨ν3|ˆaη|ν4⟩Cµη(t − t1),
+with
+Cµη
+from
+Eq. (10), and CR
+ν1ν2ν3ν4(t, t1) =
+�
+CL
+ν1ν2ν3ν4(t, t1)
+�∗.
+For brevity, we use |A⟩, |B⟩, |0⟩ as defined in the main
+text, above Eq. (11). To derive Eq. (11), we take the
+expectation value with respect to state |A⟩ on (S17) to
+obtain,
+∂t⟨A|ρs(t)|A⟩ = − i
+ℏ⟨A|Hs,−ρs(t)|A⟩
++
+�
+α,β=L,R
+�
+ν1...ν4
+(−1)α+β
+� t
+t0
+dt1Cβ
+ν1ν2ν3ν4(t, t1)
+× ⟨A| ˆAα
+ν1ν2(t)ρ(1)β
+s,ν3ν4(t, t1)|A⟩.
+(S18)
+Since the |νi⟩ are orthogonal, only certain combinations
+of states and α, β survive.
+Using the definition of the
+QNM correlation function (Eq. (10)) and the initial con-
+ditions for ρ(1), results in the first-order equation of mo-
+tion given in Eq. (11). Similarly, we obtain an equation
+for the coherence ⟨A|ρs(t)|B⟩:
+∂t⟨A|ρs(t)|B⟩ = − (γA + γB)⟨A|ρs(t)|B⟩
+− 2V ∗
+BAe−iωBτ⟨0|ρ(1)L
+s,0B(t, t − τ)|B⟩
++ c.c.(A ↔ B).
+(S19)
+The equations for the occupation in slab B and the second
+coherence term are obtained from Eq. (11) and Eq. (S19),
+respectively, by exchanging A ↔ B.
+For Eq. (12), we insert Eq. (S16) into Eq. (9). We use a
+rotating-frame representation of ρ(1) with respect to its
+time arguments, e.g.,
+⟨0|ρ(1),L
+s,0B (t, t1)|A⟩ → e−iω1(t+t1)⟨0|ρ(1),L
+s,0B (t, t1)|A⟩, (S20)
+where we have used ωA = ωB = ω1. A similar derivation
+as for the matrix elements of ρs yields fast-rotating terms.
+Within the rotating-frame, ⟨0|ρ(1),L
+s,0B (t, t1)|A⟩ evolves ac-
+cording to Eq. (12). In the same manner, we derive:
+∂t⟨A|ρ(1)R
+s,A0(t, t1)|0⟩ = δ(t − t1)⟨A|ρs(t1)|A⟩
+= −γA⟨A|ρ(1)L
+s,A0(t, t1)|0⟩
+− 2V ∗
+BAe−iωBτ⟨0|ρ(1)R
+s,0B(t1, t − τ)|A⟩
+− 2V ∗
+BAe−iωBτ⟨B|ρ(1)L
+s,A0(t − τ, t1)|0⟩.
+(S21)
+The last six matrix elements of ρ(1) are derived from
+Eq. (12) and (S21) by complex conjugation or exchang-
+ing the indices A and B. Note that ρ(1)(t, t1) vanishes for
+t < t1 or t1 < 0. Furthermore, only ρ(1)(t, t − τ) appears
+in Eq. (11) and Eq. (S19). Therefore, the last terms in
+Eq. (12) and Eq. (S21), respectively, do not contribute
+to the dynamics of ρs.
+For the two-photon coherences, we obtain (following a
+similar derivation as for the single-photon occupation):
+∂t⟨20|ρs(t)|00⟩ = −2γA⟨20|ρs(t)|00⟩
+−
+√
+8V ∗
+BAe−iωAτ⟨10|ρ(1)L
+s,0B1B(t, t − τ)|00⟩,
+(S22)
+where we use 0B and 1B to indicate that the initial
+system-bath interaction involves the transition of cavity
+B from the one-photon state to the ground state. Anal-
+ogously, ⟨02|ρs(t)|00⟩ = (⟨20|ρs(t)|00⟩)(A ↔ B) and
+∂t⟨11|ρs(t)|00⟩ = −(γA + γB)⟨11|ρs(t)|00⟩
+−
+√
+8V ∗
+BAe−iωAτ⟨01|ρ(1)L
+s,1B2B(t, t − τ)|00⟩
+−
+√
+8V ∗
+ABe−iωBτ⟨10|ρ(1)L
+s,1A2A(t, t − τ)|00⟩
+− 2V ∗
+BAe−iωAτ⟨01|ρ(1)L
+s,0B1B(t, t − τ)|00⟩
+− 2V ∗
+ABe−iωBτ⟨10|ρ(1)L
+s,0A1A(t, t − τ)|00⟩.
+(S23)
+The equations for the matrix elements of ρ(1) in the rotat-
+ing frame read (keeping only those terms that contribute
+to ρs):
+∂t⟨10|ρ(1)L
+s,0B1B(t, t1)|00⟩ = δ(t − t1)∂t⟨11|ρs(t1)|00⟩
+− γA⟨10|ρ(1)L
+s,0B1B(t, t1)|00⟩
+−
+√
+8V ∗
+BAe−iωAτ⟨01|ρ(1)L
+s,1B2B(t1, t − τ)|00⟩
+− 2V ∗
+BAe−iωAτ⟨01|ρ(1)L
+s,0B1B(t1, t − τ)|00⟩,
+∂t⟨10|ρ(1)L
+s,0A1A(t, t1)|00⟩ = −γA⟨10|ρ(1)L
+s,0A1A(t, t1)|00⟩
+− 2V ∗
+BAe−iωAτ⟨10|ρ(1)L
+s,0B1B(t1, t − τ)|00⟩,
+∂t⟨10|ρ(1)L
+s,1A2A(t, t1)|00⟩ = δ(t − t1)∂t⟨20|ρs(t1)|00⟩
+− γA⟨10|ρ(1)L
+s,1A2A(t, t1)|00⟩.
+(S24)
+The remaining three matrix elements are again obtained
+by exchanging A ↔ B.
+Wave function approach
+For initially one excitation in slab A from Fig. 1(b),
+the general wave function has the form
+|ψ⟩ = NA|A⟩|0⟩ + NB|B⟩|0⟩ +
+�
+dx
+� ∞
+0
+dωNx,ω|0⟩|x, ω⟩.
+(S25)
+The first Ket refers to the system state as defined above
+Eq. (11), and the second is the bath state with contin-
+uous spatial and frequency indices x, ω. N is the time-
+dependent amplitude of a particular state, with the ini-
+tial conditions NA(0) = 1, NB(0) = Nx,ω(0) = 0. In the
+
+S4
+interaction picture, the dynamics of the states are gov-
+erned by the Schr¨odinger equation with the system-bath
+interaction Hamiltonian from Eq. (S12). The QNM and
+bath operators carry the free evolution of the system and
+bath: ˆaµ(t) = e−iωµtˆaµ and ˆc(x, ω, t) = e−iωtˆc(x, ω).
+Multiplying the Schr¨odinger equation for (S25) with
+⟨0|⟨A| from the left yields an equation for NA:
+iℏ∂tNA(t) = ℏ
+�
+dx
+� ∞
+0
+dωNx,ωgA(x, ω)e−iωteiωAt.
+(S26)
+Similarly, we obtain the equation for Nx,ω:
+iℏ∂tNx,ω(t) = ℏ
+�
+NAg∗
+A(x, ω)e−iωAt
++NBg∗
+B(x, ω)e−iωBt�
+eiωt,
+which we integrate formally and insert the result back
+into eq. (S26) to find:
+∂tNA(t) = − 1
+ℏ2
+� ∞
+t0
+dt′
+�
+dx
+� ∞
+0
+dω (CAA(t − t′)NA(t′)
++ eiωAt−iωBt′CAB(t − t′)NB(t′)
+�
+,
+(S27)
+where we have inserted the definition of the QNM cor-
+relation function from Eq. (S13).
+Using Eq. (10) and
+ωA = ωB, we arrive at:
+∂tNA(t) = −γANA(t) − 2V ∗
+BAeiωBτNB(t − τ)Θ(t − τ).
+(S28)
+An analogous derivation for NB yields a similar equation,
+with the indices switched (A ↔ B). The density matrix
+elements are calculated by multiplying the amplitudes
+with their complex conjugates, e.g., ⟨A|ρs|A⟩ = |NA|2.
+

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+GPU-based high-precision orbital propagation of large sets of
+initial conditions through Picard-Chebyshev augmentation
+Alessandro Masat ∗ and Camilla Colombo†
+Politecnico di Milano, Milano, Italy, 20156
+Arnaud Boutonnet ‡
+European Space Agency (ESA-ESOC), Darmstadt, Germany, 64293
+The orbital propagation of large sets of initial conditions under high accuracy requirements
+is currently a bottleneck in the development of space missions, e.g. for planetary protection
+compliance analyses. The proposed approach can include any force source in the dynamical
+model through efficient Picard-Chebyshev (PC) numerical simulations. A two-level augmenta-
+tion of the integration scheme is proposed, to run an arbitrary number of simulations within the
+same algorithm call, fully exploiting high performance and GPU (Graphics Processing Units)
+computing facilities. The performances obtained with implementation in C and NVIDIA®
+CUDA® programming languages are shown, on a test case taken from the optimization of a
+Solar Orbiter-like first resonant phase with Venus.
+I. Introduction
+Complex trajectory solutions have become a standard choice for interplanetary missions, and often include several
+gravity assist maneuvers to reach remote space regions with limited fuel consumption. Two recent missions, the
+ESA/NASA mission Solar Orbiter [1] and the ESA mission JUICE [2], are meaningful ongoing cases. The former
+first features preparatory flybys of Earth and Venus, then exploits several resonant encounters with Venus to raise the
+spacecraft inclination over the ecliptic and observe the Sun’s polar regions. The latter requires multiple flybys to reach
+Jupiter, and then repeatedly swings by different Jupiter’s moons to fulfill its scientific observational objectives. Both
+missions would not be practically feasible without all the designed flybys, as every saved kilogram of fuel means more
+mass available to board scientific equipment.
+The common framework where high-energy multi-flyby trajectories are designed is the patched conics approximation,
+whose simple but effective model allows a consistent preliminary mission analysis. Feasible gravity assist maneuvers are
+identified and the consequent optimization of the initial trajectory guess can be performed, by refining the preliminary
+orbital parameters and/or adding correction maneuvers to meet the operational requirements and fit the full body
+∗PhD Candidate, Department of Aerospace Science and Technology, Via G. La Masa 34, 20156, Milano, Italy, alessandro.masat@polimi.it
+†Associate Professor, Department of Aerospace Science and Technology, Via G. La Masa 34, 20156, Milano, Italy, camilla.colombo@polimi.it
+‡Senior Mission Analyst, HSO-GFA, ESA-ESOC, Robert-Bosch-Str. 5, D-64293 Darmstadt, Germany, arnaud.boutonnet@esa.int
+arXiv:2301.03989v1  [cs.DC]  10 Jan 2023
+
+dynamics.
+Nevertheless, the real-life environment features more complex physical phenomena, whose perturbing effects may
+have significant consequences on the nominal mission. Repeated flybys also imply a higher impact risk of disposal
+objects with the planets flown by, possibly affecting the mission compliance with planetary protection requirements.
+COSPAR [3] maintains worldwide planetary protection policies, e.g. ESA demands the use of at least the N-body
+Newtonian gravitational model for the compliance assessment [4]. This accuracy requirement has so far limited the
+analysis to Monte-Carlo based techniques [5–9] with Cartesian equations of motion, whose simulations inherently carry
+a heavy computational burden that may limit the mission development time. Recent developments [10] nearly halved
+the required computational time by implementing the Kustaanheimo-Stiefel [11] formulation of the dynamical motion.
+This work takes inspiration from the results obtained in [12, 13], whose aim was to optimize a multi-flyby trajectory
+arc while taking advantage of the natural dynamics of the N-body relativistic environment. Despite the conceptual
+complexity, the combined b-plane and Picard-Chebyshev (PC) approach led to an overall computationally feasible
+algorithm, for a Matlab® sequential implementation converging to an optimal trajectory in a few minutes only. At
+the core of the efficiency lies the fixed-point nature of the PC numerical scheme, that minimizes the need of reading
+and interpolating ephemerides data. The test case is described in more detail in Section VI.A. The set of trajectories
+generated by the optimization process is used to validate the concepts proposed in this work, in terms of their parallel
+propagation with the modified version of the PC method.
+The modified PC method has been continuously developed in the past few years, both in its formulation and
+implementation and outlining possible applications for Earth orbits where it contributed to increase the efficiency
+of the numerical analyses. Junkins et al [14] analyzed the performances of the method comparing the efficiency
+against the Runge-Kutta-Nystrom 12(10) integrator, proposing also a second order version. Later, Koblick and Shankar
+[15] extended the analysis to the propagation of accurate orbits testing difference force models with NASA’s Java
+Astrodynamics toolkit. Woollands et al. [16–18] applied the method as numerical integrator for the solution of
+the Lambert two-point boundary value problem, assessing also the benefits of adopting the Kuustanheimo-Stiefel
+formulation of the dynamics and proposing a solution for the multi-revolution trajectory design. Swenson et al. [19]
+applied the modified PC method on the circular restricted three-body problem, using the differential correction approach.
+Singh et al. [20] used the method as the numerical integration scheme for their feasibility study on quasi-frozen,
+near polar and low altitude lunar orbits, including the N-bodies and the spherical harmonics perturbations. The fixed
+point nature of the method was exploited by Koblick et al. [21] to design low-thrust trajectories as an optimal control
+problem, discretizing the control impulses and also included the Earth’s oblateness J2 perturbation. Macomber et al.
+[22] introduced the concepts of cold, warm, hot starts of the method, addressing possible efficiency improvements by
+means of better initial conditions, and variable-precision force models taking advantage of the fixed-point nature of the
+algorithm. Woollands et al. [23]extended the optimal low-thrust design to a high-fidelity model for the non-spherical
+2
+
+Earth, considering an arbitrary number of spherical harmonics in the perturbing acceleration. Woollands and Junkins
+[24] developed the Adaptive PC method, including an integral error feedback that accelerates the convergence of the
+Picard iterations and an empirical law to determine segment length and polynomial degree of the method, based on
+previous stability analyses. Atallah et al. [25] compared the method with other sequential integration techniques on
+different Earth-based orbital cases.
+Despite the parallelization possibilities, the method’s instability when dealing with long integration spans and the
+limited number of nodes (about 200) required to reach machine precision levels are disadvantages for the massively
+parallel/GPU (Graphics Processing Units) implementation of the PC integration scheme. Differently from the previous
+developments of the method, this work introduces an augmentation of the dynamical system being integrated to enable
+stable massive parallelism for the short-term propagation of large sets of initial conditions. The results obtained in
+[12, 13], as all the simulated trajectories analyzed to find the optimal one, are re-run as a direct application of the
+proposed approach, discussing the parallelization options in deep detail. Within a single run, the common time nodes of
+all the trajectory arcs allow to rework the iterative refinement process of PC. Instead of the six-dimensional Cartesian
+state of the single trajectory, the propagation is performed for a two-level augmented state, made by a properly sorted
+collection of the states of multiple trajectories at the same time node. Regardless the sorting choices of the single states
+within the augmented system, the fundamental mathematical structure of the PC process remains unaltered, without
+the need of re-defining the method’s steps, coefficients and matrices. The integration of the newest developments of
+the method, particularly the second order version [14] with error feedback [24], would therefore be applicable to the
+augmented version as well. The two different augmentation levels allow to preserve the fine grain flexibility that, if
+feasible, simulating each single trajectory alone would have, without a too large sacrifice in computational efficiency.
+Based on a different approach, GPU-based programs for the propagation of large sets of initial conditions already
+exist: Geda et al. [26] developed for the European Space Agency the CUDAjectory tool, a GPU-based propagator
+that implements the Runge-Kutta-Fehlberg 7/8 scheme. The massive parallelism is exploited by taking the forward
+integration step in parallel for many different samples, parallelizing the evaluation of the dynamics function and the
+simple matrix multiplications involved in the evaluation of the intermediate sub steps. Another interesting feature
+implemented in CUDAjectory is the optimized ephemerides reading and storage technique, following the work of
+Schrammel et al. [27], which increases the data locality and minimizes the required memory transactions. Despite
+also propagating large sets of trajectories, the proposed work differs from the CUDAjectory approach in the adopted
+numerical scheme. The PC method is by itself a parallelizable routine, the augmented version proposed and implemented
+in this work contributes to enhance this aspect, resulting in a highly scalable and performing program.
+On a final note, the concept of the proposed approach may extend to the more general uncertainty propagation case,
+applied also to planetary and terrestrial system, for robust analysis or collision probability computation. The heavy
+computational burden has promoted the adoption of simplified approaches, that aim at achieving acceptable accuracy in
+3
+
+the propagated uncertainty for lower computational costs [28]. The heavy burden of accurate Monte Carlo simulations
+was found to be a major challenge, for instance, even in the design of the touchdown of the Hayabusa 2 mission [29] and
+the characterisation of the impact probability with Earth of the Asteroid Apophis [30]. It is beyond the scopes of this
+work to detail the existing uncertainty propagation techniques that have been developed to reduce the computational
+burden. The only similarity with the proposed approach is the accessible reduction of the computational time, that is
+achieved using an inexpensive GPU, whose logic could be transferred to any other uncertainty propagation method that
+requires the simulations of relatively large sets of initial conditions.
+II. Picard-Chebyshev integration method
+A. Integration concept
+Picard iterations [31] are a method that can be used to obtain an approximation of the solution of initial/boundary
+value problems. Denoting the state of dimension 𝑛 with x, the independent variable with 𝑡, the initial/boundary condition
+with x0 and the dynamics function with f(x, 𝑡), the problem is defined as:
+𝑑x
+𝑑𝑡 = f(x, 𝑡),
+x0 = x(𝑡0)
+(1)
+Starting from an initial approximation x(0) (𝑡) of the actual solution x(𝑡) in the interval
+�
+𝑡0, 𝑡
+�
+of the initial/boundary
+value problem presented in Equation (1), the 𝑖-th Picard iteration improves the previous approximation x(𝑖−1) (𝑡) of x(𝑡)
+with x(𝑖) (𝑡) as in [31]:
+x(𝑖) (𝑡) = x(0) (𝑡) +
+∫ 𝑡
+𝑡0
+f�x(𝑖−1) (𝑠), 𝑠�𝑑𝑠
+(2)
+The method converges for a good enough initial approximation x(0) (𝑡) and for 𝑖 −→ +∞ [31].
+In the analytical Picard iteration context, performing more than one iteration is in general hard. The increasingly
+complex expressions for x(𝑖) (𝑡) make it difficult to retrieve closed form solutions after the first 2-3 steps [32]. At the
+same time, numerically computing the integral functions by quadrature might not suffice in accuracy, as only the first
+few iterations in general improve the function approximation. In the attempt to develop parallelizable routines for the
+integration of the dynamical motion, the PC method was built combining the Picard iterations with the Chebyshev
+polynomial approximation [33]. A possible derivation of the method that follows the work of Fukushima [34] can be
+summarized in three steps:
+1) Select a good enough initial guess x(0) (𝑡).
+2) Approximate f(x, 𝑡) and x(0) (𝑡) with their Chebyshev polynomial expansion.
+3) Perform a Picard iteration to update the coefficients of the interpolating Chebyshev polynomials.
+The Picard iterations halt when the stopping conditions are met, based on the maximum difference between two
+4
+
+consecutive iterations dropping below some user-specified tolerance.
+The so defined method allows to easily perform several more Picard iterations than the analytical case. The involved
+expressions remains always of the same type, i.e. the Chebyshev polynomials. The function approximation becomes an
+interpolation through nodes that should be close to the true trajectory, instead of a global function whose value after the
+iterations still depend on the initial guess choice. Furthermore, few iterations suffice to drop below a low tolerance if
+the real solution x(𝑡) differs from the initial guess x(0) (𝑡) only because of small perturbations [31]. Starting from the
+unperturbed Keplerian solution for the generic weakly perturbed two body problem, a relatively fast convergence of the
+method is ensured [34]. In the context of orbital simulations, Macomber [35] referred to this type of initial guess as
+warm-starting the PC iteration method, because the analytic solution of the dominant dynamics part is used to reduce
+the number of iterations required. Differently, the cold start was defined by simply setting all the trajectory samples
+as equal to the initial condition. In general, the closer the initial guess to the true trajectory, the lower the number of
+iterations will be. Semi-analytic initial guesses or results of propagations from simpler models are also an options, and
+in the case of three-body-like perturbed trajectories would be a better choice compared to the Keplerian approximation.
+Macomber also introduced the concept of hot start in the case of time spans covering multiple Earth planetary orbits
+[35], where the first orbit was used to compute the difference between the Keplerian guess and the converged trajectory.
+The near-periodicity of the spherical harmonics perturbation was then exploited, including this difference in the starting
+trajectory, achieving a further reduction of the iterations required for convergence.
+B. Matrix form for vectorized and parallel computation
+The method is suitable for parallel or vector implementation (indeed Fukushi-ma also proposed a vectorized
+version [36]), in particular for the evaluation of the dynamics function and the execution of the matrix multiplications.
+More recent works over this technique by Bai and Junkins developed the modified PC method [37] and a CUDA
+implementation for NVIDIA GPUs [32]. For compactness and to better highlight the parallelization possibilities, the
+method is presented following the matrix formulation by Koblick et al [38].
+For 𝑁 Chebyshev nodes and the integration interval [𝑡0, 𝑡𝑁 −1], the independent variable 𝑡 is sampled for 𝑗 =
+0, 1, ..., 𝑁 − 1 up-front as
+𝑡 𝑗 = 𝜔2𝜏𝑗 + 𝜔1
+(3)
+with
+𝜏𝑗 = − cos
+�
+𝑗𝜋
+𝑁 − 1
+�
+,
+𝜔1 = 𝑡𝑁 −1 + 𝑡0
+2
+,
+𝜔2 = 𝑡𝑁 −1 − 𝑡0
+2
+(4)
+Given the 𝑛-dimensional sampled states y(𝑖−1) (𝑡 𝑗) = y(𝑖−1)
+𝑗
+,
+𝑗 = 0, ..., 𝑁 as a matrix y(𝑖−1) of dimension 𝑁 × 𝑛
+computed at the Picard iteration 𝑖 − 1, the whole process can be summarized in three sequential steps to obtain the states
+5
+
+at the iteration 𝑖. The first one collects the evaluations of the dynamics function f in the 𝑁 × 𝑛 force matrix F [38]:
+F(𝑖)
+𝑗+1 = 𝜔2 f�y(𝑖−1)
+𝑗
+, 𝑡 𝑗
+�,
+𝑗 = 0, ..., 𝑁 − 1
+(5)
+Secondly, identifying with A, C, S the method’s constant matrices whose definition can be found in [38], the 𝑁 × 𝑛
+matrix B is obtained by rows as [38]
+B1 = SAF + 2y0,
+B 𝑗 = AF,
+𝑗 = 2, ..., 𝑁
+(6)
+Third and last, the 𝑁 × 𝑛 matrix of the state guesses y(𝑖) for the 𝑖-th Picard iteration is
+y(𝑖) = CB
+(7)
+The iteration process stops when the maximum state difference between two consecutive Picard iterations y(𝑖) and
+y(𝑖−1) drops below a specified relative or absolute tolerance, upon user’s choice.
+Despite the proved theoretical convergence, large integration spans may lead to numerical instabilities, due to the
+cumulation of round-off errors even with large 𝑁 as multiple orbital revolutions take place [32, 34, 37]. Fukushima [34]
+suggests a piece-wise approach as a workaround, which has been implemented in this work and uses the modified PC
+method to integrate orbit by orbit in sequence∗ until the end of the span.
+The core steps of the proposed algorithm follow the presented scheme [32, 37], together with the automatic generation
+of the Keplerian initial guess spanning one nominal orbital period.
+III. Two-level system augmentation
+The just described instability problems make the PC algorithm not efficient for massive parallelism, since increasing
+the number of nodes would not result in improved accuracy after a certain point. Nevertheless, an increase in parallel
+efficiency can be found when more trajectories are propagated, in the way the following lines describe.
+A. One-level augmentation
+Instead of the evolution of the sole trajectory determined by the initial condition y0, the system being integrated
+can be re-written so that 𝑀 different trajectories sampled on the same 𝑁 time nodes can be processed within a unique
+iterative process. At the iteration 𝑖, the matrix Y(𝑖) collects all the samples of all the trajectories, its 𝑗-th row is related
+∗The proposed implementation automatically handles either forward or backward integration.
+6
+
+to the 𝑗-th time sample of the 𝑚-th trajectory by:
+Y(𝑖)
+𝑗
+=
+�
+y(𝑖)
+𝑗,1 · · · y(𝑖)
+𝑗,𝑚 · · · y(𝑖)
+𝑗,𝑀
+�
+,
+𝑗 = 1, ..., 𝑁
+(8)
+and similarly for the dynamics function evaluations collected in the matrix F(𝑖):
+F(𝑖)
+𝑗
+=
+�
+F(𝑖)
+𝑗,1 · · · F(𝑖)
+𝑗,𝑚 · · · F(𝑖)
+𝑗,𝑀
+�
+,
+𝑗 = 1, ..., 𝑁
+(9)
+whose elements are still computed per sample:
+F(𝑖)
+𝑗,𝑚 = 𝜔2 f�y(𝑖−1)
+𝑗,𝑚 , 𝑡 𝑗−1
+�,
+𝑗 = 1, ..., 𝑁
+(10)
+In principle, building the augmented system only requires to define Y(𝑖) by stacking the different 𝑀 trajectory
+matrices along the columns, and similarly F(𝑖) undergoes the exact same modification. The structure of the PC iterations
+remains unchanged and features the usual steps. First, evaluate the dynamics function for all the 𝑁 states of all the 𝑀
+trajectories with F(𝑖)
+𝑗,𝑚 = 𝜔2 f�y(𝑖−1)
+𝑗,𝑚 , 𝑡 𝑗−1
+�. Second, perform the matrix operations B1 = SAF + 2Y0 and B 𝑗 = AF, for
+𝑗 = 2, ..., 𝑁. Third and last, update the guesses for all the 𝑀 trajectories with Y(𝑖) = CB.
+B. Two-level augmentation
+The stack-along-column rule can be applied again, this time collecting in one single matrix 𝑃 groups of different
+𝑀𝑝 trajectories each. The augmented matrix Y(𝑖) is now built as
+Y(𝑖)
+𝑗
+=
+�
+Y(𝑖)
+𝑗,1 · · · Y(𝑖)
+𝑗,𝑝 · · · Y(𝑖)
+𝑗,𝑃
+�
+,
+𝑗 = 1, ..., 𝑁
+(11)
+with
+Y(𝑖)
+𝑗,𝑝 =
+�
+y(𝑖)
+𝑗,𝑝,1 · · · y(𝑖)
+𝑗,𝑝,𝑚 · · · y(𝑖)
+𝑗,𝑝,𝑀𝑝
+�
+,
+𝑗 = 1, ..., 𝑁
+(12)
+In principle, infinite augmentation levels could be built relying on the same logic, none of them would require
+modifications in the core PC algorithm structure. Nevertheless, a re-definition of the iteration error can be helpful for
+practical purposes, since the augmentation rationale is purely computational.
+Two strategies can be addressed. The first uses a traditional error definition, that treats the trajectory samples as if
+they were part of a unique system, whose maximum will be compared against the iteration stopping condition. The
+second introduces a more flexible per-block error definition, that treats the different trajectory blocks as independent, for
+which the augmentation has then a sole computational purpose. Both the approaches have advantages and drawbacks.
+The former would allow a simpler implementation and is inevitably computationally more efficient than the latter,
+7
+
+because of the reduced overhead compared to maintaining the group split. However, dissimilar trajectories requiring a
+significantly different number of iterations would keep the computational resources busy for already converged blocks,
+while the per-block definition allows far more flexibility on this regard.
+This work uses the two augmentation levels to build a hybrid approach, that treats the outer blocks as independent,
+and the inner ones as a single system in a more strict sense. In this way, groups of similar trajectories can be considered as
+unique but separate augmented system, allowing to maximize the integration performances. The two level augmentation
+also provides a framework to deal with single trajectories within the same high performance computing context,
+considering them as a group made of only one member. As a practical example, the outer augmentation level could
+be used to "isolate" a sub-group of trajectories experiencing a planetary flyby, since their dynamics would become
+significantly different from the rest of the samples.
+IV. Parallel computing, GPU computing and NVIDIA® CUDA® fundamentals
+Parallelizable programs are characterized by some of their parts that can be executed simultaneously. Conceptually,
+task-parallel and data-parallel routines may exist: the former features completely independent tasks that do not need to
+be executed one before the other, the latter is identified by a common and repeated task that should be executed on
+different data. This work focuses on this second aspect, whose features enable efficient GPU computing when massive
+parallelization is possible. For instance, taking the product of two matrices is a highly parallelizable task, as any element
+of the result matrix could be computed in parallel. Similarly, in the orbital propagation of large sets of initial conditions†
+the dynamics function could be evaluated in parallel for all the states.
+A. Shared memory parallelism with multiple CPUs: OpenMP®
+Prior to discussing the rationale of GPU computing, the most straightforward parallelization concept involves the
+use of multiple CPUs (Central Processing Units). Shared memory parallelism is the simplest scenario, where all the
+machine compute cores have access to the same and common memory locations. More complex supercomputers use
+however a distributed memory logic, where group of CPUs access their own independent memory. Such systems also
+include a communication network, to distribute and collect the computed data on the different nodes, and require their
+own programming paradigm that includes the data and message passing routines [39].
+OpenMP is a set of pre-processor instructions that enable simple shared memory parallelization of C, C++ and
+Fortran programs [40], which take action during the code compilation. Only minor modifications are required to
+accelerate the most intensive parts of the program, and OpenMP instructions are simply ignored and treated as comments
+if the OpenMP compilation flag is disabled. Common parallelization instructions involve for loops, which can include
+some extra functionality: for instance, perfectly nested loops can be collapsed into a single larger loop, increasing the
+†Assuming they are not interacting with each other and have negligible mass.
+8
+
+program efficiency, and specific instructions can aid the memory management. Despite the simple interface, most
+programs require at least some variables to be declared private to each worker, since OpenMP treats all the variables as
+shared by default. The typical case of variables that should be made private are the loop counters: it is fundamental that
+each thread works with its own variable, particularly for those cases where the loop counter also identifies the position in
+a shared array where to access some data. More complex programming scenarios often arise and require the programmer
+to explicitly control concurrent updates of shared variables, it is beyond the scopes of this work to tackle them all. The
+reader can refer to the OpenMP programming guide for more complete and detailed information [40]. This paragraph
+serves as a simple introduction to the tool that is used to parallelize the C version of the proposed program.
+The parallel dynamics function is implemented as the simple OpenMP for loop parallelization, collapsing all the
+states of the augmented system into a single loop. The multiple workers access the shared state array, then they compute
+the acceleration values and temporarily store them into thread-private variables, and finally they copy them back to
+a shared and global acceleration array. OpenMP exploits the flexibility of the CPU architecture, thus no significant
+modification are required to the innermost parts of the dynamics function to make an efficiently parallelized program.
+OpenMP is also used "indirectly" for the matrix multiplications of the PC method. The optimized OpenBLAS [41–43]
+libraries are used to implement this part of the program, they already include the OpenMP parallelization.
+B. Parallel reduction
+The parallelization potential is also exploited in the computation of the PC iteration error, through the parallel
+reduction mechanism. Reduction tasks are a broad category of compute operations, whose aim is to extract a single
+scalar value from an array of elements. Some examples are the sum of array elements, finding the maximum or minimum
+element in an array, "and" and "or" logical operators. Despite appearing intrinsically sequential tasks, parallelization
+possibilities do exist even in the reduction case: in principle, the whole array is split into several chunks on the parallel
+workers, which cooperate to perform the reduction task on their own chunk of elements. The cooperated process
+continues, until a single scalar reduced value is obtained. Figure 1 shows a graphical example of the parallel reduction
+logic: on an array of 16 elements, only 5 sequential steps are eventually required with 16 parallel workers. OpenMP
+implements the reduction clause among its functions, the programmer is only asked to define the final reduced scalar as
+a shared variable [40]. The compiler then ensures that all the array values are scanned through and avoids simultaneous
+overwriting of the reduced scalar.
+C. GPU computing
+The features of GPU computing arise from the hardware architecture of graphics cards, which is profoundly different
+from the traditional compute units. Figure 2 shows a graphical representation of such differences: in summary, more
+transistors are devoted to pure data processing on GPUs, instead of flow control and memory management as in the
+9
+
+Fig. 1
+Parallel reduction graphical scheme.
+CPU case [44]. Corresponding colors refer to chip elements of the same type, i.e. green for compute cores, yellow for
+control units, violet for the core-level cached memory, blue for shared cache, and orange for global memory.
+Fig. 2
+CPU vs GPU architecture difference graphical scheme. Picture from [44].
+The processing units are grouped in blocks (typically 32 processing units per block), each controlled by one controller.
+All processing units in the same block all execute the same instruction, issued only once by the controller. This aspect,
+together with the normally hundreds to thousands of processing units available in modern graphics cards, makes GPUs
+10
+
+Core
+Con
+Core
+Con
+trol
+trol
+L1 Cache
+L1 Cache
+Core
+Con
+Core
+Con
+trol
+trol
+L1 Cache
+L1 Cache
+L2 Cache
+L2 Cache
+L3 Cache
+L2 Cache
+DRAM
+DRAMprone to implement massive parallelism, although with lower flexibility and higher programming effort compared to
+CPU applications. Some key concepts are given in the following subsection, a comprehensive view can be found in the
+CUDA C++ programming guide [44].
+D. Main programming paradigms and the CUDA® API
+This section is intended to provide a brief overview and nomenclature of the CUDA language and API. Italic font is
+used to introduce CUDA-specific names and concepts. A complete description can be found in the CUDA user manual
+[44].
+The fundamental execution unit is called thread. Threads can be grouped in blocks, and some shared memory
+(dozens of kilobytes) is available to all threads in a common block. The execution of a single instruction is always
+performed by groups of 32 threads at the same time, called a warp, regardless the number of threads in a block.
+Therefore, blocks with less than 32 threads do not exploit the full hardware resources. Complex configurations can be
+achieved combining multiple CPUs and GPUs to run the same program. The program flow is always controlled by the
+CPU, which also controls the execution of the GPU. A function that is invoked by the CPU but executed on the GPU is
+called kernel.
+The first difference compared to standard programming regards the memory access: the GPU cannot read the usual
+compute memory, but data must be loaded on GPU memory prior to running kernels. In a similar manner, data must be
+retrieved to the CPU after the kernel has completed its execution, before executing other CPU tasks on the same data.
+Consequently, the flow of a GPU program/sub-program always follows a first initialization on the host, a subsequent
+data movement from the CPU to the GPU, the kernel execution, and finally the data retrieval from the GPU to the CPU,
+as represented in Figure 3.
+Fig. 3
+GPU program basic flow.
+11
+
+CPU
+GPU
+Compute data
+Compute data
+Kernel execution
+Result data
+Result dataBecause all threads execute the same instruction at the same time, kernels must be programmed in a warp-oriented
+manner. This also includes explicitly managing the data access by the various threads, and complex functions typically
+require the programmer to optimize memory access and cache utilization by hand: while it is implicitly controlled by the
+compiler for CPUs, graphics card store data by default in the so called global memory, large in size (some gigabytes) and
+referred to by default when GPU variables are initialized. This eases data movements between CPU and GPU, even in
+case of large arrays. However, the access latency is much higher compared to shared memory: this makes global memory
+not efficient for repeated read/write operations with GPU variables. To overcome this limitation, shared memory can be
+exploited for compute purposes to keep read-only values on lower latency locations, instead of only holding common
+values for all the threads in a block. Few cached bytes are also available on registers, thread-private locations to store up
+to 256 single precision floating point values. Their small size and the compute intensity of the relativistic dynamics
+function makes it difficult to use registers instead of the shared memory for the proposed application. Their use is
+limited to temporary and handle variables that aid the final acceleration computation.
+The way array elements are sorted is also fundamental for optimized data movements across global and shared
+GPU memory. While CPU-GPU data transfers are specified by the array size, the most efficient intra-GPU memory
+access happens when threads read/write values on adjacent locations. If this condition is satisfied, data are moved as a
+single memory transaction for all the threads, resulting in minimized cycles spent reading or writing on global memory.
+This memory access pattern is called coalesced, and represents a fundamental performance driver in complex GPU
+programs: even if a program has massive parallelization possibilities, non-optimal memory access may result in the
+memory access latency not compensated at all by the parallelized computational tasks.
+NVIDIA® developed and maintains the CUDA® programming language, which remarkably simplifies the use of
+NVIDIA® GPUs in computer programs. It is built as a C++ extension, with a set of keywords and API functions that
+allow programmers to build their own kernels and control the GPU execution flow. A set of optimized libraries is also
+available, for instance the basic linear algebra cuBLAS® implemented for the PC matrix multiplications of this work
+[44].
+E. Concurrency and advanced features
+The CPU-GPU duality and cooperation exposes more possibilities, other than the simple acceleration of intensive
+parts of the program. In general, kernel calls are asynchronous and do not block the work of the CPU, which allows the
+CPU to execute other tasks while a GPU kernel is still running. Furthermore, modern GPUs can manage at the same
+time two memory transfers (one per direction, CPU to GPU and GPU to CPU) while saturating its compute units for one
+or more concurrent kernel execution [44].
+CUDA® allows the queuing of a series of sequentially dependent GPU function calls with the use of streams: for
+instance, an application may require some data to be transferred to the GPU before the execution of a custom kernel,
+12
+
+which must necessarily be completed before calling a cuBLAS® function, at the end of which the processed data should
+be transferred back to the CPU. All it takes is assigning the sequentially dependent GPU function calls to the same
+stream. Multiple streams can be created and used at the same time, the obtained behaviour mimics batch job submissions
+to supercomputing facilities. CUDA® guarantees the correct execution line and synchronization within the same stream,
+whereas different streams must instead be synchronized with each other by hand. The compiler typically schedules
+executions and memory transactions so that the GPU use is maximized, superposing different GPU function calls and
+data transfers from separate streams [44].
+Despite the lower latency, optimized programs are designed to also control the way threads access shared memory
+locations. If two adjacent threads are asked to access two non-contiguous array elements, then the memory access is
+performed on two cycles instead of a single one, slightly slowing down the program execution. This issue is called bank
+conflict, and can be avoided ensuring thread-varying elements to be stored in the leading dimension of shared memory
+arrays. Bank conflicts do not happen if all threads access the same single memory location.
+V. Implementation
+Among the implementations outlined in this section, the case of independent PC runs for all the propagated
+trajectories is considered as benchmark. This allows to directly assess the performance of the augmented PC algorithm
+against the original integrator for the same test case. All the algorithms were implemented using the C language, with
+the sole exception of the GPU program that was coded in CUDA.
+A. Independent runs PC workflow
+The basic workflow of the independent PC runs is given in the block-scheme of Figure 4. The only parallelization
+possibilities, for high numbers of trajectories, apply at the highest level, inevitably introducing a considerable overhead
+for both the inner sequential execution and the still parallelizable inner functions. In fact, all the per-trajectory steps of
+the PC process would still be parallelizable algorithms per se.
+B. Sequential Augmented PC workflow
+The implementation of the augmented PC integration follows a one-level augmentation only, to highlight the pipeline
+benefits in terms of overhead that this implementation introduces. A block-scheme representation of the augmented PC
+integration workflow is given in Figure 5. The conceptual change, from the PC iteration viewpoint, is only the initial
+sampling in a single array containing all the state vectors of all the trajectories of the augmented system.
+C. OpenMP® parallelized Augmented PC workflow
+The parallelization of the augmented system integration, whose block-scheme representation is given in Figure
+6, becomes fine-grained. It acts directly on the single state vectors for the dynamics function evaluation and on the
+13
+
+Fig. 4
+Standard PC workflow.
+Fig. 5
+Augmented PC workflow.
+elementary products of the matrix operations, moreover already implemented by the OpenBLAS libraries [41]. In
+addition, reduction operations can be made through OpenMP® for a cooperated and parallel search of the maximum
+error.
+D. CUDA® Augmented PC workflow
+The block-scheme representation of the CUDA® algorithm is given in Figure 7. The two-level augmentation concept
+is exploited, assigning one higher level augmented system to each CUDA® stream and using the thread-based parallelism
+on the lower level augmented systems.
+The principal benefit is the cooperation between CPU and GPU for the overall execution, with as many operations as
+possible executed concurrently. Each lower level augmented system is initially sampled by the host and then moved to
+the GPU. The CUDA® stream management API allows to overlap the CPU sampling of the next higher level augmented
+14
+
+Do while error higher than
+tolerance
+Core iteration:
+Sample
+Evaluate the
+Sequence of
+Iteration error
+Augmented
+dynamics
+matrix
+computation
+initial guess
+function
+operations
+OpenMp? parallel
+OpenMP parallel,
+OpenMP? parallel
+OpenMP? parallel
+OpenBLAS optimized
+reduction
+librariesOpenMP? parallel
+Do for all states in
+Do while error higher than
+the system
+tolerance
+Core iteration:
+Evaluate the
+Sample initial
+Sequence of
+Iteration error
+dynamics
+matrix
+computation
+guess
+function
+operations
+OpenBLAS optimized
+librariesFig. 6
+Augmented and OpenMP® parallelized PC workflow.
+Fig. 7
+Augmented CUDA® PC workflow.
+systems with memory transfers and kernel executions of the already launched ones‡. Similarly, the last step of the PC
+iteration requires to retrieve the computed iteration error for each stream from the GPU to the CPU for loop control
+purposes, which is also subject to the stream concurrency benefits. A stream synchronization at the end of each while
+loop iteration is necessary to achieve the overlapping behaviour of all the streams. Running independent loops for each
+higher level augmented system would result in completely sequential and non-overlapped executions. If a single CUDA®
+stream is generated, the standard one-level augmented system case is reproduced, albeit with the GPU computing
+acceleration instead of the OpenMP® implementation previously described.
+The warp-centric programming model of CUDA® kernels requires a small modification on the lower level augmented
+system definition. Contiguous array elements should be of the same component type (i.e. contiguous 𝑥 coordinates,
+then contiguous 𝑦 coordinates, and so on), instead of storing state vector by state vector. This aspect might seem
+‡This is true as long as the CPU memory is allocated as paged with specific CUDA® functions [44].
+15
+
+Do while error higher than
+tolerance
+Core iteration:
+Sample
+Evaluate the
+Iteration error
+Sequence of
+Augmented
+dynamics
+matrix
+computation
+initial guess
+function
+operations
+OpenBLAS optimized
+librariesDo while error higher than
+p-th stream on p-th augmented
+tolerance
+Stream synchronization
+sub-system
+before next iteration
+Core iteration:
+Sample
+Evaluate the
+Sequence of
+Iteration error
+dynamics
+lower level augmented
+matrix
+computation
+function
+initial guess
+operations
+cuBLAS @ library
+Kernel
+Done by the host.
+Kernel with
+can be OpenMP? parallel.
+on p-th stream
+functions
+reduction + host
+initialises p-th stream
+on p-th stream
+on p-th streaman implementation detail, however it is fundamental to ensure coalesced global memory access. A too high latency
+would happen otherwise, which cannot be hidden even by intensive parallelized GPU computations. The just discussed
+modification has no effect on the overall algorithm structure, all it requires is the dynamics and error kernels to be
+implemented following this array element logic. This aspect is discussed in more detail in the following section, together
+with the implications it has especially on the evaluation of the dynamics function.
+1. Dynamics model, array sorting, and CUDA kernel
+At the core of the PC integration scheme lies the evaluation of the dynamics function at each Picard iteration. This
+task can be performed in parallel for all the states of the system being integrated, however, although conceptually
+simple, its implementation may not be straightforward in the GPU computing case. The more complex the dynamical
+model becomes, the more intertwined its implementation inevitably gets, possibly requiring to access data distributed in
+multiple arrays, possibly of notably different sizes. The accuracy requirements of the proposed application demand to
+work under the restricted relativistic N-body problem, following the Einstein-Infeld-Hoffmann equations [45], which
+has a dynamics function of the form:
+�r = f�r, �r, r𝑖, �r𝑖, �r𝑖
+�
+(13)
+with 𝑖 condensing the dependence on the states of all the major bodies in the ephemeris model, e.g. the solar system
+planets, and r, �r, �r denoting position, velocity, and acceleration, respectively. The relations are unfortunately non-linear:
+as a consequence, each CUDA® thread cannot perform simple operations on one single array element, since both
+position and velocity of each state are required to compute any acceleration component. Moreover, ephemerides data
+for r𝑖, �r𝑖, �r𝑖 also enter the dynamics function. These aspects suggest to implement the dynamics kernel having each
+CUDA® thread to process one full state vector, rather than one element. At the same time, coalescing the global memory
+access remains crucial to obtain a well-performing kernel. The PC method introduces however a partial constraint on
+the array shapes: the matrix multiplications that build the method need the sampled trajectory states to be stored as rows
+of an overall matrix, fixing the different times to identify each row. For a column-major sorted augmented state matrix,
+contiguous state elements are interrupted by the ending time nodes, likely leading to non-coalesced memory access for
+numerous warps. Row-major sorted state arrays feature instead non-coalesced access for all the state elements.
+To cope with these issues, the lower-level augmented state is reformulated by stacking along the columns the same
+components of all the state vectors in the augmented system:
+Y(𝑖)
+𝑗
+=
+�
+𝑥(𝑖)
+𝑗,1 · · · 𝑥(𝑖)
+𝑗,𝑀
+· · · 𝑝(𝑖)
+𝑗,1 · · · 𝑝(𝑖)
+𝑗,𝑀
+· · · �𝑧(𝑖)
+𝑗,1 · · · �𝑧(𝑖)
+𝑗,𝑀
+�
+,
+𝑗 = 1, ..., 𝑁, 𝑝 = 𝑦, 𝑧, �𝑥, �𝑦
+(14)
+where (𝑥, 𝑦, 𝑧) are the Cartesian components of r. The advantage is obvious in the column-major sorting case, since
+16
+
+all the common components are found in adjacent memory addresses. Since in the presented application the number
+of states in the augmented system is much larger than the number of sampled trajectory nodes, many contiguous
+state components also appear in the row-major sorted array case. In addition, the higher-level augmented system
+definition may remain unaltered, since different CUDA® streams would be processing each lower-level sub-systems.
+The augmented force matrix F(𝑖)
+𝑗
+can be adapted accordingly, without introducing any modification to the matrix
+multiplication characterizing the PC iteration. Lastly, bank conflicts (explained in Section IV.E) are automatically
+avoided [44] with this array sorting approach.
+The kernel design is tied with the array sorting strategy. In particular, a key role is played by the fixed time nodes.
+The augmented system logic is a consequence of the shared time nodes among the different state vectors, this feature
+should also be exploited to design the thread blocks to make the most of the available shared memory. In particular, the
+amount of memory required to store the planetary ephemerides is minimized if all the threads in a block process state
+vectors corresponding to the same time node. For this reason, the proposed program implements a row-major sorting
+strategy of state and force matrices. The augmented state matrix is accessed as a two-dimensional block array: one
+dimension (the rows) follows the different time nodes, whereas the other is used to split the states of a common time
+node into smaller chunks, each containing 32 states§. In this way, all the states in the same block of the two-dimensional
+block array require exactly the same ephemerides data, because they are all associated to the same time node. For
+fixed-time thread blocks bank conflict is automatically avoided also when reading ephemerides data from the shared
+memory, since all the threads are forced to access the same ephemerides item or vector component [44].
+The computation of the dynamics function is a compute-bound task: most of the effort lies on performing
+computations on a limited amount of data, rather than on the movement of a large amount of information between two
+memory locations. Furthermore, the values of the state elements need to be repeatedly accessed. For these reasons
+and because of the limited capacities of thread-private registers, shared memory is used to also temporarily store the
+state values, because of its lower latency compared to the global GPU memory [44]. The resulting dynamics kernel is
+summarized in Figure 8.
+In addition, a close look at the Einstein-Infeld-Hoffmann equations reveals that the acceleration of each propagated
+body depends not only on its state and the gravitational parameter and states of major bodies in the ephemeris model,
+but also explicitly on the gravitational potential and the acceleration of such bodies [45]. These contributions can be
+computed before the evaluation of the dynamics function itself in the restricted problem of the proposed application,
+saving the computational burden of a task that would be repeated at each Picard iteration. The values of gravitational
+potential and the acceleration of the major bodies of the ephemeris model are computed by the CPU before starting the
+Picard iterations, then moved to the global GPU memory, and eventually loaded to the shared memory for the time node
+§32 is the warp size for most NVIDIA® graphics cards [44]. For augmented systems with a number of states that is not a multiple of the block
+size, the dynamics kernel is implemented so that the last block of threads processes the remainder of the integer division between the number of states
+and the block size.
+17
+
+Fig. 8
+Dynamics kernel memory management.
+associated with each block, along with the ephemeris states and gravitational parameters.
+2. CPU-GPU cooperated iteration error computation
+All the involved CUDA® kernels are run at each Picard iteration. Their execution must be called by the CPU, which
+also stops the Picard iteration while loop as the error between two consecutive steps falls below the desired tolerance.
+Since the updated states already reside on the GPU, the GPU massive parallelism can be exploited to accelerate the
+computation of the iteration error, transferring only a limited amount of data to the CPU to be used for the loop control.
+Despite dealing with the augmented state system, the error is still defined on a per-state basis, with the maximum of the
+errors of all the states that is used to control the loop.
+The error computation process involves two separate kernels and a CPU function. The first kernel computes both
+the position and the velocity errors and stores the maximum between these two, for all the states in the augmented
+system. The second kernel computes the maximum error of groups of 4096 states: 1024 thread-sized blocks are created,
+discerning the first maximums while reading four consecutive chunks of state errors into the shared memory. Then,
+1024 threads cooperate to find the actual maximum error among the remaining 1024 state errors, with reduction-driven
+parallelism¶. Eventually, the maximum error is copied back to the global memory, in a new array consisting of reduced
+errors only. Finally, this whole array is copied back to the CPU, which finds the actual maximum with a traditional
+sequential for loop-based approach. Even with augmented states made of millions of state vectors, this approach makes
+the CPU search sequentially only over hundreds to thousands candidates at most, with a negligible computational cost
+compared to the other steps.
+¶Like the dynamics kernel, the error kernels are implemented so that the last thread block processes the remainder between the integer division
+between the number of states and the block size (4096 in the case of the reduction kernel). The reduction steps are controlled accordingly: only those
+corresponding to a number of threads less than or equal to the number of states in the block are activated.
+18
+
+Global Memory
+Shared Memory
+Shared ephemerides
+Shared ephemerides
+States of the block
+States of the block
+Acceleration computation
+Accelerations of the block
+Accelerations of the blockVI. Program performance
+A. Test case
+The test case follows what already computed in [12, 13]. The first resonant phase with Venus of a Solar Orbiter-like
+mission was reproduced, designing a continuous trajectory even during flyby injections and exits. The approach is
+based on the b-plane [46] description of flybys, and uses the PC method for efficient fixed-point simulations to surf the
+relativistic N-body environment‖ minimizing the deep space maneuver effort required to control the flyby entrance.
+Following the trajectory scheme of Figure 9, the design algorithm follows a backward, dynamic programming-like
+recursion logic, i.e. designing flyby 𝑗 + 1 before flyby 𝑗, to preserve accuracy and continuity while also fulfilling
+mission requirements. The reader can find more insight on the design algorithm and technique in [12, 13].
+Fig. 9
+Orbital scheme of the phase from flyby 𝑗 to flyby 𝑗 + 1.
+The Matlab®∗∗ implementation proposed in [12, 13] was re-run on a single core of a local workstation equipped
+with an Intel® CoreTM i7-7700 CPU (3.60 GHz), running Windows® 10 Pro. The algorithm converged running 13509
+PC integrations in 506.3 seconds, to the residual Δr∗ and impulsive action Δv∗ for the required maneuver presented
+in Table 1. To better detail the embedded PC integrations, they all feature 200 Chebyshev nodes spanning the arc
+Table 1
+Optimization results, in terms of position difference residual Δr∗ and correction effort Δv∗ at the
+maneuvering time corresponding to the apocenter of the post-flyby orbit.
+Δr∗
+Δv∗
+[m]
+[m/s]
+𝑥
+𝑦
+𝑧
+𝑥
+𝑦
+𝑧
+−0.52
+−0.52
+−1.19
+−1.28
+1.57
+0.22
+connecting the flyby exit and the maneuver point, long about 0.87 orbital periods. Figures 10a and 10b show the final
+trajectory, optimized in the relativistic N-body environment.
+‖The enforced trajectory continuity even during flybys requires extreme design precision, because the fast dynamics of close approaches would
+lead to divergent and therefore meaningless trajectories, already after the first flyby.
+∗∗Following the update to Matlab® R2021b higher runtimes were obtained for the same final results, although not affecting the presented discussion
+since the performance analysis is only made on the C and the CUDA algorithm versions.
+19
+
+Maneuver
+Pre-maneuver
+Post-maneuver
+Flyby j+1
+Flyby j
+Pre-flyby j
+Post-flyby j+l(a) Deep space correction maneuver and overview.
+(b) Zoom over Venus’ flybys V2 and V3.
+Fig. 10
+Solar Orbiter’s continuous first resonant phase with Venus. Pictures from [12, 13].
+B. Computational setup
+To build a common framework for the pure algorithm performance evaluations, the 13509 initial conditions generated
+in the optimization process to eventually obtain the results of Table 1 are re-run using C and CUDA implementations
+of the PC integration. The execution of the 13509 independent runs with the C implementation of the algorithm is
+considered as benchmark case, both completely sequential and parallelized with OpenMP [40]. The matrix operations
+featured in the PC iterations are performed using the OpenBLAS library [41–43]. All the presented runs of the C
+algorithm have been executed on a machine running Ubuntu Linux 20.04 equipped with 40 physical / 80 logical cores of
+the type Intel® Xeon™ CPU E5-4620 V4 running at 2.1 GHz, with varying number of OpenMP threads and the "o3"
+gcc compiler optimization enabled. Because of the physical machine where the GPU was available, the CUDA™ code
+has been run on the same workstation of the Matlab® optimal solution computation. In this case a four core OpenMP®
+parallelization on a Intel® CoreTM i7-7700 CPU (3.60 GHz), is combined for concurrent executions with a NVIDIA®
+GTX 1050 (1.3GHz) graphics card††. As a final remark, in all the presented cases the relative error among the different
+implementations for corresponding initial conditions always falls below the specified PC relative tolerance (10−12),
+allowing runtime-only performance comparisons. These small differences are due to the inevitably slightly different
+order of the elementary operations performed by the different machines, libraries and implementations.
+C. Accuracy comparison
+Figure 11 shows the evolution of the propagation error, measured as relative position and velocity error with respect
+to the independent runs case, for both the C augmented and the CUDA® programs. In both cases the error remains
+††This is a 2016 low-end gaming card model, whose design purpose is far from the double precision computing of this work. Modern
+gaming/professional cards could run up to 60 times faster, data center cards up to 400-500 times faster than this model for the presented application.
+20
+
+Overview
+×108
+-Pre V2
+Post V3
+-V2-V3
+0.5
+-Maneuver
+Venus'
+[km]
+Sun
+0
+-0.5
+-1
+-1.5
+-1
+-0.5
+0
+0.5
+1
+x [km]
+X108Flybys
+-8.5,×107
+..Pre V2
+..Post V3
+-8.52
+.. 2-V3
+-Flyby V2
+-Flyby V3
+-8.54
+. Venus at V2
+Y-8.56
+-8.58
+-8.6
+-6.72
+-6.7
+-6.68-6.66-6.64
+-6.62
+-6.6
+x [km]
+×107lower than the specified relative tolerance used to halt the Picard-Chebyshev iterations, set to 10−12.
+Fig. 11
+Errors of C augmented and CUDA® programs, as the average of the error of all the states in the
+augmented systems.
+Figure 12 shows instead the evolution of the error for the dynamics function only, distinguishing two different
+compilations of the CUDA® programs, with (red dashed) and without (orange) the –use-fast-math flag enabled.
+Despite the error experienced in Figure 11, the C augmented program shows no error for the computation of the
+dynamics function. That means, the error is accumulated throughout the Picard-Chebyshev iterations only because
+of the different structure of the matrix multiplications. Their implementation in the OpenBLAS library [41–43] is
+optimized on the matrix size. This is a known issue in high-performance computing problems, where the floating point
+representation of numbers interferes with a change in the order of the basic math operations on the matrix elements,
+producing small deviations with respect to a reference non-parallelized solutions. No differences were observed between
+parallelizing and not parallelizing the augmented C programs. On top of this consideration, the CUDA® program is also
+subject to the errors introduced by the different compilers. Other than being two different environments (Ubuntu Linux
+with GNU® compilers for the C augmented program, Windows 10 Pro with Microsoft® Visual Studio® and Nvidia®
+CUDA® compilers for the CUDA ® program), small differences can also be observed by setting different optimization
+flags in the compilation. The comparison between enabling and disabling the fast math optimization options are run
+21
+
+-11
+10
+Relative position error
+12
+10
+10
+-13
+14
+10
+10
+-15
+C Augmented
+16
+10
+CUDA
+-17
+10
+0
+25
+50
+75
+100
+125
+150
+175
+200
+10
+-11
+Relative velocity error
+-12
+10
+0
+13
+-14
+10
+-15
+C Augmented
+10
+-16
+CUDA
+-17
+10
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Picard-Chebsyhev nodesto exclude the possibility of implementation problems for the dynamics kernel. As it can be observed comparing the
+two compilations, the error spikes reaching 10−12 happen in a seemingly unpredictable manner and can be completely
+attributed to the compiler, because happening at different Picard-Chebyshev nodes for the two cases. Otherwise, the
+error level remains more than two orders of magnitude lower. In any case, the accumulated effect of this error source is
+not taking the overall error above the 10−12 iteration tolerance, as Figure 11 already highlighted.
+Fig. 12
+Errors of C augmented and CUDA® programs, as the average of the error of all the states in the
+augmented systems.
+D. Performance comparison
+Table 2 shows the runtime difference between the sequential C programs. The improved efficiency of the augmented
+integration can be immediately seen even in this sequential case, where the augmented system approach runs 23.87%
+faster, because of the minimized overhead experienced by sharing the outer while loop. In the considered application
+the various trajectories all require 41 or 42 PC iterations, making it negligible to keep running trajectories even if their
+PC process has already converged.
+The scalability properties of the integration of independent trajectories and the augmented system are studied on the
+C implementations, i.e. assessing how well the execution of the two programs accelerates with increasing number of
+22
+
+-11
+10
+Relative velocity error
+12
+10
+13
+10
+14
+10
+C Augmented
+10
+-15
+CUDA
+16
+10
+CUDA Fast Math
+-17
+10
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Relative acceleration error
+-11
+10
+-12
+10
+13
+10
+-14
+10
+C Augmented
+-15
+10
+CUDA
+-16
+10
+CUDA Fast Math
+-17
+10
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Picard-Chebsyhev nodesTable 2
+Sequential runtimes for the independent runs and the augmented system executions.
+Case
+Runtime
+[s]
+Independent runs
+245.02
+Augmented system
+186.54
+OpenMP® threads. Figure 13 shows that the augmented system features excellent scalability properties, for a runtime
+that keeps decreasing for increasing number of OpenMP® threads. On the contrary, the integration of independent
+trajectories experiences even higher runtimes after a certain number of threads. This happens because the parallelization
+itself introduces some overhead to the overall program execution, which cannot be compensated for by newly created
+parallel threads.
+Fig. 13
+Augmented system and independent integrations C code runtime comparison with OpenMP® paral-
+lelization and varying number of threads.
+Figure 14 shows the achieved speedup, defined as the ratio between the sequential runtime and the parallel runtime,
+varying the number of OpenMP® threads. It provides a measure of how parallelizable the algorithm is, with higher
+values underlining higher accelerations. The augmented system integration shows once again excellent scalability
+properties even at high number of threads, suggesting the efficiency of the GPU computing transition even without
+23
+
+Independent
+250
+Augmented
+200
+Runtime [s]
+150
+100
+50
+0
+10
+20
+30
+40
+50
+60
+70
+80
+Number of threadshaving assessed the performances of the CUDA® implementation yet. If the number of threads is set equal to one,
+Fig. 14
+Augmented system and independent integrations C code speedup comparison with OpenMP® paral-
+lelization and varying number of threads.
+the algorithm conceptually reduces to a sequential case. However, enabling compiler optimization and the OpenMP®
+flag in the compilation, a parallel program introduces data distribution and retrieval tasks across the possibly multiple
+workers. With OpenMP®, the number of threads is specified by an environment variable right before running the
+program. Therefore, activating one thread only exposes all the parallelization-induced overhead without having any
+computational benefit at all. For this reason the one-thread runtimes for the independent and the augmented systems
+both result higher than the benchmark, sequential runtimes, which were compiled with optimization enabled but without
+active OpenMP® flag.
+Finally, the execution of the CUDA® implementation‡‡ resulted the faster overall, taking 15.84 seconds. The whole
+trajectory set was split into 10 stre-ams§§. The stream definition guideline should in practice fit the application the
+propagator would run on, being the sole flexibility degree left by the implementation. However, from the GPU viewpoint,
+larger kernels always imply a better GPU exploitation, thus creating too many outer augmentation levels, i.e. activating a
+‡‡Only the results obtained for the compilation without the –use-fast-math options are shown. No significant runtime difference (less than 0.1
+seconds was observed).
+§§By the result of the integer division of the 13509 states by 9, with the tenth stream containing a number of states equal to the remainder of that
+division.
+24
+
+Independent
+10
+Augmented
+8
+6
+4
+2
+0
+10
+20
+30
+40
+50
+60
+70
+80
+Number of threadslarge number of independent CUDA® streams, with too few trajectories each would result in a performance degradation,
+eventually obtaining what already observed with the independent integration cases for the single trajectories.
+Table 3 summarizes the runtime results discussed in the previous lines for the different cases, for selected number of
+cores in the C implementation cases.
+Table 3
+Runtimes for the independent runs and the augmented system executions. Average of 10 different runs
+each.
+Case
+Threads
+GPU
+Runtime [s]
+C Independent
+1
+-
+245.02
+C Augmented
+1
+-
+186.54
+C Independent
+8
+-
+113.09
+C Augmented
+8
+-
+40.60
+C Independent
+40
+-
+113.54
+C Augmented
+40
+-
+22.67
+C Independent
+80
+-
+178.18
+C Augmented
+80
+-
+18.70
+CUDA® Augmented
+4
+GTX 1050
+15.84
+Figure 15 shows the increasing speedup achieved by the augmented system when compared to the sequential and
+independent simulations of all the samples. The CUDA® implementation runs more than 15 times faster compared to
+the baseline case, suggesting the suitability of the augmented PC algorithm to high performance and GPU computing
+facilities.
+E. CUDA® Kernel profiling and optimization
+The final kernel implementations are the result of a long and detailed profiling and optimization process, performed
+with the Nsight® Compute tool and a NVIDIA® GeForce® GTX 1050 graphics card. The choice of the 32 units block
+size of the dynamics kernel was driven mostly by the need of using the shared memory also for the temporary storage of
+the double precision state and the acceleration vector elements, just too large to be kept on the thread-private registers.
+The error computation kernels do not feature this bottleneck, thus the full amount of 1024 threads per block can be
+activated, minimizing the number of memory transactions and maximizing the reduction effects. The further four times
+of reduction while reading the error from ephemerides data (better detailed in Section V.D.2) are the result of multiple
+trials: adding more reduction layers resulted in a kernel slowdown not compensated by the consequent speedup in the
+CPU function final call. Analogously, removing some of them resulted in a CPU function slowdown not compensated
+by the kernel speedup. The chosen number of while-reading reductions may however be affected by the problem size of
+the selected test case. Larger (or smaller) sets of initial conditions to be propagated may feature a different optimal
+implementation of the maximum error computation. On the contrary, 32 is already the minimum block size for the
+dynamics kernel and is not related to the problem size. Other non-relativistic dynamics function may drive the use of
+25
+
+Fig. 15
+Speedup comparison among C and CUDA® implementations.
+the shared memory in a different way, possibly allowing the use of larger block sizes.
+The kernel performance is condensed into three numbers that result from the profiling process, i.e. the average
+multiprocessor occupancy for compute operations (Compute in Tables 4 and 5), the shared memory utilization (Shared
+Memory in Tables 4 and 5), and the memory throughput¶¶ (Throughput in Tables 4 and 5). Despite profiling tools
+provide more detailed information, the presented indicators already allow the description of the kernel performances in
+sufficient detail. The profiling indicators of the cuBLAS® kernels for matrix-matrix product (dgemm), matrix-vector
+product (dgemv), and element-wise summation (daxpy)∗∗∗ used for the matrix multiplications embedded in the PC
+method are also shown, providing a performance comparison against well-known and heavily optimized library functions.
+In addition, also the kernel runtime as measured in the profiling activity is added to the comparison, to highlight the
+difference for the two tested augmented system sizes.
+Table 4 shows the kernel profiling results for an augmented system made of all the 13509 states to be propagated.
+The compute-bound kernels can be easily recognized as the ones exploiting the most the GPU’s compute capability
+(Dynamics (dynamics), dgemm, error computation (errCompute), and maximum error reduction (maxReduce)). The
+matrix-matrix multiplication is more memory bound than the dynamics function, because larger arrays must be loaded
+¶¶Non-compute intensive but still highly parallelizable tasks may have the memory transfer as final bottleneck, this indicator measures how much
+of the CPU-GPU communication band width is used.
+∗∗∗For the sake of conciseness the kernels are identified with the names of the cuBLAS® API functions, although optimized kernels are called for
+each specific GPU architecture and problem size.
+26
+
+18
+15.57x
+16
+14
+13.10x
+12
+10.81x
+10
+8.30x
+8
+6.03x
+6
+4
+2
+1.31x
+1.00x
+0
+GPU
+sequential
+1 thread
+8 threads
+16 threads
+40 threads
+80 threads Augmentedat the same time on the shared memory to perform the computation. On the contrary, the dynamics kernel features a
+more relevant computational bottleneck, because of the much more complex algorithm compared to the simple products
+and summations of the dgemm case. The remaining kernels, as well as the lower-intensity but still highly parallelizable
+error computation, all feature a high memory throughput. The GTX 1050 card used as a maximum memory band width
+of 112.1 Gb/s: the closer the throughput to that value the better the memory transactions are managed, essential feature
+for memory-bound problems. If some computations are added in the kernel, some throughput is inevitably lost, as
+latency sources are introduced between transaction to/from the shared/global memory.
+Table 4
+13509-sized augmented state kernel profiling results.
+Kernel
+Compute
+Shared
+Throughput
+Runtime
+[%]
+Memory [%]
+[Gb/s]
+[𝜇s]
+dynamics
+97.11
+24.96
+1.91
+137470
+dgemm
+94.15
+26.67
+4.06
+164210
+dgemv
+47.14
+84.54
+93.24
+1400
+daxpy
+18.18
+74.78
+60.26
+32.48
+errCompute
+99.41
+53.26
+59.36
+4740
+maxReduce
+36.80
+71.36
+79.25
+279.36
+Table 5 shows the kernel profiling results for an augmented system made of 1501 states to be propagated†††.
+The more compute-bound kernels do not show a significant loss of performance compared to the 13509-sized single
+augmented state case of Table 4, whereas the other kernels do. This highlights the suitability of GPU computing to
+extremely intense and parallelizable tasks, where the kernel call overhead is heavily compensated for by the massive
+task parallelization. Nevertheless, the capability of successfully processing also smaller-sized problems remains crucial
+for the program flexibility, particularly for what concerns the adoption of the proposed two-level augmentation scheme
+comprising smaller sub-systems significantly different from each other.
+Table 5
+1501-sized augmented state kernel profiling results.
+Kernel
+Compute
+Shared
+Throughput
+Runtime
+[%]
+Memory [%]
+[Gb/s]
+[𝜇s]
+dynamics
+96.86
+24.92
+1.91
+15300
+dgemm
+93.67
+26.65
+4.08
+18390
+dgemv
+42.59
+78.74
+87.53
+170.85
+daxpy
+9.19
+33.95
+36.31
+6.62
+errCompute
+98.25
+53.44
+59.78
+529.34
+maxReduce
+26.29
+63.35
+68.20
+42.88
+Table 6 compares instead the runtimes of the CUDA® program, for the cases of 1 and 10 active streams. Theoretically,
+†††This number results from splitting the full augmented system of 13509 states into 10 CUDA® streams, each but the last with a number of states
+equal to the integer division between 13509 and 9. The tenth and last streams contains a number of states equal to the remainder of the previous
+integer division.
+27
+
+the former has the advantage of a better GPU resource exploitation, whereas the latter makes a more aggressive use of
+the CPU-GPU concurrency. Despite the lower GPU efficiency, the achieved runtimes are almost identical. Therefore,
+the two-level augmentation scheme can efficiently tackle the case of differently-sized lower-level augmented subsystems,
+showing the flexibility of the proposed computational scheme, despite the fine grain code optimization necessary for its
+successful implementation.
+Table 6
+1 and 10 streams CUDA® program executions. Average of 10 different runs each.
+Case
+Runtime
+[s]
+1 stream
+15.78
+10 streams
+15.84
+VII. Conclusion
+This work explores the benefits that high-performance CPU clusters and GPU computing architectures bring to
+the short leg orbital propagation of large sets of initial conditions. The tested case studies the runs required to design
+a Solar Orbiter-like resonant phase with Venus, optimized to surf the relativistic N-body environment, proposing a
+two-level augmentation strategy implemented in the C and the CUDA® programming languages.
+Propagating the augmented system always outperforms propagating the trajectories independently, both in the
+sequential and the parallelized case. The augmentation benefits appear in two different aspects, the first being the
+reduced overhead compared to the repeated independent runs, the second represented by a finer grain parallelization
+also exploiting optimized linear algebra libraries.
+The approach scalability allows its implementation on GPU architectures, with low end and old graphics card
+already capable of matching the performance of a common-size cluster node. Data center card models can make the
+algorithm run around 400-500 times faster, while the newest gaming GPUs should already allow a 60 times program
+acceleration. In addition, implementing the second order version of the modified PC method with error feedback should
+introduce a further two/three-fold acceleration.
+Despite being shown on a test case derived by a trajectory optimization application, the proposed scheme represents
+a completely general orbital propagator. Any application requiring the propagation of large sets of initial conditions
+could benefit by the high computational efficiency provided by this algorithm, not necessarily requiring the use of
+supercomputing facilities in favour of much less expensive graphics cards.
+Future works may keep developing the software tool, managing longer integration spans in sequence, and, as a
+general propagator, implementing flyby detection procedures. At the same time, the optimization scheme that led to the
+presented trajectory design application can be modified to accommodate massively parallel search approaches. On the
+implementation side, the benefits of using CUDA® graphs instead of the stream-based management of the outer level
+28
+
+augmented systems may also be explored.
+Acknowledgments
+The research leading to these results has received funding from the European Research Council (ERC) under the
+European Union’s Horizon2020 research and innovation programme as part of project COMPASS (Grant agreement
+No 679086), www.compass.polimi.it, and the European Space Agency (ESA) through the Open Space Innovation
+Platform (OSIP) co-funded research project "Robust trajectory design accounting for generic evolving uncertainties",
+Contract No. 4000135476/21/NL/GLC/my.
+References
+[1] European Space Agency (ESA), “Solar Orbiter Definition Study Report (Red Book),” Tech. Rep. July, 2011. URL https:
+//sci.esa.int/s/w7yO4P8.
+[2] European Space Agency (ESA), “Jupiter ICy moons Explorer Exploring the emergence of habitable worlds around gas giants.
+Definition Study Report.” Tech. Rep. 1.0, 2014. URL https://sci.esa.int/web/juice/-/54994-juice-definition-
+study-report.
+[3] COSPAR - Committee on Space Research, “COSPAR Policy on Planetary Protection,” Space Research Today, Vol. 208, 2020,
+pp. 10–22. https://doi.org/https://doi.org/10.1016/j.srt.2020.07.009, URL https://www.sciencedirect.com/science/article/pii/
+S1752929820300372.
+[4] Kminek, G., “ESA planetary protection requirements, Technical Report ESSB-ST-U-001,” Tech. rep., European Space Agency,
+2012.
+[5] Jehn, R., “Estimating the impact probability of ariane upper stages,” Tech. rep., MAS Working paper 601, European Space
+Agency, Dec 2014.
+[6] Wallace, M., “A Massively Parallel Bayesian Approach to Planetary Protection Trajectory Analysis and Design,” Proceedings
+of the 2015 AAS/AIAA Astrodynamics Specialist conference, Vol. AAS 15-535, Vail, CO, USA, 2015. URL https://trs.jpl.nasa.
+gov/handle/2014/45859.
+[7] Colombo, C., Letizia, F., and Van Der Eynde, J., “SNAPPshot ESA planetary protection compliance verification software Final
+report V1.0, Technical Report ESA-IPL-POM-MB-LE-2015-315,” Tech. rep., University of Southampton, 2016.
+[8] Romano, M., Losacco, M., Colombo, C., and Di Lizia, P., “Impact probability computation of near-Earth objects using Monte
+Carlo line sampling and subset simulation,” Celestial Mechanics and Dynamical Astronomy, Vol. 132, No. 8, 2020, p. 42.
+https://doi.org/10.1007/s10569-020-09981-5, URL https://doi.org/10.1007/s10569-020-09981-5.
+[9] Romano, M., “Orbit propagation and uncertainty modelling for planetary protection compliance verification,” Ph.D. thesis,
+29
+
+Politecnico di Milano, Supervisors: Colombo, Camilla and Sánchez Pérez, José Manuel, Feb 2020. https://doi.org/10.13140/
+RG.2.2.19692.80001.
+[10] Masat, A., Romano, M., and Colombo, C., “Kustaanheimo–Stiefel Variables for Planetary Protection Compliance Analysis,”
+Journal of Guidance, Control, and Dynamics, Vol. 45, No. 7, 2022, pp. 1286–1298. https://doi.org/10.2514/1.G006255, URL
+https://doi.org/10.2514/1.G006255.
+[11] Stiefel, E. L., and Scheifele, G., Linear and Regular Celestial Mechanics, Grundlehren der mathematischen Wissenschaften,
+Springer, Berlin, 1971.
+[12] Masat, A., Romano, M., and Colombo, C., “Combined B-plane and Picard-Chebyshev approach for the continuous design of
+perturbed interplanetary resonant trajectories,” 31st AAS/AIAA Space Flight Mechanics Meeting, Vol. AAS-21-289, Charlotte,
+NC, USA, 2021.
+[13] Masat, A., and Colombo, C., “B-plane and Picard–Chebyshev integration method: Surfing complex orbital perturbations in
+interplanetary multi-flyby trajectories,” Acta Astronautica, Vol. 194, 2022, pp. 216–228. https://doi.org/10.1016/j.actaastro.
+2022.01.045, URL https://www.sciencedirect.com/science/article/pii/S0094576522000546.
+[14] Junkins, J. L., Bani Younes, A., Woollands, R. M., and Bai, X., “Picard Iteration, Chebyshev Polynomials and Chebyshev-
+Picard Methods: Application in Astrodynamics,” Journal of the Astronautical Sciences, Vol. 60, No. 3, 2013, pp. 623–653.
+https://doi.org/10.1007/s40295-015-0061-1, URL https://doi.org/10.1007/s40295-015-0061-1.
+[15] Koblick, D., , and Shankar, P., “Evaluation of the Modified Picard-Chebyshev Method for High-Precision Orbit Propagation,”
+Journal of Aerospace Engineering, Vol. 28, No. 5, 2015, p. 04014125. https://doi.org/10.1061/(ASCE)AS.1943-5525.0000463,
+URL https://doi.org/10.1061/(ASCE)AS.1943-5525.0000463.
+[16] Woollands, R. M., Bani Younes, A., and Junkins, J. L., “New Solutions for the Perturbed Lambert Problem Using Regularization
+and Picard Iteration,” Journal of Guidance, Control, and Dynamics, Vol. 38, No. 9, 2015, pp. 1548–1562. https://doi.org/10.
+2514/1.G001028, URL https://doi.org/10.2514/1.G001028.
+[17] Woollands, R. M., Read, J. L., Probe, A. B., and Junkins, J. L., “Multiple Revolution Solutions for the Perturbed Lambert
+Problem using the Method of Particular Solutions and Picard Iteration,” Journal of the Astronautical Sciences, Vol. 64, No. 4,
+2017, pp. 361–378. https://doi.org/10.1007/s40295-017-0116-6, URL https://doi.org/10.1007/s40295-017-0116-6.
+[18] Woollands, R. M., Read, J., Hernandez, K., Probe, A., and Junkins, J. L., “Unified Lambert Tool for Massively Parallel
+Applications in Space Situational Awareness,” Journal of the Astronautical Sciences, Vol. 65, No. 1, 2018, pp. 29–45.
+https://doi.org/10.1007/s40295-017-0118-4, URL https://doi.org/10.1007/s40295-017-0118-4.
+[19] Swenson, T., Woollands, R., Junkins, J., and Lo, M., “Application of Modified Chebyshev Picard Iteration to Differential
+Correction for Improved Robustness and Computation Time,” Journal of the Astronautical Sciences, Vol. 64, No. 3, 2017, pp.
+267–284. https://doi.org/10.1007/s40295-016-0110-4.
+30
+
+[20] Singh, S. K., Woollands, R., Taheri, E., and Junkins, J., “Feasibility of quasi-frozen, near-polar and extremely low-altitude lunar
+orbits,” Acta Astronautica, Vol. 166, 2020, pp. 450–468. https://doi.org/https://doi.org/10.1016/j.actaastro.2019.10.037, URL
+https://www.sciencedirect.com/science/article/pii/S0094576519313657.
+[21] Koblick, D., Xu, S., Fogel, J., and Shankar, P., “Low Thrust Minimum Time Orbit Transfer Nonlinear Optimization Using
+Impulse Discretization via the Modified Picard-Chebyshev Method,” Computer Modeling in Engineering & Sciences, Vol. 111,
+No. 1, 2016. https://doi.org/10.3970/cmes.2016.111.001, URL https://doi.org/10.3970/cmes.2016.111.001.
+[22] Macomber, B., Probe, A. B., Woollands, R., Read, J., and Junkins, J. L., “Enhancements to Modified Chebyshev-Picard Iteration
+Efficiency for Perturbed Orbit Propagation,” Computer Modeling in Engineering & Sciences, Vol. 111, No. 1, 2016, pp. 29–64.
+https://doi.org/10.3970/cmes.2016.111.029, URL http://www.techscience.com/CMES/v111n1/27313.
+[23] Woollands, R., Taheri, E., and Junkins, J. L., “Efficient Computation of Optimal Low Thrust Gravity Perturbed Orbit Transfers,”
+Journal of the Astronautical Sciences, Vol. 67, No. 2, 2020, pp. 458–484. https://doi.org/10.1007/s40295-019-00152-9, URL
+https://doi.org/10.1007/s40295-019-00152-9.
+[24] Woollands, R., and Junkins, J. L., “Nonlinear Differential Equation Solvers via Adaptive Picard-Chebyshev Iteration:
+Applications in Astrodynamics,” Journal of Guidance, Control, and Dynamics, Vol. 42, No. 5, 2019, pp. 1007–1022.
+https://doi.org/10.2514/1.G003318, URL https://doi.org/10.2514/1.G003318.
+[25] Atallah, A. M., Woollands, R. M., Elgohary, T. A., and Junkins, J. L., “Accuracy and Efficiency Comparison of Six Numerical
+Integrators for Propagating Perturbed Orbits,” Journal of the Astronautical Sciences, Vol. 67, No. 2, 2020, pp. 511–538.
+https://doi.org/10.1007/s40295-019-00167-2, URL https://doi.org/10.1007/s40295-019-00167-2.
+[26] Geda, M., Noomen, R., and Renk, F., “Massive Parallelization of Trajectory Propagations using GPUs,” Master’s thesis, Delft
+University of Technology, 2019. URL http://resolver.tudelft.nl/uuid:1db3f2d1-c2bb-4188-bd1e-dac67bfd9dab.
+[27] Schrammel, F., Renk, F., Mazaheri, A., and Wolf, F., “Efficient Ephemeris Models for Spacecraft Trajectory Simulations on
+GPUs,” Euro-Par 2020: Parallel Processing, edited by M. Malawski and K. Rzadca, Springer International Publishing, Cham,
+2020, pp. 561–577.
+[28] Li, J.-S., Yang, Z., and Luo, Y.-Z., “A review of space-object collision probability computation methods,” Astrodynamics, Vol. 6,
+No. 2, 2022, pp. 95–120. https://doi.org/10.1007/s42064-021-0125-x, URL https://doi.org/10.1007/s42064-021-0125-x.
+[29] Yoshikawa, K., Sawada, H., Kikuchi, S., Ogawa, N., Mimasu, Y., Ono, G., Takei, Y., Terui, F., Saiki, T., Yasuda, S., Matsushima,
+K., Masuda, T., and Tsuda, Y., “Modeling and analysis of Hayabusa2 touchdown,” Astrodynamics, Vol. 4, No. 2, 2020, pp.
+119–135. https://doi.org/10.1007/s42064-020-0073-x, URL https://doi.org/10.1007/s42064-020-0073-x.
+[30] Armellin, R., Di Lizia, P., Bernelli-Zazzera, F., and Berz, M., “Asteroid close encounters characterization using differential
+algebra: the case of Apophis,” Celestial Mechanics and Dynamical Astronomy, Vol. 107, No. 4, 2010, pp. 451–470.
+https://doi.org/10.1007/s10569-010-9283-5, URL https://doi.org/10.1007/s10569-010-9283-5.
+31
+
+[31] Hairer, E., Wanner, G., and Nørsett, S. P., Solving Ordinary Differential Equations I, 2nd ed., Springer Berlin, 1993.
+https://doi.org/10.1007/978-3-540-78862-1.
+[32] Bai, X., and Junkins, J. L., “Solving initial value problems by the Picard-Chebyshev method with NVIDIA GPUs,” Advances in
+the Astronautical Sciences, 2010.
+[33] Rivlin, T. J., “The Chebyshev Polynomials,” Mathematics of Computation, Vol. 30, 1976. https://doi.org/10.2307/2005983.
+[34] Fukushima, T., “Picard Iteration method, Chebyshev Polynomial Approximation, and Global Numerical Integration of
+Dynamical Motions,” The Astronomical Journal, Vol. 113, 1997, pp. 1909–1914. https://doi.org/10.1086/118404.
+[35] Macomber, B. D., “Enhancements to Chebyshev-Picard Iteration Efficiency for Generally Perturbed Orbits and Constrained
+Dynamical Systems,” Ph.D. thesis, Texas A & M University, Supervisor: Junkins, John L., Aug 2015.
+URL https:
+//hdl.handle.net/1969.1/155745.
+[36] Fukushima, T., “Vector Integration of Dynamical Motions by the Picard-Chebyshev Method,” The Astronomical Journal, Vol.
+113, 1997, p. 2325. https://doi.org/10.1086/118443.
+[37] Bai, X., and Junkins, J. L., “Modified Chebyshev-Picard Iteration Methods for Orbit Propagation,” Journal of the Astronautical
+Sciences, Vol. 58, No. 4, 2011, pp. 583–613. https://doi.org/10.1007/BF03321533, URL https://doi.org/10.1007/BF03321533.
+[38] Koblick, D., Poole, M., and Shankar, P., “Parallel high-precision orbit propagation using the Modified Picard-Chebyshev
+Method,” ASME International Mechanical Engineering Congress and Exposition, Proceedings (IMECE), 2012. https:
+//doi.org/10.1115/IMECE2012-87878.
+[39] Forum, M. P., “MPI: A Message-Passing Interface Standard,” Tech. rep., USA, 1994.
+[40] Chandra, R., Dagum, L., Kohr, D., Menon, R., Maydan, D., and McDonald, J., Parallel programming in OpenMP, Morgan
+kaufmann, 2001.
+[41] “Xianyi/OpenBLAS,” , Last access: November 2021. URL https://github.com/xianyi/OpenBLAS#supported-cpus-and-
+operating-systems.
+[42] Wang, Q., Zhang, X., Zhang, Y., and Yi, Q., “AUGEM: Automatically generate high performance Dense Linear Algebra kernels
+on x86 CPUs,” SC ’13: Proceedings of the International Conference on High Performance Computing, Networking, Storage
+and Analysis, 2013, pp. 1–12. https://doi.org/10.1145/2503210.2503219.
+[43] Xianyi, Z., Qian, W., and Yunquan, Z., “Model-driven Level 3 BLAS Performance Optimization on Loongson 3A Processor,”
+2012 IEEE 18th International Conference on Parallel and Distributed Systems, 2012, pp. 684–691. https://doi.org/10.1109/
+ICPADS.2012.97.
+[44] NVIDIA corporation, “CUDA C++ Programming Guide,” , Nov 2021. URL https://docs.nvidia.com/cuda/cuda-c-programming-
+guide/index.html.
+32
+
+[45] Seidelmann, P. K., Explanatory Supplement To The Astronomical Almanac, University Science Books, 1305 Walt Whitman
+Road, Suite 110 Melville, NY 11747 USA, 1992.
+[46] Valsecchi, G. B., Milani, A., Gronchi, G. F., and Chesley, S. R., “Resonant returns to close approaches: Analytical theory,”
+A&A, Vol. 408, No. 3, 2003, pp. 1179–1196. https://doi.org/10.1051/0004-6361:20031039, URL https://doi.org/10.1051/0004-
+6361:20031039.
+33
+

BNAzT4oBgHgl3EQfhv2_/content/tmp_files/2301.01490v1.pdf.txt

@@ -0,0 +1,1210 @@
+Journal of Virtual Reality and Broadcasting, Volume n(200n), no. n
+Towards a Pipeline for Real-Time Visualization of Faces for
+VR-based Telepresence and Live Broadcasting Utilizing Neural
+Rendering
+Philipp Ladwig*, Rene Ebertowski*,Alexander Pech*, Ralf D¨orner†, Christian Geiger*
+*University of Applied Sciences D¨usseldorf, Germany
+Mixed Reality and Visualization Group (MIREVI)
+{philipp.ladwig@, rene.ebertowski@study.,
+alexander.pech@., geiger@}hs-duesseldorf.de
+www.mirevi.de
+†RheinMain University of Applied Sciences
+Faculty of Design – Computer Science – Media, Wiesbaden, Germany
+ralf.doerner@hs-rm.de
+Abstract
+While head-mounted displays (HMDs) for Virtual Re-
+ality (VR) have become widely available in the con-
+sumer market, they pose a considerable obstacle for a
+realistic face-to-face conversation in VR since HMDs
+hide a significant portion of the participants faces.
+Even with image streams from cameras directly at-
+tached to an HMD, stitching together a convincing im-
+age of an entire face remains a challenging task be-
+cause of extreme capture angles and strong lens dis-
+tortions due to a wide field of view. Compared to the
+long line of research in VR, reconstruction of faces
+hidden beneath an HMD is a very recent topic of re-
+search.
+While the current state-of-the-art solutions
+demonstrate photo-realistic 3D reconstruction results,
+they require high-cost laboratory equipment and large
+Digital Peer Publishing Licence
+Any party may pass on this work by electronic
+means and make it available for download under
+the terms and conditions of the current version
+of the Digital Peer Publishing Licence (DPPL).
+The text of the licence may be accessed and
+retrieved via internet at
+http://www.dipp.nrw.de/.
+First presented at the Workshop of GI Special Interest group VR/AR 2020,
+extended and revised for JVRB
+computational costs. We present an approach that fo-
+cuses on low-cost hardware and can be used on a com-
+modity gaming computer with a single GPU. We lever-
+age the benefits of an end-to-end pipeline by means of
+Generative Adversarial Networks (GAN). Our GAN
+produces a frontal-facing 2.5D point cloud based on a
+training dataset captured with an RGBD camera. In
+our approach, the training process is offline, while the
+reconstruction runs in real-time. Our results show ad-
+equate reconstruction quality within the ’learned’ ex-
+pressions. Expressions not learned by the network pro-
+duce artifacts and can trigger the Uncanny Valley ef-
+fect.
+Keywords:
+Neural Rendering, Telepresence, Face
+Reconstruction, Virtual Reality, Live Broadcasting,
+Image-to-Image Translation, Pix2Pix, Generative Ad-
+versarial Networks
+1
+Introduction
+Natural
+face-to-face
+communication
+is
+three-
+dimensional.
+A conversation includes not only
+the verbal communication channel but also the
+nonverbal channel. In particular, eye contact, facial
+expressions as well as gestures performed with arms
+and hands (kinesics), and even the physical distance
+between each other (proxemics) are essential informa-
+urn:nbn:de:0009-6-348, ISSN 1860-2037
+arXiv:2301.01490v1  [cs.CV]  4 Jan 2023
+
+Journal of Virtual Reality and Broadcasting, Volume n(200n), no. n
+Figure 1: Our conceptual pipeline: First, we capture several RGBD images with a helmet camera mount. These
+images are processed and serve as the input data for our GAN. After training, the GAN produces textured point
+clouds in real time. In this work we improve the data set processing, training and inference stage compared to
+our previous systems [LPG20, LPDG20]. Building an face-tracking HMD is not part of the present work.
+tion carriers during a conversation [LG18]. Currently,
+common mainstream technologies for computer-
+mediated communication are video conferencing
+applications such as Skype or FaceTime. While these
+allow reading the facial expressions of the counterpart,
+no ’real’ eye contact is possible, wide gestures may
+be cut off in the camera image, deictic gestures are
+difficult to interpret spatially, and perceptual physical
+body distance between the participants does not exist.
+Current head-mounted displays (HMDs) for VR are
+capable of delivering believable and immersive 3D ex-
+periences including telepresence. However, this does
+not fully apply to real-time social interactions in VR.
+When the face of a person is covered by an HMD,
+it is impossible to read its non-verbal facial commu-
+nication cues, which is, in fact, a crucial communi-
+cation channel between individuals. This is not only
+relevant for face-to-face meetings in VR (for example
+in VRChat [vrc22], Altspace [alt22] or Meta’s Horizon
+Worlds [met22]) but also in VR application scenarios
+in which only one VR user wears an HMD and tries
+to engage with their audience. For example, such a
+VR user could be an architect who presents ideas for a
+new building in VR to their clients, a Virtual YouTuber
+(VTuber), a Twitch streamer in front of a green screen
+who broadcasts themselves from inside a VR environ-
+ment, or friends playing a VR game together in a living
+room.
+The classic way for creating and rendering photo-
+realistic humans in real time is costly and requires a lot
+of manual effort such as scanning, modeling, and man-
+ual texturing from a skilled 3D artist. Furthermore,
+today’s HMDs usually lack adequate sensors for face
+tracking. Only a few research groups have so far ad-
+dressed this problem and presented methods that can
+generate authentic face avatars for VR without exten-
+sive manual modeling [TZS+18, LSSS18, WSS+19,
+RZS+21, GPL+21]. These approaches are not avail-
+able to the public, and some of them require expensive
+hardware [LSSS18, WSS+19].
+Human avatars and their perception have been stud-
+ied in multiple domains. A key issue in this context is
+the occurrence of the Uncanny Valley effect. Humans
+are markedly sensitive to minimal and unnatural dis-
+crepancies in faces. As soon as a virtual human does
+not perfectly resemble a real human, it is often subcon-
+sciously classified as unlikeable, unpleasant, or even
+creepy. One technique that has successfully bridged
+the Uncanny Valley in recent years is Generative Ad-
+versarial Networks (GANs). Today, GANs serve as the
+core technology behind Deepfakes. They enable such
+authentic results that their methods and algorithms are
+the subject of current research to distinguish fake im-
+ages from real ones, as the human eye is no longer
+able to reliably do so [RCV+19]. Therefore, we use
+algorithms in the context of this work that are also
+employed to create Deepfakes.
+We extend this ap-
+proach with an additional dimension (textured 2.5D
+point cloud instead of only an RGB image) to gener-
+ate realistic representations of 2.5D face avatars. We
+do not create a full 3D head because we only capture
+the face of a person from a static frontal position with
+an RGBD sensor. This implies that we do not generate
+realistic textures from side views. However, we main-
+tain a stereoscopic perception of the reconstructed face
+during face-to-face conversations in a virtual environ-
+ment.
+We present an end-to-end learning system that has
+low hardware cost compared to others, requires mod-
+erate computational resources, and generates results
+with frame rates suitable for VR applications.
+Our
+research contributes to GANs playing a key role in
+authentic 3D telepresence applications in the near fu-
+ture. In addition to sharing our insights in this pa-
+per, we make the code of our prototype publicly avail-
+able under:
+https://github.com/Mirevi/
+urn:nbn:de:0009-6-348, ISSN 1860-2037
+
+capture
+data set processing
+training
+inference
+textured point cloudJournal of Virtual Reality and Broadcasting, Volume n(200n), no. n
+face-synthesizer-JVRB.
+This work is an extension and improvement of our
+previous neural rendering pipeline [LPG20] and com-
+plements our recent work on how to build an HMD
+with face tracking capabilities [LPDG20]. The contri-
+butions of this work are the creation of a face capture
+pipeline as well as the introduction of a pair of Gen-
+erative Adversarial Networks (GAN) [GPAM+14] that
+are tailored for the authentic reconstruction of faces in
+three-dimensional telepresence and live broadcasting
+applications. The motivation is to use a commodity
+graphics card to capture and reconstruct the individ-
+ual characteristics of a person’s face with a high level
+of (personal) details and VR-enabled frame rates in
+order to create an authentic avatar that goes beyond
+the capabilities of today’s avatar creation tools such
+as VRChat [vrc22], Altspace [alt22] or Meta’s Horizon
+Worlds [met22].
+2
+Related Work
+Face reconstruction for telepresence and (live) broad-
+casting with an HMD occluding a person’s face is a
+young research field. Olszewski et al. [OLSL16] pre-
+sented a system that uses an RGB camera to transfer
+facial expressions from the lower face area to an avatar.
+Li et al. [LTO+15] extended this approach with pres-
+sure sensors placed in the foam of an HMD captur-
+ing a person’s facial expressions. The idea of using
+sensors within the HMD is similar to our concept, but
+we use personalized avatars that are trained in advance
+and synthesized in a final step.
+Casa et al. [CFA+16], Fr¨uh et al. [FSK17], and
+Thies et al. [TZS+18] used stationary RGBD cameras
+to create personalized avatars of users.
+In the first
+step, the user was captured by the camera without
+an HMD in order to create a virtual avatar.
+When
+the user wore the HMD, the stationary RGBD camera
+recognized facial expressions. Due to the fixed posi-
+tion of the camera, the range of head motion was lim-
+ited. Eye movements were registered by eye-tracking
+cameras and transferred to the user’s face avatar.
+The approaches of Casa et al. [CFA+16] and Fr¨uh et
+al. [FSK17] evoked the Uncanny Valley effect to vary-
+ing degrees. To mitigate this, Fr¨uh et al. [FSK17] did
+not completely remove the HMD, but rendered it as
+a semi-transparent object. These systems are similar
+to our approach in the way that they create a person-
+alized avatar using an RGBD camera and produce al-
+most photo-realistic avatars. The approach of Thies
+et al. [TZS+18] demonstrated better results by using
+a 3D morphable model (3DMM) as underlying head
+mesh template and by optimizing the visual quality
+by an analysis-by-synthesis approach [BV99]. While
+the visual quality is convincing, this approach only
+provides stereoscopic renderings without the ability to
+freely choose the perspective around the reconstructed
+face because the final results are based on a given 2D
+video. Furthermore, it explicitly does not allow for
+manipulation of the head’s rotation, scale, and posi-
+tion in the final result.
+The systems of Lombardi et al. [LSSS18], Wei et
+al. [WSS+19], and Raj et al. [RZS+21] create photo-
+realistic avatars with authentic facial expressions.
+While previous works completed the generation of
+personalized avatars within a few minutes, the sys-
+tem of Lombardi et al. requires a computational time
+of more than a day.
+The three-dimensional avatar
+is generated with the aid of a large number of high-
+resolution images from different angles and facial ex-
+pressions with an expensive hardware setup that gen-
+erates a large amount of data for further processing.
+The created face avatar can be controlled by three
+RGB cameras attached to an HMD. A key component
+of this system is the use of Variational Autoencoders
+(VAEs). Both VAEs and GANs have been proven sev-
+eral times to be suitable for authentic face reconstruc-
+tion. However, since literature shows that VAEs and
+only a L1 loss tend to produce blurry results more of-
+ten, we use GANs [JH19]. The latter concept was first
+presented by Goodfellow et al. [GPAM+14], and Rad-
+ford et al. [RMC15] improved it in a sustainable way.
+Furthermore, Karras et al. [KALL17] achieved photo-
+realistic portrait images that are indistinguishable from
+real photographs by using the principle of Progressive
+Growing GAN. However, according to Karras et al.,
+the GAN has little to no external control over the ap-
+pearance of the generated object or face because the
+input to the network is a latent vector without any di-
+rect relation to a face property such as hair color, fa-
+cial expression, or gender. In further works Karras
+et al. [KLA18] enhanced the architecture of the GAN
+and were able to automatically separate higher-level
+attributes (e.g. pose, identity) from stochastic varia-
+tions (e.g. freckles, hair). Nevertheless, this approach
+does not allow to explicitly control the facial expres-
+sion.
+Conditional GANs (cGANs) have been shown to be
+able to learn and reproduce concrete relationships be-
+tween inputs and outputs.
+For example, Mirza and
+urn:nbn:de:0009-6-348, ISSN 1860-2037
+
+Journal of Virtual Reality and Broadcasting, Volume n(200n), no. n
+Osindero [MO14] have extended the input to the gen-
+erator and discriminator with a label y, which makes it
+possible to generate images from a particular category
+y. This method for conditioning GANs was developed
+further by Radfort et al. [RMC15] with the DCGAN
+and by Isola et al. [IZZE17] with the Pix2Pix GAN.
+They replaced the noise input vector z with a user-
+defined input vector.
+Without a noise vector, there
+is no latent space Z (since z ∈ Z).
+If the stochas-
+tic aspect contained in the noise vector is not com-
+pensated, the GAN will only memorize the training
+examples. Any inputs that deviate from the training
+data would lead to inadequate results, as described
+by Isola et al. [IZZE17]. By using a U-net architec-
+ture [RFB15] with dropouts in the Pix2Pix GAN, the
+stochastic aspect as well as the missing latent space
+can be otherwise integrated into the generator. The
+discriminator of the Pix2Pix GAN receives the same
+input image x as the generator as well as its output
+image yfake = G(x) or the image yreal matching x
+from the dataset. This is basically equivalent to the
+idea of cGANs [MO14] where not only the output of
+the generator is evaluated but also its difference from
+the input. Unlike the cGAN, the output of the discrim-
+inator of the Pix2Pix GAN is not a scalar that decides
+between ’real’ or ’false’ but a matrix. By using con-
+volutional layers (cf. Radford et al. [RMC15]), each
+entry in the output matrix represents an n ∗ m-sized
+region (so-called patch) of the input image. This al-
+lows abstract representations to be admitted as matri-
+ces (e.g. images) for conditioning the network to have
+a controlled influence on the output of the generator.
+This approach was further developed by [WLZ+17]
+with the Pix2PixHD GAN to generate images with a
+higher resolution and more details. In this paper, we
+adapt the idea of cGANs, especially of the Pix2Pix and
+Pix2PixHD frameworks, and tailor them to our appli-
+cation domain.
+3
+System
+In the following, we briefly explain the process and
+structure of the proposed system, as shown in Fig. 1,
+and then discuss the steps in more detail in the subse-
+quent sections.
+Our process starts with the acquisition of a personal
+RGBD dataset. The acquired data is preprocessed by
+an automated procedure.
+A Facial Landmark Map
+(FLM) per RGB image is extracted and saved beside
+the corresponding RGB image. It decodes the facial
+expression of the respective RGB image in a binary
+image as so-called landmarks as shown in the second
+image from left in Fig.1. Our proposed GAN is trained
+with the captured RGBD images as well as with the
+corresponding FLM. After the training, the system can
+be used for real-time telepresence or live broadcasting.
+In the application scenario, the user would wear a face-
+tracking head-mounted display (HMD) that could cre-
+ate an FLM in real time, which we then feed into the
+trained generator module of our GAN, as described in
+our previous work [LPDG20]. The GAN could create
+an RGB and a D image of the ’learned’ person based
+on the FLM, and finally, we fuse the generated RGB
+and D images into a textured point cloud. Instead of
+our face-tracking HMD [LPDG20], we use raw face-
+tracking data from our evaluation dataset captured in
+the first step with the camera helmet mount. We thus
+ensure that no tracking errors of the HMD are incor-
+rectly evaluated as reconstruction errors of the GAN.
+3.1
+Training data
+Figure 2: Helmet camera mount for RGBD data acqui-
+sition. This mount ensures that head rotations are not
+included in the training dataset and, therefore, reduces
+the entropy in the dataset. Moreover, this method sig-
+nificantly reduces the training time and increases the
+visual quality of the output images. The material price
+of the helmet mount without RGBD camera is about
+60 USD.
+An RGBD dataset of the respective person forms
+the basis for the training process. For the acquisition
+process, an RGBD camera is mounted in a fixed po-
+sition to a self-made helmet mount, as shown in Fig-
+ure 1 left and Figure 2. This mount ensures that the
+entropy in the dataset is as minimal as possible. Vary-
+ing distances, positions and head rotations do not con-
+tribute to ’learning’ the user’s face. By using the hel-
+met mount, we ensure that the training time of the net-
+works is short and the reconstruction quality of the fi-
+nal results is high, as we learned from previous exper-
+iments conducted without the helmet mount.
+urn:nbn:de:0009-6-348, ISSN 1860-2037
+
+RGBD camera
+(Microsoft Azure Kinect)Journal of Virtual Reality and Broadcasting, Volume n(200n), no. n
+The RGBD sensor in the mount captures facial ex-
+pressions and stores the color information as a 3-
+channel image file (8-bit for each channel) with 2048
+to 1536 pixels, while it stores the depth information
+as a 1-channel image file (16-bit grayscale) with 640
+to 576 pixels. During the capture process, the user is
+encouraged to perform a variety of different facial ex-
+pressions. A dataset of a person should contain about
+1500 to 2000 RGBD images, and the acquisition pro-
+cess takes about 10 minutes.
+The data is then pre-
+processed for further steps. First, the foreground and
+background pixels are clipped in front of and behind
+the face and the depth resolution is reduced from 16
+to 8 bit, which reduces the depth range from 65,535
+to 255 mm.
+This speeds up the training process of
+the GAN and significantly reduces depth noise. With
+the help of the helmet mount, we ensure that a depth
+range of 255 mm is sufficient for spatially reproducing
+the frontal part of the head since the distance between
+the sensor and the face in the helmet mount is always
+fixed. Furthermore, the data is normalized and the im-
+age data is converted into values between -1 and 1 for
+the training process. Contrasts are sharpened by nor-
+malizing the histogram.
+3.2
+Determination of Facial Landmarks
+To control the output of the generator and thus the fa-
+cial expressions of a person’s face, the dataset must be
+labeled before the training process. We use 70 facial
+landmarks (68 of the Multi-PIE scheme [GMC+08]
+and two for the location of the irises) as binary images
+for each tuple of RGB and corresponding depth im-
+age in the dataset. As mentioned before, we call these
+binary images Facial Landmark Maps (FLM). The po-
+sition of the landmarks identifies the expressions of the
+person in each image of the dataset. To determine the
+landmarks, we use the Face Alignment Network (FAN)
+of Bulat and Tzimiropoulos [BT17]. The FAN is not
+able to determine the position of the person’s pupils
+within an image. Therefore, two additional landmarks
+were implemented based on an eye-tracking procedure
+by Timm and Barth [TB11, pup18]. We experimented
+with an Tobii Eye Tracker 4C, but users reported that
+additional weight on the helmet mount felt uncomfort-
+able during the capture procedure. Both the tracking
+results of the Tobii Eye Tracker and Timm and Barth
+were similar and sufficient for our application.
+When the landmarks in each image had been lo-
+cated, a rectangle of the maximal and minimal loca-
+tions of the landmarks was created and the RGB and
+D images were cropped to this area and resized to
+512 × 512 pixels.
+3.3
+Neural Network Architecture
+Previous work with neural networks, such as Wu
+et al. [WZX+16], has shown that a voxel-based ap-
+proach is associated with high training and execu-
+tion times of the model. Therefore, an RGBD-based
+solution was targeted.
+The advantage of this ap-
+proach lies in the compact representation of the data
+as a point cloud and the possibility to adapt previ-
+ous RGB-based methods. Our underlying network ar-
+chitecture is derived from the Pix2Pix GAN by Isola
+et al. [IZZE17].
+In earlier experiments, we discov-
+ered that this architecture was able to produce ac-
+ceptable results [LPG20, LPDG20], but the images
+have a low resolution of 256 × 256 pixels, often lack
+details in high frequency areas such as facial hair
+and tend to produce time-inconsistent reconstructions
+with noise.
+Therefore, we experimented with the
+Pix2PixHD framework [WLZ+17], which is an exten-
+sion of the Pix2Pix GAN [IZZE17]. Although it pro-
+duces images with a higher resolution and better qual-
+ity, the inference time is not capable of retaining real-
+time frame rates on commodity hardware. Therefore,
+we kept the Pix2Pix framework as a base and gradually
+added elements from the Pix2PixHD framework and
+further improvements from other works until we ob-
+tained an acceptable image quality with a reasonable
+processing speed for interactive frame rates. In sum-
+mary, we propose the following changes to the Pix2Pix
+framework:
+1. We added a multi-scale discriminators that re-
+ceives three different resolutions of the input im-
+age and an additional Feature Matching Loss as
+described in Pix2PixHD [WLZ+17].
+2. We changed the Sigmoid Cross Entropy Loss of
+Pix2Pix’s discriminator with the Least-Squares
+Loss of the LSGAN [MLX+17] suggested by
+Wang et al. [WLZ+17].
+3. We
+exchanged
+the
+Perceptual-VGG
+Loss [JAFF16] originally suggested by Wang
+et al. [WLZ+17] with the better performing
+Learned
+Perceptual
+Image
+Patch
+Similarity
+(LPIPS) by Zhang et al. [ZIE+18].
+urn:nbn:de:0009-6-348, ISSN 1860-2037
+
+Journal of Virtual Reality and Broadcasting, Volume n(200n), no. n
+For the discriminator we obtain the following objec-
+tive:
+min
+D1,D2,D3 VGAN(D) =
+�
+k=1,2,3
+LcLSGAN D(Dk, G)
+(1)
+while D1, D2 and D3 describe the three different res-
+olution of the input image. For the generator we end
+up with the following objective function:
+min
+G VGAN(G) =
+�
+k=1,2,3
+�
+LcLSGAN G(Dk, G) + λFMLFM(Dk, G)
+�
++ λL1LL1(G) + λLPIPSLLPIPS(y, G(x))
+(2)
+where we choose the following hyper parameters:
+λFM
+= 10, λL1 = 100, λLPIPS
+= 10.
+The
+functions LcLSGAN D and LcLSGAN G can be found
+in Mao et al. [MLX+17], whereas the function LFM
+is based on the feature matching loss of Wang et
+al. [WLZ+17]. In order to maintain faster inference
+time than the Pix2PixHD we did not implement the
+coarse to fine approach for the generator. We sacrifice
+very high resolution of the output images for compu-
+tational speed. We provide further details to the archi-
+tecture on GitHub.
+Using these improvements helps to prevent high-
+frequency details such as facial hair and significantly
+increases the reconstruction quality, as explained fur-
+ther in the results section (sec. 4). Because we mainly
+enhanced the loss function and the discriminator side,
+we did not need to change the generator. Therefore,
+we are able to maintain high frame rates during infer-
+ence since only the generator module is used in the
+telepresence and live broadcasting scenario. As a side
+effect, the training process requires more memory, but
+the overall training time decreases by more than a half
+compared to our Pix2Pix-only approach (from about
+19 hours to 8 hours) because the new loss term is more
+purposeful for our application and, therefore, helps to
+obtain better results in less time.
+The generator of our GAN receives a 512×512 pixel
+FLM of the facial landmarks as input. Compared to
+the Pix2Pix GAN, the output has been extended by a
+fourth feature map to be able to generate depth im-
+ages. In addition, the discriminator receives five fea-
+ture maps as input instead of only four. While the
+first four correspond to the four channels of the RGBD
+image, the remaining feature map contains the corre-
+sponding FLM, as visualized in Figure 3. One of our
+Figure 3: Example convolution for the discriminator
+input. Each RGBD channel and the FLM are weighted
+individually.
+early hypotheses in [LPG20] was that a higher number
+of FLMs (e.g. four times) would result in better re-
+construction results to start the training process with a
+balanced ratio between RGBD images and FLMs (cf.
+Fig. 3). This hypothesis has been disproved. Chang-
+ing the number of FLMs does not change the quality
+of the results but only increases the training time.
+3.4
+Training Process
+Before the training process, all weights of the GAN are
+initialized with a random value. The random values
+follow a Gaussian distribution with an expected value
+of 0 and a standard deviation of 0.02. The batch size of
+the input data into the GAN is 1, and the epoch count
+is 100. Both generator and discriminator are trained
+with an initial learning rate of 0.0002, which decreases
+linearly towards zero over the last 70 epochs. The dis-
+criminator LossD = (LossDreal + LossDfake) ∗ 0.5
+reduces the learning rate, making it slower to learn rel-
+ative to the generator. This is necessary because at the
+beginning of the training phase the discriminator can
+effortlessly accomplish its task. If the discriminator
+learns too quickly, the generator has no chance to learn
+how to create the desired face.
+4
+Results
+GANs are difficult to train because of their adversar-
+ial training procedure to find the balance between the
+learning rate and losses of the generator and discrim-
+inator networks. As we use many different losses, a
+long line of hyper parameter tuning was necessary to
+find the optimal settings. Fig. 5 shows the reconstruc-
+tion results for four different persons with the best
+urn:nbn:de:0009-6-348, ISSN 1860-2037
+
+R
+Featuremap
+G
+(showninsimplifiedform)
+B
+4x4 kernel
+FLM
+Convolutional resultJournal of Virtual Reality and Broadcasting, Volume n(200n), no. n
+Figure 4: Our new pipeline, network architecture and
+losses significantly improved the quality.
+Image a)
+shows a sample from the previous system of Ladwig
+et al.[LPG20, LPDG20]. Image b) illustrates the en-
+hanced resolution (from 256×256 to 512×512 pixels)
+and the improved preservation of high-frequency de-
+tails.
+training parameters described in section 3.4. We did
+not use the face-tracking HMD for the evaluation, sim-
+ilar to [LPDG20], because it can cause slight tracking
+errors and could decrease the quality of the results for
+comparison. Our intention is to directly compare the
+improvements of our pipeline to our previous system
+described in [LPG20]. For a comparison in motion, we
+refer the reader to the corresponding video that can be
+found via the GitHub link. As a quantitative metric for
+the assessment of the reconstruction quality, we use
+Structural Similarity (SSIM) [ZBSS04] and Learned
+Perceptual Image Patch Similarity (LPIPS) [ZIE+18].
+Our previous system [LPG20] performed on average
+with an SSIM of 0.851 and a value of 0.114 on LPIPS.
+Our proposed system reaches on average 0.910 for
+SSIM (higher is better) and 0.082 for LPIPS (lower
+is better). The numerical results of the SSIM metric
+are comparable to a JPEG compression of about half
+the original file size of the images in column 3. In
+contrast, the previous system [LPG20] only achieved
+a reconstruction quality of less than a quarter of the
+original file size by means of a comparison with the
+JPEG compression.
+The main issue of our previous approach was the
+sharpness of the generated images [LPG20, LPDG20],
+as visualized in Fig. 4. Due to the new architecture and
+loss functions, the system produces images with more
+details. Even skin pores are reconstructed well, e.g.
+on the user’s forehead, as can be seen in Fig. 5, rows
+E to H. Furthermore, we noticed a better reconstruc-
+tion quality in areas with high-frequency details such
+as facial hair, as shown in Fig. 4. Also, the temporal
+consistency is improved. Please see the linked video
+for details.
+The error between the generated and ground truth
+depth values is mostly below 4 mm, as depicted in col-
+umn 6. Exceptions are the reconstruction results with
+the worst SSIM and LPIPS metrics per dataset of a
+person, such as shown in Fig. 5, row D and H, as well
+as in Fig. 6, row E and J. Furthermore, outliers can be
+seen in column 8. Note that we use the raw depth im-
+age of the Kinect and the raw output of our GAN. We
+do not filter or smooth the images, therefore, we as-
+sume that the network has also learned the depth noise
+of the sensor, which causes additional depth errors.
+To compare the faces between the images of
+columns 2 and 3 without measuring background
+changes, we determined the facial area based on depth
+values and rejected all other pixels. At the border area
+between the faces and the background, large differ-
+ences in the SSIM and the depth difference visualiza-
+tion can be seen. These differences are caused by the
+fact that the cropped areas of the real and generated
+images do not always align perfectly due to minimal
+differences in the generated faces. In addition, we also
+apply erosion and clipping to the face to reject parts of
+the background, which can cause the minimal differ-
+ences.
+Although our quantitative metrics indicate better re-
+sults, our system still shows limitations in the recon-
+struction quality of the eyes, lips and oral cavity. Es-
+pecially teeth and the tongue are often reconstructed
+with noisy artifacts, as can be seen in Fig. 5, row D,
+column 2. The error increases with a graceful degra-
+dation when the expression moves towards exagger-
+ated expressions that are far from the neutral face ex-
+pression. The eyes are reconstructed with less artifacts
+than the oral cavity, but we observed that even a little
+amount of image artifacts can evoke the Uncanny Val-
+ley Effect, as can be seen especially in Fig. 5, row B,
+column 2. Please zoom in for details.
+In our experiments we used PyTorch 1.8 and Python
+3.7 on Windows 10. We trained the same data set with
+approximately 1600 elements (an element is a set com-
+urn:nbn:de:0009-6-348, ISSN 1860-2037
+
+a
+b)Journal of Virtual Reality and Broadcasting, Volume n(200n), no. n
+prised of an RGBD image and an FLM) on two differ-
+ent machines. This took 8 hours on a system equipped
+with an Nvidia RTX2080Ti, an AMD Ryzen Thread-
+ripper 2990WX and 128GB RAM. With a newer sys-
+tem, comprised of an Nvidia RTX3090, an Intel i7-
+9900K and 32GB RAM, training took only 4 hours
+and 13 minutes. Our previous system took 17-20 hours
+on the machine with the RTX2080Ti for only 600 ele-
+ments.
+The time required for a forward pass of the gener-
+ator module with an image size of 512 × 512 pixels
+is between 3 and 4 ms (333 - 250 fps) on an Nvidia
+RTX3090 and between 6 and 7 ms (167 - 143 fps)
+on an Nvidia RTX 2080. Our previous system was
+faster (between 1 and 3 ms) but generates only images
+with a size of 256 × 256 pixels. The timings of the
+present system are still suitable for VR-based appli-
+cation, where 75 to 120 fps are common frame rates.
+However, note that many face tracking systems only
+work with 60 or even 30 fps, which can limit the frame
+rate of the pipeline.
+5
+Limitations and Future Work
+As mentioned before, one of the major issues is the
+low reconstruction quality of exaggerated expressions
+of the eyes, lips and oral cavity. The artifacts can in-
+duce the Uncanny Valley Effect and must be avoided
+for telepresence or broadcasting applications.
+As a
+further step, we plan to use a 3DMM [BV99] such
+as the FLAME model [LBB+17] as a better inductive
+bias to regularize depth and color information more
+efficiently. Furthermore, we observed that landmark
+tracking is not sufficient for faithful lip movement dur-
+ing speech. Therefore, an additional input signal be-
+sides the landmarks is necessary. A conditioning of
+speech as audio signals could provide a solution. An-
+other issue to improve is the uncomfortable helmet
+mount. A solution with a stationary RGBD camera
+placed on a tripod or table is a favorable approach for
+future research.
+6
+Conclusion
+We presented an improved end-to-end pipeline com-
+pared to previous approaches [LPG20, LPDG20] and
+a new GAN architecture that can learn facial iden-
+tity and individual expressions of a user and repro-
+duce them as a textured point cloud with frame rates
+that are suitable for Virtual Reality, telepresence and
+broadcasting environments.
+We have incorporated
+and extended the architecture, losses, and process-
+ing pipelines of several approaches from the field of
+neural rendering. Compared to previous works, our
+proposed system generates higher quality image re-
+sults with slightly longer run time at inference. We
+achieved this goal by mainly changing the architec-
+ture of the discriminator while keeping the architec-
+ture of the generator lean. The reconstruction results
+partially lie in the Uncanny Valley, but they still con-
+vince with an authentic visualization of the respective
+person’s identity and individual facial expressions. We
+believe that neural rendering will be a crucial part of
+photo-realistic rendering of humans in real-time appli-
+cations in the future. Our work is a further step into
+this direction and hopefully helps to understand, im-
+prove and apply this technology.
+7
+Acknowledgments
+We thank the MIREVI group at the University of Ap-
+plied Sciences D¨usseldorf and the ’Promotionszen-
+trum Angewandte Informatik’ (PZAI) in Hessen, Ger-
+many. This work is sponsored by the German Fed-
+eral Ministry of Education and Research (BMBF)
+under the project numbers 16SV8182 (HIVE-Lab),
+13FH022IX6 (iKPT 4.0) and 16SV8756 (AniBot).
+References
+[alt22]
+AltspaceVR,
+https://altvr.
+com/, 2022, Accessed: 2022-03-04.
+[BT17]
+Adrian Bulat and Georgios Tzimiropou-
+los, How Far are We from Solving the
+2D and 3D Face Alignment Problem?,
+2017 IEEE International Conference on
+Computer Vision (ICCV) (2017).
+[BV99]
+Volker Blanz and Thomas Vetter, A
+Morphable Model for the Synthesis of
+3D Faces, Proceedings of the 26th An-
+nual Conference on Computer Graph-
+ics and Interactive Techniques, SIG-
+GRAPH, 1999.
+[CFA+16]
+Dan Casas, Andrew Feng, Oleg Alexan-
+der,
+Graham Fyffe,
+Paul Debevec,
+Ryosuke Ichikari, Hao Li, Kyle Ol-
+szewski, Evan Suma, and Ari Shapiro,
+urn:nbn:de:0009-6-348, ISSN 1860-2037
+
+Journal of Virtual Reality and Broadcasting, Volume n(200n), no. n
+Figure 5: Results 1/2: This overview shows FLMs in column 1 from our evaluation datasets, hence the results
+are based on unseen data for the neural network. The FLMs were created from the images in column 3 – a
+real image from the evaluation data set. Column 2 shows the results generated by our GAN. The GAN received
+the FLM from column 1 and generated the images in column 2. Column 4 depicts the SSIM difference. Darker
+values indicate larger differences between the images in columns 2 and 3. Column 6 visualizes the error
+between the generated depth and the ground truth depth. The combination of the generated depth and color
+data can be seen in columns 7 and 8 from an angle of 30 and 90 degrees.
+urn:nbn:de:0009-6-348, ISSN 1860-2037
+
+Color
+Depth
+Color and
+Depth
+Results of me
+Depth differences
+SSIM from
+RGB output
+RGB ground
+trics on columns
+ between ground truth
+Renderings of textured
+FLM
+of our GAN
+truth
+columns 2 and 3
+2 and 3
+and GAN depth
+depth map
+2
+3
+8
+1
+4
+5
+6
+7
+12mm
+10 mm
+SSIM: 0.906
+(higher is better)
+8 mm
+LPIPS: 0.086
+A
+(lower is better)
+6 mm
+4mm
+2 mm
+0 mm
+12mm
+SSIM: 0.903
+(higher is better)
+8 mm
+LPIPS: 0.076
+B
+(lower is better)
+4 mm
+2 mm
+12 mm
+10mm
+SSIM: 0.923
+(higher is better)
+8 mm
+LPIPS: 0.068
+C
+(lower is better)
+Best result in
+4 mm
+data set
+12 mm
+10 mm
+SSIM: 0.836
+(higher is better)
+8 mm
+LPIPS: 0.146
+D
+6 mm
+(lower is better)
+4 mm
+Worst result in
+data set
+2 mm
+12 mm
+10 mm
+SSIM: 0.924
+(higher is better)
+LPIPS: 0.068
+E
+6mm
+(lower is better)
+4 mm
+2 mm
+12mm
+0mm
+SSIM: 0.909
+(higher is better)
+LPIPS: 0.086
+F
+(lower is better)
+ 0mm
+10mm
+SSIM: 0.933
+(higher is better)
+LPIPS: 0.067
+G
+6 mm
+(lower is better)
+4 mm
+Best result in
+data set
+0 mm
+12 mm
+10 mm
+SSIM: 0.882
+(higher is better)
+6 mm
+H
+LPIPS: 0.110
+(lower is better)
+4 mm
+Worst result in
+2 mm
+data setJournal of Virtual Reality and Broadcasting, Volume n(200n), no. n
+Figure 6: Results 2/2: Two further subjects are shown. The two last samples of each person summarizes the
+best and the worst images (measured by SSIM and LPIPS) and reflects the range of the reconstruction quality.
+urn:nbn:de:0009-6-348, ISSN 1860-2037
+
+Color
+Depth
+Color and Depth
+ Results of me
+Depth differences
+RGB output
+RGB ground
+sSIM from
+between ground truth
+ Renderings of textured
+trics on columns
+ depth map
+FLM
+of our GAN
+truth
+columns 2 and 3
+2 and 3
+and GAN depth
+3
+2
+5
+6
+4
+7
+8
+0m
+12m
+LPIPS: 0.082
+B
+(lower is better)
+C
+D
+8mm
+LPIPS: 0.180
+E
+(lower is better)
+G
+SSIM ; 0.92
+H
+0mr
+mm
+mm
+werisb
+4mm
+10mm
+6mm
+LPIPS: 0.120Journal of Virtual Reality and Broadcasting, Volume n(200n), no. n
+Rapid Photorealistic Blendshape Mod-
+eling from RGB-D Sensors, Computer
+Animation and Social Agents, CASA
+’16, 2016.
+[FSK17]
+Christian Frueh,
+Avneesh Sud,
+and
+Vivek Kwatra, Headset Removal for
+Virtual and Mixed Reality, ACM SIG-
+GRAPH, 2017.
+[GMC+08]
+Ralph Gross, Iain Matthews, Jeffrey
+Cohn, Takeo Kanade, and Simon Baker,
+Multi-pie, 2008 8th IEEE International
+Conference on Automatic Face Gesture
+Recognition, 2008.
+[GPAM+14] Ian Goodfellow, Jean Pouget-Abadie,
+Mehdi Mirza, Bing Xu, David Warde-
+Farley, Sherjil Ozair, Aaron Courville,
+and Yoshua Bengio, Generative adver-
+sarial nets, Advances in Neural In-
+formation Processing Systems (NIPS),
+2014.
+[GPL+21]
+Philip-William Grassal,
+Malte Prin-
+zler, Titus Leistner, Carsten Rother,
+Matthias Nießner, and Justus Thies,
+Neural head avatars from monocular
+RGB videos, 2021, Accessed:
+2022-
+03-21 from https://arxiv.org/
+abs/2112.01554.
+[IZZE17]
+P. Isola, J. Zhu, T. Zhou, and A. A.
+Efros, Image-to-Image Translation with
+Conditional
+Adversarial
+Networks,
+IEEE Conference on Computer Vision
+and Pattern Recognition (CVPR), 2017.
+[JAFF16]
+Justin Johnson, Alexandre Alahi, and
+Li Fei-Fei, Perceptual losses for real-
+time style transfer and super-resolution,
+ECCV, 2016.
+[JH19]
+Greg Walters John Hany, Hands-On
+Generative Adversarial Networks with
+PyTorch 1.x,
+Packt Publishing Ltd.,
+2019.
+[KALL17]
+Tero Karras, Timo Aila, Samuli Laine,
+and
+Jaakko
+Lehtinen,
+Progressive
+Growing of GANs for Improved Quality,
+Stability,
+and Variation,
+2017,
+Ac-
+cessed:
+2022-07-07 from https:
+//arxiv.org/abs/1710.10196.
+[KLA18]
+Tero Karras, Samuli Laine, and Timo
+Aila,
+A Style-Based Generator Ar-
+chitecture for Generative Adversarial
+Networks,
+2018,
+Accessed:
+2022-
+07-07 from https://arxiv.org/
+abs/1812.04948.
+[LBB+17]
+Tianye Li, Timo Bolkart, Michael. J.
+Black, Hao Li, and Javier Romero,
+Learning a model of facial shape and
+expression from 4D scans, ACM Trans-
+actions on Graphics (2017).
+[LG18]
+Philipp Ladwig and Christian Geiger,
+A Literature Review on Collabora-
+tion in Mixed Reality, Smart Indus-
+try and Smart Education, 15th Interna-
+tional Conference on Remote Engineer-
+ing and Virtual Instrumentation (REV
+’18), Springer International Publishing,
+2018.
+[LPDG20]
+P. Ladwig, A. Pech, R. Dorner, and
+C. Geiger, Unmasking Communication
+Partners: A Low-Cost AI Solution for
+Digitally Removing Head-Mounted Dis-
+plays in VR-Based Telepresence, 2020
+IEEE International Conference on Ar-
+tificial Intelligence and Virtual Real-
+ity (AIVR) (Los Alamitos, CA, USA),
+IEEE Computer Society, 2020.
+[LPG20]
+Philipp Ladwig, Alexander Pech, and
+Christian Geiger,
+Auf dem Weg zu
+Face-to-Face-Telepr¨asenzanwendungen
+in Virtual Reality mit generativen neu-
+ronalen Netzen, GI VR / AR Workshop
+(Benjamin Weyers, Christoph L¨urig,
+and Daniel Zielasko, eds.), Gesellschaft
+f¨ur Informatik e.V., 2020.
+[LSSS18]
+Stephen
+Lombardi,
+Jason
+Saragih,
+Tomas
+Simon,
+and
+Yaser
+Sheikh,
+Deep Appearance Models for Face
+Rendering, ACM Trans. Graph., 2018.
+[LTO+15]
+Hao Li, Laura Trutoiu, Kyle Olszewski,
+Lingyu Wei, Tristan Trutna, Pei-Lun
+Hsieh, Aaron Nicholls, and Chongyang
+Ma, Facial Performance Sensing Head-
+Mounted Display, ACM Trans. Graph.,
+2015.
+urn:nbn:de:0009-6-348, ISSN 1860-2037
+
+Journal of Virtual Reality and Broadcasting, Volume n(200n), no. n
+[met22]
+Meta
+Horizon
+Worlds,
+https:
+//www.oculus.com/
+horizon-worlds/?locale=
+de_DE, 2022, Accessed: 2022-03-04.
+[MLX+17]
+X. Mao, Q. Li, H. Xie, R. K. Lau,
+Z. Wang, and S. Smolley, Least squares
+generative adversarial networks, 2017
+IEEE International Conference on Com-
+puter Vision (ICCV) (Los Alamitos,
+CA, USA), 2017.
+[MO14]
+Mehdi
+Mirza
+and
+Simon
+Osin-
+dero,
+Conditional
+Generative
+Adversarial
+Nets,
+2014,
+Ac-
+cessed:
+2020-07-07 from https:
+//arxiv.org/abs/1411.1784.
+[OLSL16]
+Kyle Olszewski, Joseph J. Lim, Shun-
+suke Saito, and Hao Li, High-Fidelity
+Facial and Speech Animation for VR
+HMDs, ACM Trans. Graph. (2016).
+[pup18]
+Locating eye centers using means of
+gradients, https://github.com/
+jonnedtc/PupilDetector, 2018,
+Accessed: 2021-12-28.
+[RCV+19]
+A. R¨ossler, D. Cozzolino, L. Verdo-
+liva, C. Riess, J. Thies, and M. Niess-
+ner, FaceForensics++: Learning to De-
+tect Manipulated Facial Images, 2019
+IEEE/CVF International Conference on
+Computer Vision (ICCV), 2019.
+[RFB15]
+Olaf Ronneberger, Philipp Fischer, and
+Thomas Brox, U-Net:
+Convolutional
+Networks for Biomedical Image Seg-
+mentation, Medical Image Computing
+and Computer - Assisted Intervention -
+MICCAI, 2015.
+[RMC15]
+Alec Radford, Luke Metz, and Soumith
+Chintala, Unsupervised Representation
+Learning with Deep Convolutional Gen-
+erative Adversarial Networks,
+2015,
+Accessed: 2022-07-07 from https:
+//arxiv.org/abs/1511.06434.
+[RZS+21]
+Amit Raj, Michael Zollhofer, Tomas Si-
+mon, Jason Saragih, Shunsuke Saito,
+James Hays, and Stephen Lombardi,
+Pixel-aligned volumetric avatars, Pro-
+ceedings of the IEEE/CVF Conference
+on Computer Vision and Pattern Recog-
+nition (CVPR), 2021.
+[TB11]
+Fabian Timm and Erhardt Barth, Accu-
+rate eye centre localisation by means of
+gradients, Computer Vision Theory and
+Applications (VISAPP), 2011.
+[TZS+18]
+Justus Thies, Michael Zollh¨ofer, Marc
+Stamminger, Christian Theobalt, and
+Matthias Nießner, FaceVR: Real-Time
+Gaze-Aware Facial Reenactment in Vir-
+tual Reality, ACM Trans. Graph. (2018).
+[vrc22]
+VRChat,
+https://hello.
+vrchat.com/,
+2022,
+Accessed:
+2022-03-04.
+[WLZ+17]
+Ting-Chun Wang, Ming-Yu Liu, Jun-
+Yan Zhu, Andrew Tao, Jan Kautz, and
+Bryan Catanzaro, High-Resolution Im-
+age Synthesis and Semantic Manipu-
+lation with Conditional GANs, CVPR,
+2017.
+[WSS+19]
+Shih-En Wei, Jason Saragih, Tomas Si-
+mon, Adam W. Harley, Stephen Lom-
+bardi, Michal Perdoch, Alexander Hy-
+pes,
+Dawei Wang,
+Hernan Badino,
+and Yaser Sheikh, Vr facial animation
+via multiview image translation, ACM
+Trans. Graph. (2019).
+[WZX+16]
+Jiajun Wu, Chengkai Zhang, Tianfan
+Xue, William T Freeman, and Joshua B
+Tenenbaum, Learning a Probabilistic
+Latent Space of Object Shapes via 3D
+Generative-Adversarial Modeling, Ad-
+vances in Neural Information Process-
+ing Systems, 2016.
+[ZBSS04]
+Zhou Wang, A. C. Bovik, H. R. Sheikh,
+and E. P. Simoncelli, Image quality as-
+sessment: from error visibility to struc-
+tural similarity, IEEE Transactions on
+Image Processing (2004).
+[ZIE+18]
+Richard Zhang, Phillip Isola, Alexei A
+Efros, Eli Shechtman, and Oliver Wang,
+The unreasonable effectiveness of deep
+features as a perceptual metric, CVPR,
+2018.
+urn:nbn:de:0009-6-348, ISSN 1860-2037
+

DNE0T4oBgHgl3EQfQQC5/content/tmp_files/2301.02191v1.pdf.txt

@@ -0,0 +1,1849 @@
+Physics informed neural network for charged
+particles surrounded by conductive boundaries
+A Preprint
+Fatemeh Hafezianzade
+Department of Physics
+Institute for Advanced Studies in Basic Sciences
+Zanjan, 45195-1159, Iran
+fahafe98@gmail.com
+Morad Biagooi
+Intelligent Data Aim Ltd (IDA Ltd)
+Science and Technology Park of Institute for Advanced studies in Basic Sciences
+Zanjan 45137-65697, Iran
+SeyedEhsan Nedaaee Oskoee∗
+Department of Physics
+Institute for Advanced Studies in Basic Sciences
+Zanjan, 45195-1159, Iran
+nedaaee@iasbs.ac.ir
+January 6, 2023
+Abstract
+In this paper, we developed a new PINN-based model to predict the potential of point-charged
+particles surrounded by conductive walls. As a result of the proposed physics-informed neural
+network model, the mean square error and R2 score are less than 7% and more than 90%
+for the corresponding example simulation, respectively. Results have been compared with
+typical neural networks and random forest as a standard machine learning algorithm. The
+R2 score of the random forest model was 70%, and a standard neural network could not be
+trained well. Besides, computing time is significantly reduced compared to the finite element
+solver.
+Keywords Poisson · Laplace · Physics-informed neural network · charged particles · Conductive boundaries ·
+supercapacitor
+1
+Introduction
+Computational Electromagnetic Simulation (CES) plays a significant role in many areas of science and
+engineering, such as soft matter, electrical engineering, biomedical engineering and chemistry. In addition, it
+has numerous applications in industry. For example, it is one of the main tools in investigating and designing
+the process of supercapacitors, which are porous energy storage devices with many applications in industry,
+especially when high power consumption or transfer is neededMiller and Simon [2008]. Here, studying the
+physical mechanisms arising from charge storage in supercapacitors is essential for further technological
+developmentSalanne et al. [2016], Simon and Gogotsi [2008].
+∗Corresponding author
+arXiv:2301.02191v1  [physics.comp-ph]  5 Jan 2023
+
+arXiv Template
+A Preprint
+Solving Maxwell’s equation, especially the Poisson equation in this study, is an essential part of computational
+electromagnetic algorithmsJackson [1962]. Solving the Poisson equation can help scientists to calculate the
+potential of electrical sources in any system. However, many difficulties arise in practice due to the long-range
+nature of electrical interactions. In particular, estimating the potential of point-charged components in an
+environment with conductive walls is challenging because of the induced charges presented on the boundaries.
+Generally, there are two approaches to solving the Poisson equation: analytical solutionJackson [1962]
+and numerical methods. There are limited techniques for solving analytically, like image charges methods
+applicable for cases with regular geometries; however, there is no guarantee to achieve practical results. If,
+for example, a particle is placed in a cubic conductive container, the image charges method will produce an
+infinite series. On the other hand, numerical methods lead to approximate solutions based on discretizing
+space and/or time domains. One of the typical numerical methods is the Finite Element Method (FEM)Jin
+[2014] which discretizes the continuous partial differential equations (PDEs) and forms a linear set of algebraic
+equationsS. et al. [1991]. Nevertheless, even FEM fails in calculating the potential in a charged particle’s
+position since the electrical potential is singular at the place of charges. There are a number of methods
+and algorithms that have been developed to address this problem, including Induced Charge MMM2D
+(ICMMM2D)Tyagi et al. [2007] for 2D, ELCICTyagi et al. [2008] for 2D + h, Induced Charge Computation
+(ICC∗ )Tyagi et al. [2010], Kesselheim et al. [2010], Arnold et al. [2013] for 3D periodicity, and a method
+introduced by Reed et al.Reed et al. [2007] have been developed. In addition, recently, there has been
+another algorithm named PLT. It was first demonstrated for a partially periodic system constrained between
+two metallic plates in Rostami et al. [2016], and then it was applied to CAVIAR Biagooi et al. [2020], a
+molecular dynamics simulation package for charged particles surrounded by non-trivial conductive boundaries.
+Numerical solving of these problems with the CAVIAR package is accurate; moreover, it took less time than
+ICC∗ Biagooi et al. [2020] but is still time and memory-consuming.
+Recently another data-driven approach to solving the PDEs based on deep machine learning is also of great
+current interest. For instance, Shan et al. Shan et al. [2020] present a CNN to predict the electric potential
+with different excitations and permittivity distribution in 2D and 3D models. It is fast and efficient compared
+with FEMJin [2014]. However, a couple of problems prevent it from utilizing as a Poisson solver in the MD
+simulation process; first, it could not work in the case of discrete density functions such as those of point
+charges, and second, it is a physics-free approach which makes it hard to consider boundary conditions.
+To overcome the first problem, one can use the PLT algorithm. Additionally, Raissi et al. introduced the
+physics-informed neural network (PINN) that the loss function defined by Raissi et al. [2019] is an excellent
+alternative to the conventional deep learning method because of the governing equations, boundary conditions,
+and initial conditions used in its definition.
+In this paper, we applied a new PINN-based model to predict the potential of point-charged particles
+surrounded by conductive walls. We then compared the results with typical neural networks and random
+forests as a standard machine learning algorithm. For instance, we tried to implement these models for a
+charged particle in a spherical container. The reason for utilizing this simple example was that there is an
+exact solution to this problem through the analytical method, the image charges method. As a starting point,
+we used the PLT algorithm to transfer the Poisson equation into the Laplace equation with modified boundary
+conditions. Then we trained the model to solve the Laplacian equation with new boundary conditions. The
+input data is included the position in which we want to evaluate the potential on it and the modified boundary
+conditions; the output data is the corresponding electrical potential of that position.
+2
+Methods
+This article aims to build a machine-learning model (ML-Model) to predict the potential of point-charged
+particles surrounded by conductive walls. The potential of charged particles is calculated by solving the
+Poisson equation, which can be written as Jackson [1962]:
+∇2φ = −ρ/ϵ0 = −
+N
+�
+i=1
+qiδ(x − xqi)/ϵ0,
+(1)
+where φ is the potential and ρ is a charge distribution. The first and straightforward ML-Model that jumps
+to mind is a model that includes xq and x as an input, and φ(xq, x) as an output. Here xq is the position of
+point-charged particle, x is the position in which we want to calculate the potential on it, and φ(xq, x) is the
+corresponding potential. So the number of input features depends on the number of charged particles; for
+instance, in 3 dimensions, if there is N charged particles, the input features have to be 3 + 3 × N. Therefore,
+2
+
+arXiv Template
+A Preprint
+this kind of model could only predict the potential of fix number of charged particles. In many applications of
+this method, such as the molecular dynamic simulation, this number is not fixed and could even increase or
+decrease during the simulation. We use the PLT algorithm to transpose the Poisson equation into the Laplace
+equation with new boundary conditions to overcome this problem. This algorithm will be discussed in more
+detail in 2.1. So we can train a model which includes x and modified boundary conditions as input features
+and φ(φb, x) as an output. We define the boundary conditions only on Nb fixed points on the boundary
+{φ1, φ2, . . . , φNb}. In this case, with the PLT algorithm, we can build a model with a fixed number of input
+features that can predict any charged particles’ potential.
+Creating reference data set with PLT algorithm 
+ 
+Train Set
+Test Set
+Training the models: 
+RF 
+ANN 
+PINN
+Testing and hyperparameter tuning 
+all models 
+Chosing the best model
+Spliting reference set into  
+the train and test set
+Figure 1: Methodology flow chart, The blue part: Preparing the data, in which the reference data set is
+created based on the PLT algorithm. The red part: Training models process, first the reference set is split
+to train and test set, then RF, ANN, and PINN model were applied on the train set, after tuning the
+hyperparameters the best model were chose.
+2.1
+Poisson to Laplace Transformation (PLT)
+According to the PLT algorithm, the electrical potential is divided into two parts: singular potential (φsi)
+and smooth potential (φsm); φ(⃗x) = φsi(⃗x) + φsm(⃗x). It is important to note that the smooth part here is
+the solution of the Laplace equation with modified boundary conditions,
+∇2φsm(⃗x) = 0,
+(2)
+3
+
+formula_4_flowcharts - Word
+fatemeh hafezian
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+AaBbCcD AaBbC( AaBbCc /
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+↑ Normal
+I No Spac...
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+T - {x ya,za, {1,P2...,
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+8/27/2022arXiv Template
+A Preprint
+while φsi obeys the famous Columb laws
+φsi(⃗x) =
+N
+�
+i=1
+qi
+4πϵ0∥⃗x − ⃗xi∥.
+(3)
+It can be seen that the modified boundary condition for φsm is represented by
+φsm∥⃗xbc = φ∥⃗xbc − φsi∥⃗xbc,
+(4)
+where φ∥⃗xbc corresponds to the initial electrical potential on the boundaries. Finally, with the PLT algorithm,
+we could transfer the Poisson to the Laplace equation with new modified boundary conditions, then train an
+ML-Model with these modified boundary conditions as an input parameter and the smooth potential as an
+output. Afterward with the summation of singular and predicted smooth potential, we can reach the total
+potential. Advantage of utilizing PLT algorithm is that it leads to having a fixed number of input data since
+the number of input data would be independent of the number of point-charged particles.
+(a)
+65
+70
+75
+80
+85
+90
+95
+100
+sm
+(b)
+65
+70
+75
+80
+85
+90
+95
+100
+sm
+(c)
+Figure 2: Schematic of the PLT method: a) the main system which had point charges inside of it b) the new
+system without any point charges and the boundaries were modified c) Nb points on the boundary are shown
+to be used as our model input.
+2.2
+Data Engineering
+For training a highly accurate model, having a nice train set is crucial. In this work, the reference set is
+Γ =
+�
+xi, yi, zi, ⃗
+ϕi
+bc, φi�N
+i=1, where the input is concluded {x, y, z, ⃗ϕbc = {φ1, φ2, ..., φNb}} and φ is the target.
+x, y, z is the coordinate of a point in the container on which we want to calculate their potential, φ is the
+numeric value of the potential at this point, and {φ1, φ2, ..., φNb} is the boundary condition on Nb points on
+the boundary. First, in the container, Np positions are chosen to predict the potential in their situations.
+In fact, for each boundary condition, {φ1, φ2, ..., φNb}, there are Np points that we want to calculate the
+potential at their positions. Then, the reference set could be created for Nq different boundary conditions.
+So the reference set consists of N = Nq × Np samples which could split to train and test set. In this case, our
+container is a sphere, we also set Np = 78, Nb = 26, and Nq = 100. So, our reference consists of 100 different
+boundary conditions and for each boundary condition {φ1, φ2, ..., φ26} there are 78 points in the sphere on
+which we want to calculate the potential on it. We use the solution of the image charges method (Eq.5) to
+calculate targets of the reference set:
+φ(⃗x) =
+1
+4πϵ0
+{
+q
+∥⃗x − ⃗xq∥ +
+q′
+∥⃗x − ⃗
+xq′∥},
+q′ = −rq
+a ,
+⃗
+xq′ = a2
+r
+⃗xq
+∥⃗xq∥,
+(5)
+where a is the conductive spherical shell radius and r is the distance of a point charge q from its center.
+The numeric value of potential is minimal (∼ 10−9), which conducts to significant rounding error during
+computation; therefore, the potential of an electron in a 1m distance of it, 1.44 × 10−9[V ], is used as a unit
+4
+
+ΦtotalarXiv Template
+A Preprint
+to make equation 5 dimensionless. We randomly chose 5000 and 1000 samples from the reference set to
+create a train and test set. The train and test set have no samples in common. In addition, the best model
+could adequately predict the potential of test samples and samples with different boundary conditions from
+the train and test set. So to evaluate the model better, we prepare an extrapolation set that includes 1000
+samples with 55 distinct boundary conditions.
+2.3
+ML Algorithms
+In this work, three different supervised learning methods have been used, and their regression accuracy,
+based on the metrics presented in 2.4, has been evaluated. Mainly, we stick to Physics-Informed Neural
+Networks (PINN, Raissi et al. [2019]), but to compare our results with other ML algorithms, we use Random
+Forest (RF, Breiman [2001]) and Artificial Neural Networks (ANN). All the models are briefly introduced,
+the hyperparameters are fine-tuned, and their performance is reported. Scikit-learn Pedregosa et al. [2011],
+Tensorflow Abadi [2016], Keras Chollet [2015], and NumPy Walt et al. [2011] are all the python libraries that
+have been used in this project.
+2.3.1
+Random Forest(RF)
+RF is one of the most popular machine learning algorithms in regression problems for many reasons, but in
+this project, this model has been chosen since
+a) It is speedy to learn.
+b) It is robust against over-fitting.
+over-fitting is detected when the performance of train samples is perfect while the performance of test samples
+is poor. RF is an ensemble model in which an average of many uncorrelated trees determines the predicted
+potential for the target data set. Although each tree is a weak learner, they make a strong learner when
+many trees are grouped. The RF randomizes the trees by choosing a subset of training data and features for
+each tree. Here we use scikit-learn Pedregosa et al. [2011] RF implementation.
+2.3.2
+ANN
+Typical neural network architecture consists of the input layer, multiple hidden layers, and the output layer
+with several neurons in each layer. Totally:
+• Input layer: The neurons in the input layer are the input features.
+• Hidden layers: The value of every neuron in the hidden layers is a linear combination of the neurons
+in the previous layer followed by the implementation of an activation function(Eq.6); in most cases,
+the activation function is non-linear.
+an = σl (an−1wn + bn) .
+(6)
+n is the layer number, w and b are the model parameters, weights and bias respectively, and σl is
+the activation function based on Goodfellow et al. [2016].
+• Output layer: The neurons in the output layer are the model targets and they are calculated with
+Eq.6 with linear activation function.
+• Loss function: There is a function in all neural networks that must be minimized over the model
+parameters during the training stage via back-propagation, typically the loss function is the mean
+square error between the true and the predicted values.
+Loss(w) = MSEd + λ
+�
+w
+w2,
+(7)
+MSEd = 1
+N
+N
+�
+i=1
+[U (Xi, w) − Ti]2 .
+(8)
+U and T are the predicted output and true target values, respectively, X is the input data, and w is the
+parameter of neural networks, weights, and biases. The first sentence in Eq.7 is a mean square error, and
+5
+
+arXiv Template
+A Preprint
+The second sentence exists to prevent over-fitting, namely L2 regularizationa. K. Connect et al. [1992], that
+is used in order to reduce the effects of the large weights.
+2.3.3
+PINN
+PINNRaissi et al. [2019] enforces the Laplace equation, a physical law of the electromagnetic system, as a
+constraint on the neural network. This study proposes a PINN-based approach to solve the Laplace equation
+with changeable boundary conditions. Fig.3 shows a schematic of the neural network layout for this approach.
+PINN-based models are neural networks with modified loss functions:
+Loss = λ1MSEd + λ2MSEf + λ3MSEb + λ4
+�
+w
+w2,
+(9)
+The first and the last term of Eq.9 are the same as typical neural networks in Eq.7. The second term
+corresponds to the governing physical equation, i.e., the is Laplace, and the third term corresponds to the
+boundary conditions;
+MSEd = 1
+Nd
+Nd
+�
+i=1
+∥u(xi
+d, ⃗ϕi
+d; w) − φi
+d∥2,
+(10)
+MSEf =
+1
+Nf
+Nf
+�
+i=1
+∥f(xi
+f, ui
+f; w)∥2,
+(11)
+and
+MSEb = 1
+Nb
+Nb
+�
+i=1
+∥B(xi
+b, ⃗ϕi
+b, ui
+b; w)∥2.
+(12)
+Here we define f (x, u; w)
+f (x, u; w) = 0,
+x ∈ Γf
+= ∇2u
+= ∂2u
+∂x2 + ∂2u
+∂y2 + ∂2u
+∂z2
+= ∂w
+∂x
+∂
+∂w
+�∂w
+∂x
+∂u
+∂w
+�
++ ∂w
+∂y
+∂
+∂w
+�∂w
+∂y
+∂u
+∂w
+�
++ ∂w
+∂z
+∂
+∂w
+�∂w
+∂z
+∂u
+∂w
+�
+,
+(13)
+with Dirichlet boundary conditions
+B (x, ⃗ϕ, u; w) = 0,
+x ∈ Γb
+=
+26
+�
+j=1
+(u(xj, ⃗ϕ; w) − ϕj
+b).
+(14)
+λ1, λ2, λ3 in Eq.9 correspond to the weight coefficients for the data contributions, Laplace equation, and
+boundary losses. We use the weight coefficient by motivating from the study of Kag et al. Kag et al. [2022].
+The last sentence is the L2 regularizationa. K. Connect et al. [1992]. Notice that the model with λ2 = λ3 = 0.0
+is exactly a typical neural network described in the previous subsection.
+2.4
+Evaluating Metrics
+The performance evaluation of different algorithms for potential estimation depends on different metrics,
+∆φ = φT rue − φP red,
+(15)
+σ =
+�
+�
+�
+� 1
+n
+n−1
+�
+i=0
+(∆φ)2,
+(16)
+6
+
+arXiv Template
+A Preprint
+𝜑
+ƶ𝑢
+I
+f(𝑥)
+𝜕2 ƶ𝑢
+𝜕𝑥2 +
+𝜕2 ƶ𝑢
+𝜕𝑦2 +
+𝜕2 ƶ𝑢
+𝜕𝑧2 - 0
+෍
+𝑗=1
+26
+ƶ𝑢(𝑥𝑗, 𝜑; 𝑤) − 𝜑𝑏
+𝑗)
+ƶ𝑢(𝑥, 𝜑; 𝑤) − Φ(𝑥)
+NN(𝑥; 𝜃)
+BC
+𝑑𝑎𝑡𝑎
+𝜕2
+𝜕𝑥2
+𝜕2
+𝜕𝑧2
+𝜕2
+𝜕𝑦2
+I
+Loss=
+𝝀𝟏𝑴𝑺𝑬𝒅 + 𝝀𝟐𝑴𝑺𝑬𝒇
++
+𝝀𝟑 𝑴𝑺𝑬𝒃+ 𝝀𝟒 σ𝒘 𝒘𝟐
+𝑋
+Figure 3: Physics-informed neural network scheme for solving Laplace equation with variable boundaries
+R2 = 1 −
+�n
+i=1 (φTrue − φPred )2
+�n
+i=1
+�
+φTrue − ¯φTrue
+�2 ,
+(17)
+MSE =< (∆φ)2 > .
+(18)
+Where φT rue is the true potential, φP red is the predicted potential, and ¯φT rue is the mean true potential of a
+given test sample. In this study we used scatter σ, R2 score and MSE as our evaluating metrics.
+3
+Result
+In this paper, we predict the smooth potential of a point-charged particle in a spherical conductive container.
+First, we set the train and test set with 5000 and 1000 samples (2.2), then we train our models to predict
+smooth potential. We can calculate total potential by summating smooth and singular potential (more
+detailed in 2.1). However, in this work, to compare our results with CAVIARBiagooi et al. [2020], we
+investigate the smooth potential.
+3.1
+Random Forest
+3.1.1
+hyperparameter for RF
+We optimize over the only hyperparameter, the number of trees in the forest that influences the fitting of the
+random forest model. In Fig.4, we plot MSE (the left panel) and R2 score (the right one) as a function of
+the number of trees for the test set to determine the optimal hyperparameter, which we find 100 trees since
+progress after 100 trees are negligible. Afterward, we trained the RF model using 100 trees.
+7
+
+arXiv Template
+A Preprint
+0
+50
+100
+150
+200
+250
+300
+Number of trees
+0.08
+0.10
+0.12
+0.14
+0.16
+Mean Square Error
+0
+50
+100
+150
+200
+250
+300
+Number of trees
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+R2 score
+Figure 4: MSE (left), and R2 score (right) for 1000 different sample of Test set
+3.1.2
+RF Prediction
+Fig.5 is illustrated the RF model with 100 trees. It shows that prediction is acceptable when the numeric
+value of potential is less than 0.3 (φT rue < 0.3) while it could not predict precisely in the case of φT rue >= 0.3.
+The graphs of Fig.5 compare the true potential φT rue and RF model predicted potential φRF for the train
+data set(left picture) and test data set(right image). The RF method is relatively fast; however, it works when
+the predicted potential is smooth and relatively small, it is not suitable in the case of point-charged particles
+(where a gradient of potential as well as its numeric value is high at the position of the charge). Furthermore,
+it fails to predict the potential near the boundaries since the gradient of the potential is considerable.
+0.5
+1.0
+1.5
+2.0
+2.5
+RF
+0.5
+1.0
+1.5
+2.0
+2.5
+True
+: 0.02
+Training data : 5000 targets
+0.02
+0.04
+0.06
+0.08
+0.10
+0.12
+0.14
+Density distribution
+(a)
+0.2
+0.3
+0.4
+0.5
+0.6
+RF
+0.2
+0.3
+0.4
+0.5
+0.6
+True
+: 0.06
+Testing data : 1000 targets
+0.05
+0.10
+0.15
+0.20
+0.25
+Density distribution
+(b)
+Figure 5: Potential estimation of RF model: a) show the train data set with 5000 samples with scatter of
+σ = 0.02, b) show the test data set with 1000 samples with scatter of σ = 0.07. The dashed red line shows
+where the predicted potential equals the true potential. The pink-shaded region marks 1σ scatter of potential
+errors.
+8
+
+arXiv Template
+A Preprint
+3.2
+PINN based model and NN
+By setting both λ2 and λ3 to zero in PINN-based model, one can get exactly NN model. Therefore we
+investigate both models together and report their reulsts at the same time in the following section.
+3.2.1
+hyperparameter for PINN and NN
+Unlike the RF model, we define several hyperparameters: a number of neurons, a number of layers, λ2, and
+λ3. To tune all hyperparameters, we train the model up to 100000 epochs, using the L-BFGS-B optimizerLiu
+and Nocedal [1989], until the model’s tolerance reaches the level of machine epsilon. For all layers except the
+last one, we use a tanh activation function. Table 1 is reported the MSE between the predicted and the same
+potential for a different value of hyperparameters; λ2 = [0, 0.1, 0.2, 0.3], λ3 = [0, 0.1, 0.2, 0.3, 0.4], number of
+hidden layers= [1, 3, 5, 7] and number of neurons per hidden layer= [10, 30, 50] for 1000 samples of the test
+set. As shown in Table 1, we observe that a model with one hidden layer could not predict the potential well.
+Also, a model with ten neurons per layer could not work well. So, Table 2 and Table 3 reported the results
+for just [3, 5, 7] layers as well as for [30, 50] neurons per layer. We chose λ4 = 0.0001 to prevent over-fitting.
+3.2.2
+PINN and NN Prediction
+In Contrast with Table 1, Table 2 is reported not only the MSE but also the R2score of the test set. As can
+be seen in Table 2, the model with seven layers and 50 neurons per layer resulted better when λ2 and λ3 are
+0.3, 0.3, or 0.2, 0.4, respectively. When λ2 and λ3 are zero, a standard neural network, the model has not
+worked well; it is observed from Tabel 2 and Fig.4.
+0.5
+1.0
+1.5
+2.0
+2.5
+ANN
+0.5
+1.0
+1.5
+2.0
+2.5
+True
+: 0.07
+Training data : 5000 targets
+0.2
+0.4
+0.6
+0.8
+1.0
+Density distribution
+(a)
+0.5
+1.0
+1.5
+2.0
+2.5
+PINN
+0.5
+1.0
+1.5
+2.0
+2.5
+True
+: 0.01
+Training data : 5000 targets
+0.01
+0.02
+0.03
+0.04
+0.05
+Density distribution
+(b)
+Figure 6: Potential estimation of best-tuned (a) NN with scatter of σ = 0.07 and (b) PINN model on the 5000
+samples of the train set with scatter of σ = 0.01. The dashed red line shows where the predicted potential
+equals the true potential. The pink-shaded region marks 1σ scatter of potential errors.
+Both plots in Fig.6 compare the true and predicted potentials for the best-tuned NN and the best-tuned
+PINN model with 7 layers and 50 neurons per layer, λ2 = 0.3 and λ3 = 0.3- on the train set. As can be seen,
+the NN model is not trained well, while the PINN-based model could predict the potential precisely with a
+scatter of 0.01. Although the PINN-based model predicts the train set well, aiming to clarify that over-fitting
+has not been accrued, we also evaluate the model on the test set, Fig.7.
+9
+
+arXiv Template
+A Preprint
+0.2
+0.3
+0.4
+0.5
+0.6
+PINN
+0.2
+0.3
+0.4
+0.5
+0.6
+True
+: 0.02
+Testing data : 1000 targets
+0.05
+0.10
+0.15
+0.20
+Density distribution
+Figure 7: Potential estimation of best-tuned PINN model on the 1000 samples of the test set with scatter
+of σ = 0.02, MSE = 0.069 and R2score = 0.851. The dashed red line shows where the predicted potential
+equals the true potential. The pink-shaded region marks 1σ scatter of potential errors.
+10
+
+arXiv Template
+A Preprint
+MSEtest
+MSEtest
+λ2
+λ3
+Neurons=10 Neurons=30
+Neurons=50
+Neurons=10
+Neurons=30
+Neurons=50
+Num of hidden layer:1
+Num of hidden layer:5
+0.0
+0.0
+0.246
+0.246
+0.246
+0.246
+0.246
+0.246
+0.1
+0.162
+0.145
+0.124
+0.099
+0.145
+0.078
+0.2
+0.191
+0.212
+0.132
+0.136
+0.062
+0.072
+0.3
+0.175
+0.159
+0.171
+0.14
+0.069
+0.124
+0.4
+0.24
+0.171
+0.18
+0.191
+0.064
+0.076
+0.1
+0.0
+0.246
+0.247
+0.247
+0.246
+0.246
+0.246
+0.1
+0.222
+0.143
+0.117
+0.107
+0.245
+0.192
+0.2
+0.201
+0.165
+0.129
+0.243
+0.079
+0.062
+0.3
+0.22
+0.159
+0.16
+0.112
+0.094
+0.071
+0.4
+0.249
+0.177
+0.243
+0.205
+0.088
+0.07
+0.2
+0.0
+0.246
+0.246
+0.246
+0.246
+0.246
+0.246
+0.1
+0.158
+0.144
+0.154
+0.235
+0.161
+0.083
+0.2
+0.225
+0.174
+0.174
+0.198
+0.097
+0.071
+0.3
+0.223
+0.143
+0.158
+0.192
+0.081
+0.075
+0.4
+0.197
+0.185
+0.189
+0.216
+0.065
+0.118
+0.3
+0.0
+0.246
+0.246
+0.246
+0.246
+0.246
+0.246
+0.1
+0.21
+0.132
+0.129
+0.245
+0.088
+0.098
+0.2
+0.199
+0.139
+0.157
+0.209
+0.117
+0.065
+0.3
+0.216
+0.241
+0.202
+0.182
+0.168
+0.08
+0.4
+0.225
+0.182
+0.196
+0.239
+0.083
+0.097
+Num of hidden layer:3
+Num of hidden layer:7
+0.0
+0.0
+0.246
+0.246
+0.246
+0.246
+0.246
+0.246
+0.1
+0.157
+0.104
+0.086
+0.244
+0.237
+0.247
+0.2
+0.151
+0.088
+0.086
+0.238
+0.087
+0.105
+0.3
+0.13
+0.078
+0.089
+0.244
+0.07
+0.067
+0.4
+0.202
+0.173
+0.102
+0.188
+0.068
+0.063
+0.1
+0.0
+0.246
+0.246
+0.246
+0.246
+0.246
+0.246
+0.1
+0.099
+0.089
+0.132
+0.244
+0.244
+0.244
+0.2
+0.198
+0.093
+0.077
+0.23
+0.221
+0.069
+0.3
+0.188
+0.088
+0.083
+0.224
+0.223
+0.082
+0.4
+0.262
+0.104
+0.079
+0.213
+0.072
+0.074
+0.2
+0.0
+0.246
+0.246
+0.246
+0.246
+0.246
+0.246
+0.1
+0.119
+0.146
+0.082
+0.244
+0.244
+0.245
+0.2
+0.172
+0.09
+0.086
+0.244
+0.244
+0.071
+0.3
+0.179
+0.084
+0.085
+0.225
+0.251
+0.08
+0.4
+0.197
+0.185
+0.086
+0.256
+0.166
+0.067
+0.3
+0.0
+0.246
+0.246
+0.246
+0.246
+0.246
+0.246
+0.1
+0.17
+0.125
+0.082
+0.244
+0.244
+0.244
+0.2
+0.199
+0.117
+0.096
+0.244
+0.083
+0.08
+0.3
+0.224
+0.107
+0.088
+0.202
+0.235
+0.069
+0.4
+0.199
+0.222
+0.152
+0.236
+0.097
+0.075
+Table 1: MSE between the predicted and the exact potential φ(x) for a different value of λ2, and λ3, and
+the different number of hidden layers, and neurons per hidden layer in PINN for 1000 different sample of
+the Test set. Here, λ4 = 0.0001 is fixed and λ1 = 1 − (λ2 + λ3 + λ4). In this table, the bold number means
+MSE < 0.1.
+11
+
+arXiv Template
+A Preprint
+Numlayer=3
+Numlayer=5
+Numlayer=7
+λ2
+λ3
+MSE
+R2
+MSE
+R2
+MSE
+R2
+Numneuron=30
+0.0
+0.0
+0.246
+0
+0.246
+0
+0.246
+0
+0.1
+0.104
+0.774
+0.145
+0.365
+0.237
+-14.7
+0.2
+0.088
+0.85
+0.062
+0.889
+0.087
+0.807
+0.3
+0.078
+0.826
+0.069
+0.884
+0.07
+0.904
+0.4
+0.173
+0.551
+0.064
+0.863
+0.068
+0.868
+0.1
+0.0
+0.246
+0
+0.246
+0
+0.246
+0
+0.1
+0.089
+0.796
+0.245
+-1478
+0.244
+0
+0.2
+0.093
+0.825
+0.079
+0.852
+0.221
+-2.40
+0.3
+0.088
+0.793
+0.094
+0.714
+0.223
+0.094
+0.4
+0.104
+0.719
+0.088
+0.793
+0.072
+0.887
+0.2
+0.0
+0.246
+0
+0.246
+0
+0.246
+0
+0.1
+0.146
+0.64
+0.161
+0.566
+0.244
+0
+0.2
+0.09
+0.769
+0.097
+0.75
+0.244
+-3167
+0.3
+0.084
+0.819
+0.081
+0.857
+0.251
+-2.32
+0.4
+0.185
+0.521
+0.065
+0.874
+0.166
+0.07
+0.3
+0.0
+0.246
+0
+0.246
+0
+0.246
+0
+0.1
+0.125
+0.626
+0.088
+0.747
+0.244
+0
+0.2
+0.117
+0.666
+0.117
+0.689
+0.083
+0.863
+0.3
+0.107
+0.67
+0.168
+0.59
+0.235
+0
+0.4
+0.222
+0.25
+0.083
+0.837
+0.097
+0.804
+Numneuron=50
+0.0
+0.0
+0.246
+0
+0.246
+0
+0.246
+0
+0.1
+0.086
+0.832
+0.078
+0.865
+0.247
+-254
+0.2
+0.086
+0.838
+0.072
+0.873
+0.105
+0.711
+0.3
+0.089
+0.767
+0.124
+0.66
+0.067
+0.875
+0.4
+0.102
+0.734
+0.076
+0.846
+0.063
+0.849
+0.1
+0.0
+0.246
+0
+0.246
+0
+0.246
+0
+0.1
+0.132
+0.64
+0.192
+-0.48
+0.244
+0
+0.2
+0.077
+0.85
+0.062
+0.888
+0.069
+0.897
+0.3
+0.083
+0.837
+0.071
+0.895
+0.082
+0.774
+0.4
+0.079
+0.847
+0.07
+0.887
+0.074
+0.863
+0.2
+0.0
+0.246
+0
+0.246
+0
+0.246
+0
+0.1
+0.082
+0.756
+0.083
+0.859
+0.245
+-1774
+0.2
+0.086
+0.796
+0.071
+0.88
+0.071
+0.908
+0.3
+0.085
+0.799
+0.075
+0.888
+0.08
+0.87
+0.4
+0.086
+0.8
+0.118
+0.727
+0.067
+0.902
+0.3
+0.0
+0.246
+0
+0.246
+0
+0.246
+0
+0.1
+0.082
+0.75
+0.098
+0.753
+0.244
+0
+0.2
+0.096
+0.777
+0.065
+0.894
+0.08
+0.798
+0.3
+0.088
+0.785
+0.08
+0.834
+0.069
+0.902
+0.4
+0.152
+0.557
+0.097
+0.818
+0.075
+0.867
+Table 2: MSE, and R2 score between the predicted and the exact potential φ(x) for a different value of λ2,
+and λ3, and the different number of hidden layers, and neurons per hidden layer in PINN for 1000 different
+samples of the Test set. Here, λ4 = 0.0001 is fixed and λ1 = 1 − (λ2 + λ3 + λ4). In this table bold numbers
+show cases with MSE < 0.1 and R2score > 0.9.
+12
+
+arXiv Template
+A Preprint
+Numlayer=3
+Numlayer=5
+Numlayer=7
+��2
+λ3
+MSE
+R2
+MSE
+R2
+MSE
+R2
+Numneuron=30
+0.0
+0.0
+0.281
+0
+0.281
+0
+0.281
+0
+0.1
+0.14
+0.71
+0.148
+0.364
+0.275
+-15.28
+0.2
+0.118
+0.764
+0.077
+0.847
+0.126
+0.646
+0.3
+0.104
+0.737
+0.09
+0.782
+0.089
+0.835
+0.4
+0.208
+0.587
+0.072
+0.842
+0.083
+0.822
+0.1
+0.0
+0.281
+0
+0.281
+0
+0.281
+0
+0.1
+0.105
+0.808
+0.28
+-1801
+0.28
+0
+0.2
+0.112
+0.775
+0.09
+0.845
+0.259
+-2.137
+0.3
+0.105
+0.793
+0.123
+0.697
+0.264
+0.07
+0.4
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+Table 3: MSE and R2 score between the predicted and the exact potential φ(x) for a different value of λ2,
+and λ3, and the different number of hidden layers, and neurons per hidden layer in PINN for 1000 different
+samples of the Extrapolate set. Here, λ4 = 0.0001 is fixed and λ1 = 1 − (λ2 + λ3 + λ4). In this table bold
+number shows the best hyperparameters for our PINN-based model.
+13
+
+arXiv Template
+A Preprint
+3.3
+Comparison
+We evaluate RF, NN, and PINN to estimate the potential of point-charged particles surrounded by conductive
+walls. According to Fig.6, NN was not trained well, while RF and PINN-based models could predict potential
+precisely. However, RF did not work well to estimate φT rue > 0.3. Apart from this, the best model could
+estimate not only the potential of the train and the test sets but also the potential of point-charged particles
+that are not in the train or test set. So we evaluate the best tuned-PINN model and RF on the extrapolation
+samples; the results are reported in Table 3. As can be seen in Fig.8 PINN-based model could predict the
+potential of newly charged particles better than the RF model, where PINN could predict φT rue > 0.3 by far
+better than RF.
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+RF
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+True
+: 0.07
+Extrapolation data : 1000 targets
+0.05
+0.10
+0.15
+0.20
+0.25
+Density distribution
+(a)
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+PINN
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+True
+: 0.02
+Extrapolation data : 1000 targets
+0.02
+0.04
+0.06
+0.08
+0.10
+0.12
+0.14
+0.16
+Density distribution
+(b)
+Figure 8: Potential estimation of best-tuned (a) RF with scatter of σ = 0.07, and (b) PINN-based model
+with scatter of σ = 0.02 on the 1000 samples of the Extrapolation set. The dashed red line shows where the
+predicted potential equals the true potential. The pink-shaded region marks 1σ scatter of potential errors.
+3.4
+Generalization (Multi charged particles)
+For generalization, we test the PINN-based model with λ2 = 0.3 and λ3 = 0.3 for the case of more than one
+charged particle surrounded with conductive boundaries. Since the Laplace equation is a linear function,
+we predict the potential of each charged particle and then calculate the total potential with a superposition
+of the corresponding predicted potential. After that, we report the MSE between the predicted smooth
+potential and the exact smooth solution, which is calculated by the image charges method. Fig.9 shows the
+relation between MSE and N, the number of charged particles. As expected, the MSE is independent of
+the number of charged particles. Therefore, it leads to the fact that we can also use this method for problems
+with any desired particles.
+14
+
+arXiv Template
+A Preprint
+2
+3
+4
+5
+6
+7
+8
+9
+10
+N
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+MSE
+Figure 9: MSE between true and predicted potential as a function of charged particles number; in 100
+different problems
+4
+Conclusion
+In this study, we have trained a machine to predict the smooth potential of charged components surrounded
+by conductive boundaries. In this scene, the total potential could be easily calculated by the summation
+of predicted smooth potential with singular potential due to the PLT algorithm. The reference set consists
+of analytic solutions, the solution of the image charge method, which is split into a train set with 5000
+samples and a test set with 1000 samples. To check the accuracy of our model, we set another data set called
+extrapolation set consisting of 1000 samples with different boundary conditions which were not in the train
+or even test set. Our main conclusion can be summarized as follows:
+• We find that the PINN-based model trained better than RF and NN models. RF could not predict
+high potential; on the other hand, the NN could not be trained well at all.
+• our PINN-based model could predict the potential of the test set with MSE = 0.069, R2score = 0.902,
+and scatter σ = 0.02. It also could predict the potential of the extrapolation set with MSE = 0.089,
+R2score = 0.851, and scatter σ = 0.02.
+• Since the Laplace equation is a linear equation, the trained model could predict the potential of more
+than one charged particle by summating every particle’s predicted potential. Besides, we show that
+the MSE of more than one particle is independent of a number of particles.
+References
+John R Miller and Patrice Simon. Electrochemical capacitors for energy management. science, 321(5889):
+651–652, 2008.
+Mathieu Salanne, Benjamin Rotenberg, Katsuhiko Naoi, Katsumi Kaneko, P-L Taberna, Clare P Grey, Bruce
+Dunn, and Patrice Simon. Efficient storage mechanisms for building better supercapacitors. Nature Energy,
+1(6):1–10, 2016.
+Patrice Simon and Yury Gogotsi. Materials for electrochemical capacitors, 2008. ISSN 14761122.
+John David Jackson. Jackson - classical electrodynamics, 1962. ISSN 00029505.
+Jian-Ming Jin. The finite element method in electromagnetics, 3rd edition. Journal of Chemical Information
+and Modeling, 2014.
+G. W. S., Gene H. Golub, and Charles F. Van Loan. Matrix computations. Mathematics of Computation, 56,
+1991. ISSN 00255718. doi:10.2307/2008552.
+15
+
+arXiv Template
+A Preprint
+Sandeep Tyagi, Axel Arnold, and Christian Holm. Icmmm2d: An accurate method to include planar
+dielectric interfaces via image charge summation. Journal of Chemical Physics, 127, 2007. ISSN 00219606.
+doi:10.1063/1.2790428.
+Sandeep Tyagi, Axel Arnold, and Christian Holm. Electrostatic layer correction with image charges: A linear
+scaling method to treat slab 2d+h systems with dielectric interfaces. Journal of Chemical Physics, 129,
+2008. ISSN 00219606. doi:10.1063/1.3021064.
+Sandeep Tyagi, Mehmet Süzen, Marcello Sega, Marcia Barbosa, Sofia S. Kantorovich, and Christian Holm.
+An iterative, fast, linear-scaling method for computing induced charges on arbitrary dielectric boundaries.
+Journal of Chemical Physics, 132, 2010. ISSN 00219606. doi:10.1063/1.3376011.
+S. Kesselheim, M. Sega, and C. Holm. The icc* algorithm: A fast way to include dielectric boundary effects
+into molecular dynamics simulations. arXiv:1003.1271, 2010.
+Axel Arnold, Olaf Lenz, Stefan Kesselheim, Rudolf Weeber, Florian Fahrenberger, Dominic Roehm, Peter
+Košovan, and Christian Holm. Espresso 3.1: Molecular dynamics software for coarse-grained models.
+volume 89 LNCSE, 2013. doi:10.1007/978-3-642-32979-1_1.
+Stewart K. Reed, Oliver J. Lanning, and Paul A. Madden. Electrochemical interface between an ionic liquid and
+a model metallic electrode. Journal of Chemical Physics, 126, 2007. ISSN 00219606. doi:10.1063/1.2464084.
+Samare Rostami, S Alireza Ghasemi, and Ehsan Nedaaee Oskoee. A highly accurate and efficient algorithm
+for electrostatic interactions of charged particles confined by parallel metallic plates. The Journal of
+chemical physics, 145(12):124118, 2016.
+Morad Biagooi, Mohammad Samanipour, S. Alireza Ghasemi, and Seyedehsan Nedaaee Oskoee. Caviar:
+A simulation package for charged particles in environments surrounded by conductive boundaries. AIP
+Advances, 10, 2020. ISSN 21583226. doi:10.1063/1.5140052.
+Tao Shan, Wei Tang, Xunwang Dang, Maokun Li, Fan Yang, Shenheng Xu, and Ji Wu. Study on a fast solver
+for poisson’s equation based on deep learning technique. IEEE Transactions on Antennas and Propagation,
+68(9):6725–6733, 2020. doi:10.1109/TAP.2020.2985172.
+M. Raissi, P. Perdikaris, and G. E. Karniadakis. Physics-informed neural networks: A deep learning framework
+for solving forward and inverse problems involving nonlinear partial differential equations. Journal of
+Computational Physics, 378, 2019. ISSN 10902716. doi:10.1016/j.jcp.2018.10.045.
+Leo Breiman. Random forests. Machine Learning, 45, 2001. ISSN 08856125. doi:10.1023/A:1010933404324.
+Fabian Pedregosa, Gael Varoquaux, Alexandre Gramfort, Vincent Michel, Bertrand Thirion, Olivier Grisel,
+Mathieu Blondel, Peter Prettenhofer, Ron Weiss, Vincent Dubourg, Jake Vanderplas, Alexandre Passos,
+David Cournapeau, Matthieu Brucher, Matthieu Perrot, and Édouard Duchesnay. Scikit-learn: Machine
+learning in python. Journal of Machine Learning Research, 12, 2011. ISSN 15324435.
+Martín Abadi. Tensorflow: learning functions at scale. ACM SIGPLAN Notices, 51, 2016. ISSN 0362-1340.
+doi:10.1145/3022670.2976746.
+François Chollet. Keras: The python deep learning library. Keras.Io, 2015.
+Stéfan Van Der Walt, S. Chris Colbert, and Gaël Varoquaux.
+The numpy array: A structure for ef-
+ficient numerical computation.
+Computing in Science and Engineering, 13, 2011.
+ISSN 15219615.
+doi:10.1109/MCSE.2011.37.
+Ian Goodfellow, Yoshua Bengio, and Aaron Courville. Deep Learning. MIT Press, 2016. http://www.
+deeplearningbook.org.
+a. K. Connect, a. Krogh, and J. a. Hertz. A simple weight decay can improve generalization. Advances in
+Neural Information Processing Systems, 4, 1992.
+Vijay Kag, Kannabiran Seshasayanan, and Venkatesh Gopinath. Physics and data informed neural networks
+for two-dimensional turbulence. arXiv preprint arXiv:2203.02555, 2022.
+Dong C Liu and Jorge Nocedal. On the limited memory bfgs method for large scale optimization. Mathematical
+programming, 45(1):503–528, 1989.
+16
+

I9AyT4oBgHgl3EQffvhT/content/tmp_files/2301.00345v1.pdf.txt

@@ -0,0 +1,1829 @@
+MTNeuro: A Benchmark for Evaluating
+Representations of Brain Structure Across
+Multiple Levels of Abstraction
+Jorge Quesada1∗, Lakshmi Sathidevi1∗, Ran Liu1, Nauman Ahad1, Joy M. Jackson1,
+Mehdi Azabou1, Jingyun Xiao1, Christopher Liding1, Matthew Jin1, Carolina Urzay1,
+William Gray-Roncal2, Erik C. Johnson2,†, Eva L. Dyer1,†
+1- Georgia Institute of Technology
+2 - Johns Hopkins University Applied Physics Laboratory
+Abstract
+There are multiple scales of abstraction from which we can describe the same
+image, depending on whether we are focusing on fine-grained details or a more
+global attribute of the image. In brain mapping, learning to automatically parse
+images to build representations of both small-scale features (e.g., the presence of
+cells or blood vessels) and global properties of an image (e.g., which brain region
+the image comes from) is a crucial and open challenge. However, most existing
+datasets and benchmarks for neuroanatomy consider only a single downstream
+task at a time. To bridge this gap, we introduce a new dataset, annotations, and
+multiple downstream tasks that provide diverse ways to readout information about
+brain structure and architecture from the same image. Our multi-task neuroimag-
+ing benchmark (MTNeuro) is built on volumetric, micrometer-resolution X-ray mi-
+crotomography images spanning a large thalamocortical section of mouse brain,
+encompassing multiple cortical and subcortical regions. We generated a num-
+ber of different prediction challenges and evaluated several supervised and self-
+supervised models for brain-region prediction and pixel-level semantic segmenta-
+tion of microstructures. Our experiments not only highlight the rich heterogeneity
+of this dataset, but also provide insights into how self-supervised approaches can
+be used to learn representations that capture multiple attributes of a single image
+and perform well on a variety of downstream tasks. Datasets, code, and pre-trained
+baseline models are provided at: https://mtneuro.github.io/.
+1
+Introduction
+Our understanding of our natural surroundings requires multiple levels of perceptual processing: we
+can recognize a macroscopic object (e.g., a tree), while also identifying finer-grain structures within
+it (e.g., leaves and branches), and context-relevant features (e.g., leafiness, height, or season). This
+multi-level perception scheme also translates to the medical image domain: the process of interrogat-
+ing medical images (either by a human expert or an algorithm) involves combining macrostuctural
+insights (such as a region of interest) with context-relevant microstructure information and human-
+interpretable features (e.g., the density of a given cell type in a microscopy image) in order to derive
+a diagnosis or characterize a target sample.
+∗Equal contribution. Contact authors: (ELD, JQ, LS) {evadyer, jpacora3, lsathidevi3}@gatech.edu; (ECJ)
+erik.c.johnson@jhuapl.edu; † Both senior authors contributed equally.
+35th Conference on Neural Information Processing Systems (NeurIPS 2022) Track on Datasets and Bench-
+marks.
+arXiv:2301.00345v1  [cs.CV]  1 Jan 2023
+
+Figure 1: Overview of MTNeuro Benchmark. Task 1: Brain region (macrostructure) classification (3 config-
+urations which vary in data availability and testing schemes); Task 2: Pixel-level microstructure segmentation
+(4 configurations which vary in sample dimensionality, span and target class count); Task 3: Probing semantic
+attributes from image-level embeddings obtained in Task 1.
+In particular, the ongoing effort to understand the connections and dynamics of the brain involves
+analyzing both macroscopic level properties such as region-level structures (1; 2) as well as detailed
+microstructures like the size or shape of a given cell type (3). While significant advances have
+been achieved in unveiling the properties and structures within the brain through several imaging
+modalities at different scales (4; 5; 6; 7), most existing neuroimaging benchmarks are designed for
+evaluation at a single spatial scale, or geared towards a particular downstream task. This can be
+attributed to several causes, including the prohibitive cost of manual annotation for data spanning
+multiple scales (8; 9), the associated computational cost of processing multi-scale data, and the fact
+that neuroimaging technologies have only recently progressed towards pipelines that can capture
+multi-area volumes at high resolutions.
+To fill this gap, we present the MTNeuro benchmark (Figure 2): a multi-task, multi-scale benchmark
+based on a large 3D X-ray microtomography image dataset spanning multiple areas from a mouse
+brain. Code and access to the data are provided at: https://mtneuro.github.io/. We host our
+dataset in the Brain Observatory Storage Service and Database (BossDB, a specialized interactive
+database (10)) and provide an integrated dataloader to facilitate transfer experiments and analysis.
+This benchmark provides a unified framework that allows for evaluating models and representations
+arising in three distinct tasks:
+• Task 1 - Image-level classification of brain region: prediction of the brain region (so-
+matosensory cortex, striatum, thalamus, zona incerta) to which a given image belongs.
+• Task 2 - Pixel-level segmentation of microstructures: prediction of neural microstruc-
+tures (axons, cell bodies, blood vessels, background) at the pixel-level across the four core
+brain regions in the dataset.
+• Task 3 - Probing multiple semantic features from learned image-level embeddings:
+estimation of semantic (human-interpretable) features (such as the average cell size or axon
+density) from the representation of a given image, obtained after “freezing” the weights of
+a trained encoder.
+To understand how current models perform on these different tasks, we evaluate a family of dif-
+ferent supervised and self-supervised models. Our results in Tasks 1 and 3 highlight a significant
+generalization gap between self-supervised and supervised approaches, which opens up interesting
+opportunities for further evaluation and development of self-supervised learning (SSL) methods for
+these tasks. Through testing across a family of different models across a variety of tasks, our pro-
+posed benchmark provides both an exciting platform for evaluating self-supervised learning (SSL)
+methods, and a rich tool in the effort to extract fundamental insights into brain architecture at both
+the micro- and macro-scale.
+2
+
+Task 1 - ROl Classification
+Task 2 - Microstructure segmentation
+Task 3 - Probing semantic attributes
+Probe models
+2D
+3-class
+Segmentation
+256x256
+Brain region
+Model
+image
+Classifer
+> Axon density
+ cell vessel  bkgd
+Cortex
+Striatum
+Thalamus 
+ Zona Incerta
+3D
+Model
+4-class
+Sub-tasks based on different train/test splits
+> Cell count
+train test
+model trained
+C1
+１1′
+on task 1
+C2 
+3,4
+latent
+ Cell density
+representation
+>
+C3
+1,23,4
+ cell vessel  axon 
+4 different configurations
+ bkgd
+/ per class2
+Background and Related Work
+2.1
+The need for a benchmark in brain mapping and connectomics
+Over the past decade, there have been major advances in our ability to resolve fine-scale neu-
+roanatomical structures in the brain. With these advances, we have generated large amounts of
+brain data that span many spatial scales, and can reveal different features of brain organization.
+At the nanoscale, electron microscopy has provided detailed wiring diagrams of small portions of
+cortex (11). At micron scale, microscopy techniques have provided detailed pictures of cytoarchi-
+tecture - or how neurons and cells are organized (5). Efforts at even larger scales to capture many
+brain areas simultaneously, like connectivity atlas and X-ray microtomographic datasets (12), have
+provided information about the interplay between long-range connections across brain areas and
+microstructures, such as cell body densities and other morphological features of brain structure.
+Accompanying these new tools for data generation have been major advances in machine learning
+and computational approaches for modeling and analyzing these datasets, for problems such as
+object detection, segmentation, and classification. While the information provided by these methods
+are incredibly rich and have a great deal of structure at many scales, any given method is typically
+tested on an individual challenge at a particular scale. The expense of annotating and proofreading
+can be considerable, and significant neuroanatomy knowledge is typically required of annotators
+(9). Moreover, many efforts to provide high quality data, such as (5), have not focused on building
+ML-oriented benchmarks, but rather on providing reference datasets and resources.
+Using machine learning tools to understand these emerging brain datasets at different spatial scales
+is both a challenge and an increasingly critical need. As a result, large tera- and peta-scale con-
+nectomics datasets are being collected using electron microscopy and X-ray microtomography, in-
+cluding data from the entire brain of Drosophila (13; 14), large portions of the mouse brain (15; 3)
+and even a cubic millimeter of human cortex (6). Advances in imaging technologies promise to
+continually increase the spatial extent, number of species, and number of imaged individuals. These
+datasets have the micro- or nanoscale resolution and large spatial extents required to resolve sub-
+cellular structures (e.g., mitochondria and synapses), microstructures (e.g., glia, neurons, and vascu-
+lature), and macrostructure (e.g., brain regions, cortical layer structure, and long-range white matter
+projections). The multi-scale nature and large size of these datasets requires new ML tools (16),
+which drives the need for benchmarks that can extract representations of neural structure at different
+scales.
+2.2
+Existing datasets and benchmarks for resolving brain structure
+Due to the large variety of spatial scales, neuroanatomical structure, and imaging modalities, a wide
+range of segmentation and classification problems have been formulated for neuroimaging data. At
+the macroscale, there has been a long history of developing benchmarks for different datasets in
+MRI and related modalities like DTI and fMRI. For example, the BraTS dataset (17) focuses on the
+MRI of brain tumor and motivated many segmentation works (18; 19; 20; 21). The ADNI (22) and
+MIRIAD (23) datasets provide MRI-based Alzheimer’s disease imaging that focuses on tracking
+disease progression (24; 25; 26; 27; 28; 29; 30). The UK Biobank (31) offers a huge collection of
+Brain MRI data from 5,00,000 human participants. The low spatial resolution of MRI and related
+methods, however, limit the ability to observe microscale structures at the cellular or subcellular
+level.
+For microscale structure, there are fewer benchmarking efforts due to the scale and complexity of
+the annotation and processing. Example problems include segmentation of synapses in the CREMI
+challenge (32), which provides high-resolution imaging of synapses in 5 cubic microns of non-
+isotropic EM with nanometer resolution. Other problem formulations include 2D image segmenta-
+tion of cell membranes (ISBI 2012 Challenge), with data from (33), and 3D segmentation of cells
+(34; 35). Specific benchmarks have also been developed for axon instance segmentation (36) and
+mitochondria segmentation (37). Different forms of microscopy with micrometer resolution, in-
+cluding calcium fluorescence microscopy, are suitable for segmenting cell bodies and extracting
+functional time-series data, but lack other microstructure information. Benchmarks have also been
+established for estimation of functional traces from two-photon calcium fluorescence microscopy,
+including spikefinder (38). These benchmarks have been instrumental in driving progress on specific
+problems at specific spatial scales. In general, however, most benchmarks at the microscale focus on
+3
+
+relatively small spatial extents, and lack the multi-scale macrostructure, which is also present in the
+brain. To the best of our knowledge, there are no public microtomography datasets of brain structure
+with both dense microstructure and macroscale annotations currently available.
+Encompassing larger spatial extents, projects such as the BigBrain atlas provides high-resolution
+sections from the mouse brain with Nissl contrast (39), but the resolution only allows resolution of
+cells around 20 µm. Large-scale EM datasets are also being used to benchmark performance and seg-
+ment neurons, augmented with iterative human proofreading (40; 41; 6), which are leading to large
+segmented datasets with increasingly complex annotation. These data, however, are not suitable for
+large-scale benchmarking and algorithm development in the general machine learning community
+due to their size and ongoing refinement. To accelerate progress towards machine learning tools
+which can operate at multiple levels of spatial abstraction within the same high-resolution dataset,
+benchmarks are needed which encompass large spatial extents at high resolution. This will enable a
+broader scientific community to apply state-of-the-art methods for these important applications.
+3
+Dataset and Tasks
+3.1
+Overview of dataset
+We build our benchmark on a large open access high-resolution (1.17µm isotropic) 3D X-ray micro-
+tomography imaging dataset that provides fine-scale information about brain microstructure as well
+as diverse regions of interest that give more distinct global attributes (15). The dataset thus con-
+tains a uniquely rich set of both macroscale (region of interest) and microscale (cells, blood vessels,
+axons) structures that can be interrogated throughout the dataset (42; 16).
+The full volumetric dataset provides micron resolution of an intact brain sample totalling 5805 ×
+1420 × 720 pixels. The dataset spans four regions of interest: somatosensory cortex (CTX), stria-
+tum (STR), ventral posterior region of thalamus (VP), or the zona incerta (ZI) (see Figure 2 A-B)
+and provides pixel-level microstructure (cell, axons, blood vessels, background) and macrostructure
+labels over 2D slices distributed over the dataset.
+To provide more data for training and validation, we expanded the pixel-level microstructure and
+macrostructure labels provided in the original data resource (15) to larger contiguous volumes in the
+data. To expand the pixel-level labels, we trained a 4-class Unet model on the sparse 2D annotations
+and had a trained expert proofread the annotations to create dense pixel-level microstructural labels
+(see Figure 2B), identifying each point as either part of an axon, cell, blood vessel, or background.
+The final curated pixel-level labels span 4 ROIs, with each ROI consisting 360 256x256 densely
+labelled images.
+To examine semantic attributes of the different images, in Task 3, we leveraged the dense pixel-
+level labels to compute a number of semantic features from the reconstructions: (i) the density of
+blood vessels, (ii) axon density, (iii) the number of cells, (iv) size of cells, and (v) the average
+inter-cell distance in each slice. These semantic labels provide information into different features
+of the cytoarchitecture that can be used to interpret the embeddings learned by models tested on
+this dataset. In addition to these microstructure annotations, we have expanded the macrostructure
+annotations from the original dataset to include interpolations of the region labels across all 720
+slices of size 5805 × 1420. From these interpolated sections, we extracted 12 new subvolumes
+(three for each of the four regions of interest) for examining the generalization of models in Task 1.
+3.2
+Data access
+The dataset and all corresponding labels are stored in BossDB (10), the Brain Observatory Storage
+Service and Database. The dataset project page can be found at: http://bossdb.org/project/
+prasad2020. BossDB is a specialized spatial database for Electron Microscopy and X-Ray Micro-
+tomography Datasets, with seamless visualization through Neuroglancer, which enables interactive
+visualization of large-scale 3D annotated volumes and annotations. All data are available publicly,
+using public log on credentials (no account creation required). The project page documents project
+metadata, citation instructions, and the data creators and curators.
+For benchmarking, data are accessed through the Python intern API (43). This API allows a remote
+connection to the BossDB system, including downloads of arbitrary, on-demand 3D cutouts of data,
+4
+
+500 um
+A
+ZI
+VP
+STR
+CTX
+intern SDK
+Image Cube
+Label Cube
+BossDBDataset
+DataLoader
+__getitem__
+images
+labels
+C
+Figure 2:
+Overview of datasets and annotations used in MTNeuro.
+In A we show how the dataset spans
+multiple brain areas including the somatosensory cortex (CTX), striatum (STR), thalamus (VP), and zona in-
+certa (ZI). Each of these areas contains annotations of pixel-level microstructures like axons, blood vessels, and
+cells visualized as dense X-ray microCT volumetric imaging data (1.17 micron isotropic resolution), as seen
+in B. The pipeline and BossDB data access are shown in C: channels are accessed through the Intern API to
+access cutouts without the need to download the entire dataset. This enables creating a data loader for specific
+task-relevant cutouts.
+including raw images and annotations, without the need to download the entire dataset to disk. To
+facilitate the use of this API, we provide a Pytorch DataLoader for rapid algorithm development
+and testing. We also provide sample Jupyter notebooks to demonstrate how the task cutouts can be
+dowloaded, and saved as Numpy files for development in other frameworks. The tasks are defined
+with task-specific JSON files specifying the metadata for each task. The use of this dataloader is
+illustrated in Fig. 2C, where we detail how the dataset can be efficiently accessed through BossDB.
+The data are structured into channels. Raw images, macrostructure annotations, and microstructure
+annotations, each have their own separate channel. This is detailed below.
+Raw images: All tasks utilize the same raw images, which are single color (grayscale) with 8-bit
+unsigned integer values. The total raw data volume is 720 × 1420 × 5805 voxels at a resolution of
+1.17µm. This data is available through the raw images channel: https://api.bossdb.io/v1/
+mgmt/resources/prasad/prasad2020/image.
+Macrostructure annotations: Tasks 1 and 3 utilize dense pixel-level annotations of different
+brain regions (macrostructure) : CTX (label 0); STR (label 1); VP (label 2); ZI (label 3). At
+this scale/level, there is an equal number of samples of each class and hence the classes are bal-
+anced. This label data is represented with 64-bit unsigned integers and is accessible through the
+macrostructure annotations channel: https://api.bossdb.io/v1/mgmt/resources/prasad/
+prasad_analysis/roi_labels.
+Microstructure annotations: Task 2 and Task 3 utilize pixel-level annotations of brain microstruc-
+ture. Volumes of 256 × 256 × 360 are densely annotated with microstructure labels. The labels are
+0: no label (background); 1: blood vessels; 2: cells; 3: mylenated axons. Represented with 64-bit
+unsigned integers, it is accessible through the microstructure annotations channel: https://api.
+bossdb.io/v1/mgmt/resources/prasad/prasad_analysis/pixel_labels.
+A key aspect of our approach is portability of data access and training infrastructure across neu-
+roimaging datasets in BossDB. This allows for easy extension of code and baselines to new vol-
+umetric imaging datasets stored in BossDB. These include data from new species, with new mi-
+crostructure labels (synapses, membranes, mitochondria), and with new macrostructure labels (brain
+regions, experimental state). By modifying the dataloader JSON, developers can specify different
+BossDB datasets, spatial regions, and annotation (label) sources. This allows for the flexibility re-
+quired to support the different tasks in this dataset, and will enable further training and deployment
+5
+
+A
+B
+STR
+CTX
+7
+Cortex
+Striatum
+Thalamus
+Zona
+500 um
+ axon
+(CTX)
+(STR)
+(VP)
+Incerta (ZI)
+vessel 
+Task 1 - Classify ROl
+Task 2 - Segment Microstructure
+Task 3 - Predict Attributes
+BrainRegion
+classification
+2D
+BloodVessel
+regression
+Cell Count
+regression
+3D
+Frozen
+C3)
+Cell Density
+Network
+regression
+Train
+Test
+3-class
+4-class500 μmof these baseline models to new datasets. This is an important contribution of this work towards
+developing machine learning tools for emerging large-scale neuroimaging datasets.
+3.3
+Tasks
+3.3.1
+Task 1: Image-level classification of brain region-of-interest (ROI)
+When analyzing imaging data that spans many different brain regions, one important question is
+the degree to which global image features correlate with the brain region from which the sample
+is drawn (16). Thus, we can pose this as a classification problem, where we pull a small patch (or
+small region) from the data and estimate which of the 4 brain regions the sample was drawn from.
+Given the abundance of samples available for this task across the entire volume, we can sub-divide
+this task into three different training schemes (see Figure 1), detailed below. Taken together, these
+three training schemes allow us to evaluate how different types of models are able to generalize as
+more data becomes available during training.
+ROI-C1.
+For this sub-task, we use the 4 densely-annotated cubes (shown in blue in Figure 1, Task
+1), each corresponding to one of the regions of interest (CTX, STR, VP, or ZI). We divide each of
+the four subvolumes by selecting the first 300 (256 × 256) images slices for training, and the last 50
+images for testing, leaving 10-slices between the train and test data to avoid any structural overlap.
+The resulting overall sample size for this sub-task is thus 1400 images, with 1200 images for the
+train set and 200 for the test set.
+ROI-C2.
+In this sub-task, we evaluate the performance of the models when tested on new areas
+within the larger 3D context of the dataset. We extract two additional 256 × 256 × 360 cubes per
+class (shown in Figure 1, Task 1), to serve as additional test data for the models. This sub-task
+employs 4080 images, with 1200 in the train set, and 2880 in the test set.
+ROI-C3.
+In this sub-task, we evaluate the performance of the models when allowed to learn from
+a larger set of data. We extracted one additional cube per class to serve as an additional source of
+training data for the models and use the same test set as that employed in ROI-C2. This sub-task
+employs 5520 images, with 2640 in the train set, and 2880 in the test set.
+3.3.2
+Task 2: Pixel-level segmentation of microstructures
+Another important requirement for a comprehensive mapping of brain data is correctly identifying
+brain microstructures such as cells and blood vessels. In this task, we utilize the dense pixel-level
+microstructure labels of the volumetric cutouts from 4 brain regions of interest (macrostructure; Cor-
+tex, Striatum, VP & ZI), and evaluate different baselines on their prediction accuracy in classifying
+each pixel from test volumes across the brain regions into appropriate brain microstructure classes
+(blood vessel, cell, axon & background).
+3-class segmentation task.
+We first consider the pixel-level segmentation of images into one of
+three classes: either cell bodies, blood vessels, or other (background and axons). We use the same
+train and test split as in ROI-C1 in Task 1 on the main subvolumes that are densely annotated at the
+pixel-level (300 images for train, 50 for test, with a gap of 10 slices between datasets). This sub-task
+employs 1200 images for training, and 200 images for testing.
+4-class segmentation task.
+In this task, we consider the pixel-level segmentation of images into
+one of four classes: either cell bodies, blood vessels, background or axons. When we consider dense
+axonal segmentation, we remove the ZI region from our training and testing set due to the difficulty
+to reliably segment axons in this subvolume even for human annotators. This sub-task employs 900
+images for training, and 150 images for testing.
+3.3.3
+Task 3: Probing multiple semantic features from learned image-level embeddings
+In this task, we wanted to explore the possibility of decoding semantic and human-interpretable
+features from the image-level representations learned in Task 1. If possible, it could provide a
+way to build interpretable image-level feature maps that contain information about microstructure
+without needing the expensive pixel-level labels necessary to compute these attributes in most brain
+mapping settings.
+6
+
+Table 1: Results on image classification accuracy for brain region prediction (Task 1).
+ROI - C1
+ROI - C2
+ROI - C3
+Supervised
+0.88 ± 0.03
+0.77 ± 0.03
+0.88 ± 0.02
+Sup w/ Mixup
+0.90 ± 0.04
+0.78 ± 0.03
+0.90 ± 0.02
+BYOL
+0.88 ± 0.02
+0.76 ± 0.02
+0.97 ± 0.01
+MYOW
+0.90 ± 0.02
+0.78 ± 0.05
+0.98 ± 0.01
+MYOW-m
+0.94 ± 0.02
+0.78 ± 0.03
+0.98 ± 0.01
+PCA
+0.59
+0.25
+0.07
+NMF
+0.62
+0.27
+0.50
+Specifically, we try to predict the following global properties of an image: (i) blood vessels density,
+(i) cell count, (iii) average cell size, (iv) axon density, and (v) average distance between cells, all
+through a simple linear readout. In the case of supervised models, these models are trained to classify
+images into its respective brain region; However, in the case of SSL methods, where the region-level
+labels aren’t used to guide learning, it may be possible that other global attributes of the images may
+be encoded in the latent space of the model.
+4
+Results
+4.1
+Task 1: Image-level classification of brain ROI
+Experiment setup.
+In this task, we consider the classification of different images into a number
+of candidate brain areas (CTX, STR, VP and ZI) using representations learned through supervised
+and self-supervised approaches. All of the models in this task are trained using a Resnet18 encoder
+(44). We benchmarked two supervised models: the first trained using standard approaches for reg-
+ularization (20% dropout and weight decay factor of 0.3) and the other trained using mixup (45).
+Given the underlying shared structure of image samples in volumetric data, we also consider a num-
+ber of self-supervised learning methods suitable for this task: (i) BYOL (46), (ii) MYOW (47), (iii)
+and a variant of MYOW that we tested with a single projector and predictor (MYOW-merged, or
+MYOW-m). For the SSL models, we follow the standard procedure of freezing the network weights
+after training, and then training a linear layer on top of the representations. This tells us how well
+the SSL loss captures the classes in the data after only a linear transformation. All models have a
+latent dimensions of 256. As additional baselines, we also extracted 256-dimensional embeddings
+from our data using Principal Component Analysis (PCA) and Non-Negative Matrix Factorization
+(NMF), and trained a linear layer on these representations.
+In our experiments, we evaluate all methods across 5 training instances with different random seeds,
+and report the overall mean accuracy and standard deviation. All models are trained for 100 epochs
+using an SGD optimizer with a learning rate of 0.03. For more details on our experimental setup
+and models, see Section 3 in the Appendix.
+Results in classifying brain ROIs.
+We report the results of the three subtasks for ROI classifica-
+tion in Table 1. In our first subtask (ROI-C1), we find that many of the SL and SSL models achieve
+comparable accuracy, with the MYOW-m model achieving the highest accuracy in this limited train-
+ing regime. In ROI-C2, we test the generalization capabilities of the models trained in ROI-C1 by
+evaluating how well they performed on subvolumes in other parts of the larger dataset (Table 1,
+ROI-C2). There is considerable heterogenity across brain regions and thus we can consider this as
+a form of domain generalization. In this case, we observe a significant decrease in accuracy that
+can be attributed to the domain shift in data. In ROI-C3, we add another set of training data (see
+Figure 2) and test on the same volumes as in our last subtask. In this case, we can observe that SSL
+methods significantly outperform the SL models, with SSL models achieving even higher accuracy
+than in the small-scale case (97-98%) and supervised models achieving a much more modest im-
+provement (88-90%) with the new training data. We can thus observe a significant gap in SSL over
+SL models, showing that exposure to additional data (in this case, the additional cube with respect
+to task ROI-C1) drastically improves the generalization of the SSL models to unobserved data.
+4.2
+Task 2: Pixel-level segmentation of microstructures
+In Task 2, we consider four different variants of pixel-level segmentation. The first variant employs
+2D models for pixel-level segmentation. The second variant employs 3D models instead. The third
+7
+
+Table 2: F1 & IoU scores for models trained on the pixel-level segmentation task (Task 2).
+I. 2D Pixel-level Segmentation
+3-Class
+4-Class
+Method
+Metric Bg + Axons Vessels Cells
+Avg.
+Bg
+Vessels Cells Axons
+Avg.
+2D U-Net
+F1
+0.99
+0.76
+0.85
+0.87 ± 0.012 0.97
+0.82
+0.87
+0.94
+0.90 ± 0.003
+2D U-Net
+IoU
+0.98
+0.64
+0.75
+0.79 ± 0.014 0.89
+0.70
+0.77
+0.60
+0.74 ± 0.008
+MA-Net
+F1
+0.99
+0.79
+0.87
+0.88 ± 0.003 0.97
+0.83
+0.87
+0.94
+0.90 ± 0.002
+MA-Net
+IoU
+0.98
+0.68
+0.78
+0.81 ± 0.003 0.89
+0.71
+0.78
+0.76
+0.78 ± 0.011
+FPN
+F1
+0.99
+0.72
+0.84
+0.85 ± 0.01
+0.96
+0.73
+0.84
+0.93
+0.86 ± 0.004
+FPN
+IoU
+0.97
+0.59
+0.73
+0.76 ± 0.015 0.87
+0.59
+0.72
+0.72
+0.72 ± 0.021
+U-Net++
+F1
+0.99
+0.79
+0.87
+0.89 ± 0.002 0.97
+0.81
+0.85
+0.93
+0.89 ± 0.015
+U-Net++
+IoU
+0.98
+0.68
+0.78
+0.81 ± 0.002 0.88
+0.68
+0.75
+0.73
+0.76 ± 0.036
+PAN
+F1
+0.98
+0.60
+0.80
+0.79 ± 0.035 0.95
+0.69
+0.80
+0.93
+0.84 ± 0.007
+PAN
+IoU
+0.96
+0.46
+0.66
+0.69 ± 0.039 0.85
+0.53
+0.67
+0.76
+0.70 ± 0.014
+PSPNet
+F1
+0.97
+0.48
+0.74
+0.73 ± 0.013 0.94
+0.54
+0.71
+0.91
+0.78 ± 0.012
+PSPNet
+IoU
+0.94
+0.39
+0.61
+0.65 ± 0.043 0.82
+0.38
+0.55
+0.74
+0.62 ± 0.015
+II. 3D Pixel-level Segmentation
+3-Class
+4-Class
+Method
+Metric Bg + Axons Vessels Cells
+Avg.
+Bg
+Vessels Cells Axons
+Avg.
+3D U-Net
+F1
+0.99
+0.77
+0.87
+0.88 ± 0.006 0.93
+0.76
+0.80
+0.87
+0.84 ± 0.032
+3D U-Net
+IoU
+0.98
+0.65
+0.76
+0.80 ± 0.007 0.81
+0.62
+0.67
+0.50
+0.65 ± 0.045
+VNetLight
+F1
+0.99
+0.75
+0.83
+0.85 ± 0.012 0.90
+0.65
+0.73
+0.76
+0.76 ± 0.063
+VNetLight
+IoU
+0.97
+0.61
+0.70
+0.76 ± 0.013 0.78
+0.46
+0.58
+0.43
+0.56 ± 0.061
+HighResNet
+F1
+0.99
+0.74
+0.84
+0.85 ± 0.019 0.89
+0.51
+0.73
+0.77
+0.72 ± 0.083
+HighResNet
+IoU
+0.97
+0.61
+0.72
+0.77 ± 0.026 0.73
+0.35
+0.58
+0.42
+0.52 ± 0.075
+variant is a 4-class setting where ZI is removed from the brain regions involved (as axons in this
+region are difficult to distinguish accurately for even human annotators). The fourth variant is a
+3-class setting where all 4 brain regions (Cortex, Striatum, VP and ZI) are utilized but only 3 classes
+are considered (blood vessels, cells, background+axons; avoids axon segmentation).
+Experiment setup.
+In our experiments, we perform pixel-level segmentation using a selected set
+of 2D and 3D models. Each model is put through a separate hyper-parameter tuning process for
+finding an optimal learning rate and batch size by training on the train set and evaluating on the
+validation split. Each model is trained for 20 epochs with its optimal learning rate and batch size
+and evaluated across 5 training instances (each with its own random seed). The class-wise F1-
+score, the class-wise IoU, and the overall mean and standard deviation of both metrics are reported
+for each model (Table 2). We do not report accuracy because for this particular task of pixel-
+level segmentation, it does not aptly represent the model performance due to class imbalance (see
+Appendix Section 4.2.3 for breakdown of different classes in the training and test sets).
+The models we used for the 2D segmentation task are the standard 2D U-Net model (48; 49) and
+selected models from the ’segmentation models.pytorch’ library (50): MAnet (51), FPN (52), U-
+Net++ (53), PAN (54) and PSPNet (55). The models we used for the 3D segmentation task are the
+standard 3D U-Net model (48) and selected models from ‘MedicalZooPytorch’ (56): VNetLight
+(57; 56) and HighResNet (58). For more details refer Section 4 in the Appendix.
+Pixel-level segmentation in 2D.
+The results from the selected models on our 2D pixel-level seg-
+mentation task are tabulated in Part I of Table 2 and visualized in Figure 3. The individual slices are
+the input to the models and they are fed in a batched manner during training. For training and eval-
+uation, we consider both the 3-class and 4-class settings and we use EfficientNet-b7 encoder for all
+the models as it was seen to give the best performance among the 25 encoders that were attempted.
+From our results we see that MA-Net performed the best overall among the 2D models with an av-
+erage IoU of 0.78 followed by U-Net++ with an average IoU of 0.76 in the 4-class setting (Table 2).
+As can be seen from the class-wise IoU and class-wise F1-scores, for most of the best performing
+models, the most challenging components to differentiate are cells and blood vessels, which are also
+difficult for human annotators to identify from 2D slices without further 3D context. Also, we can
+note, the average IoU (across classes and models) increases from 0.72 to 0.79 upon moving from
+4-class to 3-class setting. This indicates an expected performance improvement as there are more
+slices (more data) and fewer classes (easier) in the 3-class setting.
+8
+
+Figure 3: Visualization of predictions from pixel-level segmentation task (Task 2). From left to right: original
+image, ground truth overlaid, prediction from U-Net model in the 3-class and 4-class settings (top row is
+Striatum, bottom is VP). Axons are visualized in cyan, blood vessels in red, and cell bodies are in yellow. B)
+3D reconstruction from the predictions for Striatum for the 4-class (top) and 3-class (bottom) settings.
+Pixel-level segmentation in 3D.
+We provide the same breakdown and accuracy measures as in
+the 2D case, for this 3D case (Part II of Table 2). For providing 3D input to the models, we pass
+in consecutive slices (8 slices) as a single subvolume and build a prediction over all slices jointly in
+3D. The U-Net model performed best with 0.80 IoU (3-class) which is only competent with the best
+2D models. This would be because we define the subvolumes in a non-overlapping manner, so even
+though we have improved 3D context, the model also sees fewer samples/inputs. Due to the same
+reason, moving from 3-class to 4-class setting, we can note a drop in performance since there are
+fewer samples. Larger sample counts / allowing overlap could enable 3D models to perform better.
+4.3
+Task 3: Probing multiple semantic features from learned image-level embeddings
+In Task 3, we explore the prediction of different semantic attributes (e.g., density of blood vessels or
+cells) from the latent space learned by the models trained in Task 1.
+To do this, we leveraged the high-quality dense microstructure annotations to extract information
+about the attributes of each image that included: the proportion of pixels that are either blood vessels
+or axons, the cell count and size, and average distance between cells and their nearest neighbor.
+Further details on the experimental setup for this task can be found in the Appendix in Section 5.
+We use the trained models from Task 1, freeze their weights, and compute the representations of
+all of the annotated images used in ROI-C1 (both train and test, 360 slices). From these latent
+Table 3: Task 3: R2 scores on multi-task feature readout for supervised and SSL models trained in Task 1.
+I. Linear Readouts from Models Trained on a Single Subvolume (ROI-C1)
+Methods
+Vessels
+Axons
+Cell Count
+Cell Size
+Dist (k=1)
+Supervised
+0.77 ± 0.06
+0.94 ± 0.01
+0.67 ± 0.06
+0.61 ± 0.05
+0.48 ± 0.05
+Sup w/ Mixup
+0.82 ± 0.02
+0.95 ± 0.00
+0.71 ± 0.02
+0.67 ± 0.03
+0.47 ± 0.02
+BYOL
+0.85 ± 0.01
+0.94 ± 0.01
+0.75 ± 0. 01
+0.69 ± 0.01
+0.49 ± 0.01
+MYOW
+0.85 ± 0.01
+0.94 ± 0.01
+0.74 ± 0.01
+0.69 ± 0.01
+0.50 ± 0.02
+MYOW-m
+0.87 ± 0.01
+0.95 ± 0.01
+0.77 ± 0.01
+0.69 ± 0.01
+0.51 ± 0.01
+PCA
+0.75
+0.82
+0.55
+0.47
+0.31
+NMF
+0.81
+0.85
+0.59
+0.55
+0.34
+II. Linear Readouts from Models Trained on Two Subvolumes (ROI-C3)
+Methods
+Vessels
+Axons
+Cell Count
+Cell Size
+Dist (k=1)
+Supervised
+0.79 ± 0.02
+0.94 ± 0.02
+0.73 ± 0.02
+0.63 ± 0.04
+0.49 ± 0.02
+Sup w/ Mixup
+0.75 ± 0.04
+0.88 ± 0.04
+0.64 ± 0.04
+0.54 ± 0.07
+0.37 ± 0.05
+BYOL
+0.88 ± 0.00
+0.96 ± 0.00
+0.79 ± 0.00
+0.73 ± 0.01
+0.53 ± 0.02
+MYOW
+0.88 ± 0.01
+0.96 ± 0.00
+0.79 ± 0.01
+0.72 ± 0.01
+0.52 ± 0.01
+MYOW-m
+0.87 ± 0.01
+0.96 ± 0.01
+0.78 ± 0.01
+0.72 ± 0.01
+0.53 ± 0.01
+PCA
+0.75
+0.82
+0.53
+0.46
+0.29
+NMF
+0.75
+0.83
+0.56
+0.49
+0.31
+9
+
+original
+ground truth
+prediction -3 class
+prediction - 4 class
+A
+B
+ axon
+cell
+vessel
+original
+prediction - 3 class
+prediction - 4 class
+StriatumFigure 4: Visualization of the representations learned by MYOW-m in Task 1 (ROI-C1) with global semantic
+attributes for each image visualized as different colors. From left to right, latents are colored by brain area
+(class), % blood vessels, % axons, cell count, and cell size.
+features, we fit a simple linear regression model on these representations to predict each of the
+desired semantic attributes. We have limited data for this task and thus report the R2 on all of the
+latents (no train and test split in this case).
+We report the R2 values in Table 3 for all 360 slices in the densely annotated volume used in Task 1
+and 2 for the models trained in ROI-C1 (Table 3 I) and on more data in ROI-C3 (Table 3 II). In both
+cases, we find that the SSL models significantly outperform the rest of the baseline models. The gap
+is more pronounced in the two volume training condition, highlighting the generalization difference
+between supervised and SSL approaches (59) in providing good representations for a wide range of
+tasks. In Figure 4, we project the MYOW-m (ROI-C1) embeddings using UMAP (60), and overlay
+the semantic features.
+5
+Discussion
+In this paper, we introduced a multi-task benchmark for analysis of brain structure from high-
+resolution neuroimaging data spanning many brain areas. In addition, we also built general in-
+frastructure for training models from datasets in the BossDB framework, like dataloaders which can
+be adapted to different datasets. This will expand the use of large-scale volumetric neuroimaging
+data for machine learning tool development.
+Limitations and future work. In neuroscience, often obtaining high quality labeled data (especially
+for dense segmentation tasks like those provided in Task 2), is very costly (61). The intensive nature
+of manual annotation and proofreading data thus limits the amount of labeled and annotated data
+that we provide to train models on, or the amount of distributional shift that can be assessed. While
+this is a limitation of the work, it also is an accurate reflection of the challenges faced in the field,
+and thus requires more label-efficient approaches for learning like the SSL methods we highlight.
+Moving forward, we hope to leverage the models tested in this work to generate even more high
+quality annotated data to further improve model performance and segmentation.
+When designing our current benchmark, we focused on building a multi-scale challenge where vari-
+ability was due to changes in brain structure and not different preparations or imaging parameters.
+Therefore we focused on a single animal where we have a large intact brain volume that spans many
+heterogeneous brain regions. While this may limit the generalization of the models to new sam-
+ples or datasets, it also addresses the heterogeneous nature of different brain regions that is often
+overlooked. Our results show that even within a single brain, there is rich heterogeneity across dif-
+ferent brain regions that makes it difficult for some models to generalize. This also reveals important
+generalization gaps between SL and SSL models.
+In the future, we hope to expand this effort to learn models of brain structure from other non-
+convolutional architectures (e.g., point cloud-based models of neural structure (62)), and deploy our
+tools on new multi-scale brain datasets, perhaps using new lightsheet (63) or whole-brain scaling
+(64; 65) techniques.
+Broader impacts. The high heterogeneity of brain data, coupled with the variability in the scale
+and nature of the tasks presented in this benchmark make it a challenging and useful resource for
+the broader ML community. Furthermore, we hope the provided dataset and tasks, as well as the
+supporting codebase and BossDB infrastructure will help accelerate development of ML techniques
+for emerging, high-resolution neuroimaging datasets being collected in the broader community.
+10
+
+Brain Area
+Blood
+Axons (%)
+Cell Count
+Cell Size
+234
+.
+12
+40
+16
+50
+28
+32
+Cortex
+Striatum
+VP
+ZI
+Brain Area
+Blood Vessels (%)
+Axons (%)
+Cell Size
+Cell Count
+15
+10
+1(
+20
+30
+12
+AO
+16
+Cortex
+50
+Striatum
+60
+C
+24
+VP
+80
+32Acknowledgements
+This project was supported by NSF award IIS-2039741, NIH award 1R01EB029852-01, NIH award
+R01MH126684, NIH award R24MH114785 in addition to generous gifts from the Alfred Sloan
+Foundation, the McKnight Foundation, and the CIFAR Azrieli Global Scholars Program.
+References
+[1] Danielle S Bassett and Edward T Bullmore, “Small-world brain networks revisited,” The
+Neuroscientist, vol. 23, no. 5, pp. 499–516, 2017.
+[2] Ran Liu, Cem Subakan, Aishwarya H. Balwani, Jennifer Whitesell, Julie Harris, Sanmi
+Koyejo, and Eva L. Dyer, “A generative modeling approach for interpreting population-level
+variability in brain structure,” in Medical Image Computing and Computer Assisted Interven-
+tion – MICCAI 2020, Cham, 2020, pp. 257–266, Springer International Publishing.
+[3] Casey M Schneider-Mizell, Agnes L Bodor, Forrest Collman, Derrick Brittain, Adam A Bleck-
+ert, Sven Dorkenwald, Nicholas L Turner, Thomas Macrina, Kisuk Lee, Ran Lu, et al., “Chan-
+delier cell anatomy and function reveal a variably distributed but common signal,” bioRxiv,
+2020.
+[4] Timothy EJ Behrens and Olaf Sporns, “Human connectomics,” Current opinion in neurobiol-
+ogy, vol. 22, no. 1, pp. 144–153, 2012.
+[5] Michael Hawrylycz, Lydia Ng, David Feng, Susan Sunkin, Aaron Szafer, and Chinh Dang,
+“The allen brain atlas,” Springer Handbook of Bio-/Neuroinformatics, pp. 1111–1126, 2014.
+[6] Alexander Shapson-Coe, Michał Januszewski, Daniel R Berger, Art Pope, Yuelong Wu, Tim
+Blakely, Richard L Schalek, Peter H Li, Shuohong Wang, Jeremy Maitin-Shepard, et al., “A
+connectomic study of a petascale fragment of human cerebral cortex,” bioRxiv, 2021.
+[7] Federico Scala, Dmitry Kobak, Matteo Bernabucci, Yves Bernaerts, Cathryn Ren´e Cadwell,
+Jesus Ramon Castro, Leonard Hartmanis, Xiaolong Jiang, Sophie Laturnus, Elanine Miranda,
+et al., “Phenotypic variation of transcriptomic cell types in mouse motor cortex,” Nature, vol.
+598, no. 7879, pp. 144–150, 2021.
+[8] Jeff W Lichtman, Hanspeter Pfister, and Nir Shavit, “The big data challenges of connectomics,”
+Nature neuroscience, vol. 17, no. 11, pp. 1448–1454, 2014.
+[9] Alessandro Motta, Meike Schurr, Benedikt Staffler, and Moritz Helmstaedter, “Big data in
+nanoscale connectomics, and the greed for training labels,” Current opinion in neurobiology,
+vol. 55, pp. 180–187, 2019.
+[10] Robert Hider Jr, Dean Kleissas, Timothy Gion, Daniel Xenes, Jordan Matelsky, Derek Pryor,
+Luis Rodriguez, Erik C Johnson, William Gray-Roncal, and Brock Wester, “The brain observa-
+tory storage service and database (bossdb): A cloud-native approach for petascale neuroscience
+discovery,” Frontiers in Neuroinformatics, vol. 16, 2022.
+[11] Narayanan Kasthuri, Kenneth Jeffrey Hayworth, Daniel Raimund Berger, Richard Lee
+Schalek, Jos´e Angel Conchello, Seymour Knowles-Barley, Dongil Lee, Amelio V´azquez-
+Reina, Verena Kaynig, Thouis Raymond Jones, et al., “Saturated reconstruction of a volume
+of neocortex,” Cell, vol. 162, no. 3, pp. 648–661, 2015.
+[12] Eva L Dyer, William Gray Roncal, Judy A Prasad, Hugo L Fernandes, Doga G¨ursoy, Vin-
+cent De Andrade, Kamel Fezzaa, Xianghui Xiao, Joshua T Vogelstein, Chris Jacobsen, et al.,
+“Quantifying mesoscale neuroanatomy using x-ray microtomography,” eNeuro, vol. 4, no. 5,
+2017.
+[13] C Shan Xu, Michal Januszewski, Zhiyuan Lu, Shin-ya Takemura, Kenneth J Hayworth, Gary
+Huang, Kazunori Shinomiya, Jeremy Maitin-Shepard, David Ackerman, Stuart Berg, et al., “A
+connectome of the adult drosophila central brain,” bioRxiv, 2020.
+11
+
+[14] Zhihao Zheng, J Scott Lauritzen, Eric Perlman, Camenzind G Robinson, Matthew Nichols,
+Daniel Milkie, Omar Torrens, John Price, Corey B Fisher, Nadiya Sharifi, et al., “A complete
+electron microscopy volume of the brain of adult drosophila melanogaster,” Cell, vol. 174, no.
+3, pp. 730–743, 2018.
+[15] Judy A Prasad, Aishwarya H Balwani, Erik C Johnson, Joseph D Miano, Vandana Sampathku-
+mar, Vincent De Andrade, Kamel Fezzaa, Ming Du, Rafael Vescovi, Chris Jacobsen, et al., “A
+three-dimensional thalamocortical dataset for characterizing brain heterogeneity,” Scientific
+Data, vol. 7, no. 1, pp. 1–7, 2020.
+[16] Aishwarya Balwani, Joseph Miano, Ran Liu, Lindsey Kitchell, Judy A Prasad, Erik C Johnson,
+William Gray-Roncal, and Eva L Dyer, “Multi-scale modeling of neural structure in x-ray
+imagery,” in 2021 IEEE International Conference on Image Processing (ICIP). IEEE, 2021,
+pp. 141–145.
+[17] Bjoern H Menze, Andras Jakab, Stefan Bauer, Jayashree Kalpathy-Cramer, Keyvan Farahani,
+Justin Kirby, Yuliya Burren, Nicole Porz, Johannes Slotboom, Roland Wiest, et al., “The mul-
+timodal brain tumor image segmentation benchmark (brats),” IEEE Transactions on Medical
+Imaging, vol. 34, no. 10, pp. 1993–2024, 2014.
+[18] Liang Chen, Paul Bentley, Kensaku Mori, Kazunari Misawa, Michitaka Fujiwara, and Daniel
+Rueckert, “Self-supervised learning for medical image analysis using image context restora-
+tion,” Medical Image Analysis, vol. 58, pp. 101539, 2019.
+[19] Dheerendranath Battalapalli, B. Rao, P. Yogeeswari, Chandrasekharan Kesavadas, and
+Venkateswaran Rajagopalan,
+“An optimal brain tumor segmentation algorithm for clinical
+mri dataset with low resolution and non-contiguous slices,” BMC Medical Imaging, vol. 22,
+05 2022.
+[20] Javeria Amin, Muhammad Anjum, Muhammad Sharif, Saima Jabeen, Seifedine Kadry, and
+Pablo Ger,
+“A new model for brain tumor detection using ensemble transfer learning and
+quantum variational classifier,” Computational Intelligence and Neuroscience, vol. 2022, pp.
+1–13, 04 2022.
+[21] Ghazanfar Latif, Ghassen Ben Brahim, D. N. F. Awang Iskandar, Abul Bashar, and Jaafar Al-
+ghazo, “Glioma tumors’ classification using deep-neural-network-based features with
+svm classifier,” Diagnostics, vol. 12, no. 4, 2022.
+[22] C.R. Jack Jr., M.A. Bernstein, N.C. Fox, P. Thompson, G. Alexander, D. Harvey, B. Borowski,
+P.J. Britson, J.L. Whitwell, C. Ward, A.M. Dale, J.P. Felmlee, J.L. Gunter, D.L.G. Hill, R. Kil-
+liany, N. Schuff, S. Fox-Bosetti, C. Lin, C. Studholme, C.S. DeCarli, G. Krueger, H.A. Ward,
+G.J. Metzger, K.T. Scott, R. Mallozzi, D. Blezek, J. Levy, J.P. Debbins, A.S. Fleisher, M. Al-
+bert, R. Green, G. Bartzokis, G. Glover, J. Mugler, and M.W. Weiner, “The alzheimer’s disease
+neuroimaging initiative (adni): Mri methods,” Journal of Magnetic Resonance Imaging, vol.
+27, no. 4, pp. 685–691, 2008.
+[23] Ian B Malone, David Cash, Gerard R Ridgway, David G MacManus, Sebastien Ourselin,
+Nick C Fox, and Jonathan M Schott,
+“Miriad—public release of a multiple time point
+alzheimer’s mr imaging dataset,” NeuroImage, vol. 70, pp. 33–36, 2013.
+[24] Manhua Liu, Danni Cheng, Kundong Wang, and Yaping Wang, “Multi-modality cascaded
+convolutional neural networks for alzheimer’s disease diagnosis,” Neuroinformatics, vol. 16,
+pp. 295–308, 2018.
+[25] Donghuan Lu, Karteek Popuri, Gavin Weiguang Ding, Rakesh Balachandar, and Mirza Faisal
+Beg, “Multiscale deep neural network based analysis of fdg-pet images for the early diagnosis
+of alzheimer’s disease,” Medical Image Analysis, vol. 46, pp. 26–34, 2018.
+[26] Jyoti Islam and Yanqing Zhang, “Brain mri analysis for alzheimer’s disease diagnosis using
+an ensemble system of deep convolutional neural networks,” Brain informatics, vol. 5, no. 2,
+pp. 1–14, 2018.
+12
+
+[27] Mingxia Liu, Jun Zhang, Ehsan Adeli, and Dinggang Shen, “Joint classification and regression
+via deep multi-task multi-channel learning for alzheimer’s disease diagnosis,” IEEE Transac-
+tions on Biomedical Engineering, vol. 66, no. 5, pp. 1195–1206, 2019.
+[28] Alejandro Puente-Castro, Enrique Fernandez-Blanco, Alejandro Pazos, and Cristian R.
+Munteanu, “Automatic assessment of alzheimer’s disease diagnosis based on deep learning
+techniques,” Computers in Biology and Medicine, vol. 120, pp. 103764, 2020.
+[29] Eman N. Marzban, Ayman M. Eldeib, Inas A. Yassine, Yasser M. Kadah, and for the
+Alzheimer’s Disease Neurodegenerative Initiative, “Alzheimer’s disease diagnosis from dif-
+fusion tensor images using convolutional neural networks,” PLOS ONE, vol. 15, no. 3, pp.
+1–16, 03 2020.
+[30] N. Deepa and S.P. Chokkalingam, “Optimization of vgg16 utilizing the arithmetic optimiza-
+tion algorithm for early detection of alzheimer’s disease,” Biomedical Signal Processing and
+Control, vol. 74, pp. 103455, 2022.
+[31] Sudlow C, Gallacher J, Allen N, Beral V, Burton P, Danesh J, et al., “Uk biobank: An open
+access resource for identifying the causes of a wide range of complex diseases of middle and
+old age,” PLoS Med 12(3): e1001779., 2015.
+[32] Zhihao Zheng, Cam G. Robinson, Daniel Milkie, Eric Perlman, John Price, Davi Bock, Misha
+Kazhdan, Khaled Khairy, Bill Karsh, Eric Trautman, Peter Li, Chris Ordish, Chong Zhang, Ju-
+lia Buhmann, Jan Funke, and Stephan Saalfeld, “MICCAI challenge on circuit reconstruction
+from electron microscopy images,” https://cremi.org/data/, 2022.
+[33] Albert Cardona, Stephan Saalfeld, Stephan Preibisch, Benjamin Schmid, Anchi Cheng, Jim
+Pulokas, Pavel Tomancak, and Volker Hartenstein, “An integrated micro-and macroarchitec-
+tural analysis of the drosophila brain by computer-assisted serial section electron microscopy,”
+PLOS Biology, vol. 8, no. 10, pp. e1000502, 2010.
+[34] Kisuk Lee, Jonathan Zung, Peter Li, Viren Jain, and H Sebastian Seung, “Superhuman accu-
+racy on the snemi3d connectomics challenge,” arXiv preprint arXiv:1706.00120, 2017.
+[35] Stephen M Plaza, “Focused proofreading to reconstruct neural connectomes from em images at
+scale,” in Deep Learning and Data Labeling for Medical Applications, pp. 249–258. Springer,
+2016.
+[36] Donglai Wei, Kisuk Lee, Hanyu Li, Ran Lu, J Alexander Bae, Zequan Liu, Lifu Zhang,
+M´arcia dos Santos, Zudi Lin, Thomas Uram, et al., “Axonem dataset: 3d axon instance seg-
+mentation of brain cortical regions,” in International Conference on Medical Image Computing
+and Computer-Assisted Intervention. Springer, 2021, pp. 175–185.
+[37] Donglai Wei, Zudi Lin, Daniel Franco-Barranco, Nils Wendt, Xingyu Liu, Wenjie Yin, Xin
+Huang, Aarush Gupta, Won-Dong Jang, Xueying Wang, et al., “Mitoem dataset: large-scale
+3d mitochondria instance segmentation from em images,”
+in International Conference on
+Medical Image Computing and Computer-Assisted Intervention. Springer, 2020, pp. 66–76.
+[38] Philipp Berens, Jeremy Freeman, Thomas Deneux, Nikolay Chenkov, Thomas McColgan, Ar-
+tur Speiser, Jakob H Macke, Srinivas C Turaga, Patrick Mineault, Peter Rupprecht, et al.,
+“Community-based benchmarking improves spike rate inference from two-photon calcium
+imaging data,” PLOS Computational Biology, vol. 14, no. 5, pp. e1006157, 2018.
+[39] Katrin Amunts, Claude Lepage, Louis Borgeat, Hartmut Mohlberg, Timo Dickscheid, Marc-
+´Etienne Rousseau, Sebastian Bludau, Pierre-Louis Bazin, Lindsay B Lewis, Ana-Maria Oros-
+Peusquens, et al., “Bigbrain: an ultrahigh-resolution 3d human brain model,” Science, vol.
+340, no. 6139, pp. 1472–1475, 2013.
+[40] Michał Januszewski, J¨orgen Kornfeld, Peter H Li, Art Pope, Tim Blakely, Larry Lindsey,
+Jeremy Maitin-Shepard, Mike Tyka, Winfried Denk, and Viren Jain, “High-precision auto-
+mated reconstruction of neurons with flood-filling networks,” Nature Methods, vol. 15, no. 8,
+pp. 605–610, 2018.
+13
+
+[41] Peter H Li, Larry F Lindsey, Michał Januszewski, Mike Tyka, Jeremy Maitin-Shepard, Tim
+Blakely, and Viren Jain, “Automated reconstruction of a serial-section em drosophila brain
+with flood-filling networks and local realignment,” Microscopy and Microanalysis, vol. 25,
+no. S2, pp. 1364–1365, 2019.
+[42] Aishwarya H Balwani and Eva L Dyer, “Modeling variability in brain architecture with deep
+feature learning,” in 2019 53rd Asilomar Conference on Signals, Systems, and Computers.
+IEEE, 2019, pp. 1186–1191.
+[43] Jordan Matelsky, Luis Rodriguez, Daniel Xenes, Timothy Gion, Robert Hider, Brock Wester,
+and William Gray-Roncal, “Intern: Integrated toolkit for extensible and reproducible neuro-
+science,” bioRxiv, 2020.
+[44] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun, “Deep residual learning for image
+recognition,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern
+Recognition (CVPR), 2016, pp. 770–778.
+[45] Hongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz, “mixup: Beyond
+empirical risk minimization,” arXiv preprint arXiv:1710.09412, 2017.
+[46] Jean-Bastien Grill, Florian Strub, Florent Altch´e, Corentin Tallec, Pierre Richemond, Elena
+Buchatskaya, Carl Doersch, Bernardo Avila Pires, Zhaohan Guo, Mohammad Ghesh-
+laghi Azar, et al., “Bootstrap your own latent-a new approach to self-supervised learning,”
+Advances in Neural Information Processing Systems, vol. 33, pp. 21271–21284, 2020.
+[47] Mehdi Azabou, Mohammad Gheshlaghi Azar, Ran Liu, Chi-Heng Lin, Erik C Johnson, Kiran
+Bhaskaran-Nair, Max Dabagia, Bernardo Avila-Pires, Lindsey Kitchell, Keith B Hengen, et al.,
+“Mine your own view: Self-supervised learning through across-sample prediction,”
+arXiv
+preprint arXiv:2102.10106, 2021.
+[48] Olaf Ronneberger, Philipp Fischer, and Thomas Brox, “U-net: Convolutional networks for
+biomedical image segmentation,” CoRR, vol. abs/1505.04597, 2015.
+[49] Johannes Schmidt, “Pytorch-2d-3d-unet-tutorial,” https://github.com/johschmidt42/
+PyTorch-2D-3D-UNet-Tutorial, 2021.
+[50] Pavel Yakubovskiy,
+“Segmentation models pytorch,” https://github.com/qubvel/
+segmentation_models.pytorch, 2020.
+[51] Tongle Fan, Guanglei Wang, Yan Li, and Hongrui Wang, “Ma-net: A multi-scale attention
+network for liver and tumor segmentation,” IEEE Access, vol. 8, pp. 179656–179665, 2020.
+[52] Tsung-Yi Lin, Piotr Doll´ar, Ross Girshick, Kaiming He, Bharath Hariharan, and Serge Be-
+longie, “Feature pyramid networks for object detection,” in Proceedings of the IEEE/CVF
+Conference on Computer Vision and Pattern Recognition (CVPR), 2017, pp. 936–944.
+[53] Zongwei Zhou, Md Mahfuzur Rahman Siddiquee, Nima Tajbakhsh, and Jianming Liang,
+“Unet++: A nested u-net architecture for medical image segmentation,” in Deep Learning in
+Medical Image Analysis and Multimodal Learning for Clinical Decision Support, Danail Stoy-
+anov, Zeike Taylor, Gustavo Carneiro, Tanveer Syeda-Mahmood, Anne Martel, Lena Maier-
+Hein, Jo˜ao Manuel R.S. Tavares, Andrew Bradley, Jo˜ao Paulo Papa, Vasileios Belagiannis,
+Jacinto C. Nascimento, Zhi Lu, Sailesh Conjeti, Mehdi Moradi, Hayit Greenspan, and Anant
+Madabhushi, Eds., Cham, 2018, pp. 3–11, Springer International Publishing.
+[54] Hanchao Li, Pengfei Xiong, Jie An, and Lingxue Wang, “Pyramid attention network for se-
+mantic segmentation,” CoRR, vol. abs/1805.10180, 2018.
+[55] Hengshuang Zhao, Jianping Shi, Xiaojuan Qi, Xiaogang Wang, and Jiaya Jia, “Pyramid scene
+parsing network,” CoRR, vol. abs/1612.01105, 2016.
+[56] Adaloglou Nikolaos, “Deep learning in medical image analysis: a comparative analysis of
+multi-modal brain-mri segmentation with 3d deep neural networks,” M.S. thesis, University of
+Patras, 2019, https://github.com/black0017/MedicalZooPytorch.
+14
+
+[57] Fausto Milletari, Nassir Navab, and Seyed-Ahmad Ahmadi, “V-net: Fully convolutional neural
+networks for volumetric medical image segmentation,” CoRR, vol. abs/1606.04797, 2016.
+[58] Wenqi Li, Guotai Wang, Lucas Fidon, S´ebastien Ourselin, M. Jorge Cardoso, and Tom Ver-
+cauteren, “On the compactness, efficiency, and representation of 3d convolutional networks:
+Brain parcellation as a pretext task,” CoRR, vol. abs/1707.01992, 2017.
+[59] Atharva Tendle and Mohammad Rashedul Hasan, “A study of the generalizability of self-
+supervised representations,” Machine Learning with Applications, vol. 6, pp. 100124, 2021.
+[60] Leland McInnes, John Healy, and James Melville, “Umap: Uniform manifold approximation
+and projection for dimension reduction,” arXiv preprint arXiv:1802.03426, 2018.
+[61] David Rolnick and Eva L Dyer,
+“Generative models and abstractions for large-scale neu-
+roanatomy datasets,” Current Opinion in Neurobiology, vol. 55, pp. 112–120, 2019, Machine
+Learning, Big Data, and Neuroscience.
+[62] Joy M Jackson, Ran Liu, and Eva L Dyer, “Building representations of different brain areas
+through hierarchical point cloud networks,” in Medical Imaging with Deep Learning, 2022.
+[63] Elizabeth M.C. Hillman, Venkatakaushik Voleti, Wenze Li, and Hang Yu, “Light-sheet mi-
+croscopy in neuroscience,” Annual Review of Neuroscience, vol. 42, no. 1, pp. 295–313, 2019,
+PMID: 31283896.
+[64] Scott Trinkle, Sean Foxley, Narayanan Kasthuri, and Patrick La Rivi`ere, “Synchrotron x-ray
+micro-ct as a validation dataset for diffusion mri in whole mouse brain,” Magnetic Resonance
+in Medicine, vol. 86, no. 2, pp. 1067–1076, 2021.
+[65] Sean Foxley, Vandana Sampathkumar, Vincent De Andrade, Scott Trinkle, Anastasia Sorokina,
+Katrina Norwood, Patrick La Riviere, and Narayanan Kasthuri, “Multi-modal imaging of a
+single mouse brain over five orders of magnitude of resolution,” NeuroImage, vol. 238, pp.
+118250, 2021.
+[66] Paul A. Yushkevich, Joseph Piven, Heather Cody Hazlett, Rachel Gimpel Smith, Sean Ho,
+James C. Gee, and Guido Gerig, “User-guided 3D active contour segmentation of anatomical
+structures: Significantly improved efficiency and reliability,” Neuroimage, vol. 31, no. 3, pp.
+1116–1128, 2006.
+15
+
+Appendix
+1
+Dataset Details
+1.1
+Spatial autocorrelation between images
+We computed the cross covariance of the images in our main subvolumes to see how much cor-
+relation exists between consecutive or nearby image slices. This analysis confirmed that after 5-7
+slices, the cross-correlation between images drops off consistently (see Figure A1). Based upon this
+analysis, we crop out 10 slices between the training and testing sets in all the subvolumes.
+cortex 
+striatum
+VP 
+ZI
+Full
+zoomed in
+Figure A1: Cross-covariance matrices. Per-class cross-covariance of all 4 considered regions (right), zoomed-
+in view of cortex cross-covariance(top left) and cross-covariance of entire dataset (bottom left).
+1.2
+Dense pixel-level annotations
+Utilizing the sparse annotations from (15), we trained a 2D UNet to segment the image data into
+4 classes (blood vessels, cell bodies, axons, and background) and applied it to the slices in the
+four volumes which were not already annotated. A proofreader then reviewed the annotations for
+each slice in the volume using ITK-Snap (66) in the y-z plane. During the proofreading process
+of a slice, any structure with labels that were split between classes (errors in the UNet) were first
+corrected. Then, any structures without split labels were reviewed for correctness and changed if
+incorrect. Finally, the annotator checked the following Z slice to ensure continuity of components.
+This process continued for each slice in the volume. To ensure consistency of the annotations, the
+volume was proofread by the same annotator a second time, now going through the x-z plane. After
+proofreading we generated densely annotated volumes of size 256x256x360 in each of the four
+ROIs.
+1.3
+Interpolated ROI annotations
+A full ROI annotation of the x-ray microtomography dataset was interpolated from manually anno-
+tated z-level slices that were roughly 50 pixels apart. To generate the interpolated layers, a recursive
+algorithm was implemented to linearly shift the boundary between two annotation classes from one
+pre-existing layer to the next: once a shifting boundary region was located, the midpoint of the shift
+was identified and full, one voxel-wide annotations were created along the midpoint spanning all
+necessary z levels. This process split transition regions into two roughly equal spaces, in which the
+recursive process repeated for the two respective transition spaces until the entire transition region
+was fully annotated.
+16
+
+0
+1.0
+200
+0.8
+400
+600
+0.6
+800
+0.4
+1000
+1200
+0.2
+1400
+250
+500
+0.0
+0
+750
+1000
+12500
+10
+1.0
+50
+50
+0.8
+0.8
+100
+100
+150
+0.6
+150
+0.6
+200
+200
+0.4
+0.4
+250
+250
+300
+0.2
+300
+0.2
+350
+0.0
+350
+300
+OOE
+0.0
+0
+100
+200
+0
+100
+200
+0
+10
+0
+10
+50
+50
+80
+0.8
+100
+100
+150
+0.6
+150
+0.6
+200
+200
+0.4
+250
+250
+300
+0.2
+300
+0.2
+350
+0.0
+350
+200
+300
+100
+200
+300
+0.0
+0
+100
+0However, this process was not able to account for all border transitions. Thus, about 10% of the
+border transitions were annotated manually using the visualization software application ITK-SNAP.
+Fully interpolated ROI annotations are available for z-levels between 109 and 459, inclusive.
+Figure A2: The Recursive Process of the Interpolation of ROI Annotations. A four step recursive process
+was implemented to produce interpolated ROI annotation layers from manually annotated layers. This process
+recursively repeated until the entire transition region was interpolated with ROI annotations.
+2
+Data Access
+To allow for easy, publicly accessible data, the dataset is stored in the BossDB system. Critically,
+this database enables efficient access of arbitrary cutouts of large volumetric datasets. This project
+page is available at https://bossdb.org/project/prasad2020. No user account is required.
+Public credentials will allow read-only access of the data without the need for user account creation.
+The raw images are available with the resource identifier bossdb://prasad/prasad2020/image,
+which is the “prasad” experiment, “prasad2020” experiment, and “image” channel. The raw image
+data are in an unsigned 8-bit integer format, with preferred indexing in ZY X format. Areas with
+invalid data (e.g. outside the image volume) are assigned the value 0. The maximum and minimum
+indices are z = [0, 720], y = [0, 1420], and x = [0, 5805]. The data have 1.17µm isotropic
+resolution.
+The annotation images are available with the resource identifier bossdb://prasad/prasad_
+analysis/pixel_labels and bossdb://prasad/prasad_analysis/roi_labels, which is
+the “prasad” collection, “prasad analysis” experiment, and “pixel labels” and “roi labels” chan-
+nels. The annotation data are in an unsigned 64-bit integer format, with preferred indexing in ZY X
+format. Areas with invalid data (e.g. outside the image volume) are assigned the value 0. As for
+the raw images channel, the maximum and minimum indices are z = [0, 720], y = [0, 1420], and
+x = [0, 5805]. The data have 1.17µm isotropic resolution.
+Access is available through the Python intern array API, which provides numpy-like referencing.
+Using this library, the user creates an intern array, image_array = array(boss_url), where
+boss_url is one of the resource identifiers listed above. The user can then access data from the
+channel using the numpy-like syntax, demonstrated by this code example to download a cutout
+corresponding to cortex.
+from intern import array
+image = array(boss_url)
+data = image[110:379, 900:1156, 4600:4856]
+A data loader is provided for rapid development of Pytorch models, which is used for all mod-
+els tested in this work. On creation, the data loader loads a task configuration ‘.json’ file which
+specifies the parameters of the task. Two modes of operation are allowed, download=true and
+download=false. For the former, on creation of the data loader, data are pulled locally and stored
+17
+
+as a numpy file. If the numpy file already exists, it is instead loaded from disk. In the later case,
+data are downloaded to memory and not written to disk. The data are stored in a numpy tensor
+containing the concatenated slices from cortex, striatum, vp, zi (for four class problems). The item
+retrieval function can serve up integer region labels (for task 1), or microstructure masks (for task
+2), as dictated by the configuration. The basic input data configuration parameters are:
+• image chan - BossDB channel from which to pull the raw image data.
+• annotation chan - BossDB channel from which to pull the image annotations (either
+macro- or micro-structures).
+• xrange cor - range along the x-axis (on the full data) for the slices from Cortex.
+• yrange cor - range along the y-axis (on the full data) for the slices from Cortex.
+• xrange stri - range along the x-axis (on the full data) for the slices from Striatum.
+• yrange stri - range along the y-axis (on the full data) for the slices from Striatum.
+• xrange vp - range along the x-axis (on the full data) for the slices from VP.
+• yrange vp - range along the y-axis (on the full data) for the slices from VP.
+• xrange zi - range along the x-axis (on the full data) for the slices from ZI.
+• yrange zi - range along the y-axis (on the full data) for the slices from ZI.
+• z train - the range (slices) along the z-axis to use as the train set.
+• z val - the range (slices) along the z-axis to use as the val set.
+• z test - the range (slices) along the z-axis to use as the test set. A buffer of 10 slices is
+recommended between the train/val set and the test set.
+• volume z - the number of slices to include in a volume slice. For the 2D cases, the individ-
+ual slices are the input to the models (number of slices in a volume slice is 1). In the 3D
+case, volume slices are the input to the models.
+The repository, https://github.com/MTNeuro/MTNeuro, contains scripts and python notebooks
+for data access and running models. Python notebooks which download the numpy files for im-
+ages and annotations are provided for users not using Pytorch. These are based on the original data
+access notebooks developed for this dataset https://github.com/nerdslab/xray-thc. The
+repository and requirements can be installed via the pip package manager. The repository is struc-
+tured as
+• MTNeuro: the core code for running pytorch dataloaders and pytorch models
+• Notebooks: access notebooks for users who do not use pytorch
+• Scripts: example scripts for running pytorch models for each task, which form the basis for
+new algorithm development
+3
+Further Details on Task 1
+3.1
+Models
+3.1.1
+Supervised models
+• Resnet18: 18-layer version of a model that reformulates the layers as learning residual
+functions with reference to the layer inputs in order to efficiently train deeper architectures
+(44).
+• Resnet18 + Mixup: same model architecture described above (Resnet18), but trained using
+mixup, an augmentation that mixes inputs as well as their corresponding labels through a
+convex combination, and has been shown to yield significant improvement in supervised
+classification models (45)
+18
+
+3.1.2
+Self-supervised models
+We test several contrastive-based SSL methods that do not need negative examples to generate good
+representations of data (46). All SSL models are trained using a Resnet18 encoder backbone in
+order to make them comparable to the evaluated supervised models.
+We tested the following SSL models:
+• BYOL: BYOL relies on a mirrored structure of online and target networks that learn from
+each other (46). In particular, the online network tries to predict the encoding of a target
+image view x by minimizing the distance in latent space to an augmented view xa.
+• MYOW: MYOW builds on BYOL by also minizing the distance in latent space between
+views, but incorporates a view mining approach in order to search the dataset for neigh-
+boring samples in representations space. The augmented and mined samples are then inte-
+grated into a unified latent space through the use of an additional predictor network (47).
+• MYOW-m: MYOW-m is an extension of MYOW that integrates MYOW’s sample mining
+procedure, but uses a single predictor on both the augmented and mined views rather than
+using the cascaded projector design proposed in MYOW.
+Since all of these SSL methods rely on some form of transformation to process the views during
+training, we choose the simplest possible augmentation for all three methods: randomly cropping
+patches half the width and height of each image sample to generate augmented and mined views (see
+Figure A3). In order to standardize the evaluation of the methods, we evaluate the trained encoder’s
+performance in classifying which class each of the four corners of an image sample belongs to.
+Figure A3: View generation for self-supervised approaches. In A we show how augmented views are gener-
+ated by extracting random crops from a given image sample. In B, we schematize the view mining process,
+involving comparing different random crops from a pool of samples to a target view (red) and choosing the
+closest one in representation space as the mined view (Image 3, highlighted).
+3.2
+More Details on Training
+3.2.1
+Configuration Files
+Two types configuration files are provided as input for training:
+• Network Configuration - a ’.json’ file containing settings for the optimizer, augmentations
+to use, the parameters for the corresponding supervised or self-supervised models, and the
+seed to use for the training.
+• Task Configuration - a ’.json’ file specifying information regarding the data slices to ac-
+cess for the training. This includes the database channel to access for the data and region-
+level annotations, the x and y ranges for the slices from each region, the size of the slices
+(corresponding to the level of downsampling applied), the train-val-test split, the depth of
+each slice sample (1, since we focus on 2D processing for this task).
+19
+
+random
+A
+crop
+resize
+view augmentation
+random
+crop
+B Image 1
+Image 2
+Image 3
+Image 4
+view miningSetting
+Values cube set A
+xrange cor
+[4600,4856]
+yrange cor
+[900,1156]
+xrange stri
+[3700,3956]
+yrange stri
+[500,756]
+xrange vp
+[3063,3319]
+yrange vp
+[850,1106]
+xrange zi
+[1543,1799]
+yrange zi
+[650,906]
+z train
+[110, 379]
+z val
+[380, 409]
+z test
+[420, 470]
+Table A1: Dataset configuration used for task 1 (ROI-C1) and task 2
+Setting
+Values cube set B
+Values cube set C
+Values cube set D
+xrange cor
+[5312,5568]
+[5056,5312]
+[4600,4856]
+yrange cor
+[388,644]
+[644,900]
+[400,656]
+xrange stri
+[3828,4084]
+[3344,3600]
+[3800,4056]
+yrange stri
+[912,1168]
+[400,656]
+[244,500]
+xrange vp
+[2551,2807]
+[2151,2407]
+[2295,2551]
+yrange vp
+[850,1106]
+[950,1206]
+[694,950]
+xrange zi
+[1287,1543]
+[1031,1287]
+[1799,2055]
+yrange zi
+[906,1162]
+[906:1162]
+[650:906]
+Table A2: Coordinates for the additional cube sets used in ROI-C2 (C and D) and ROI-C3 (B, C and D)
+3.2.2
+Model Settings
+The basic model settings are:
+• model - either the classifier (if supervised) or encoder backbone (if SSL) to train for the
+classification task.
+• classes - the number of output classes of the model.
+• method - (only for SSL) the self-supervised approach to train.
+3.2.3
+Input Data Configuration
+The dataset configurations used for the results in Table 1, column ROI-C1 are listed in Table A1.
+The coordinates for the additional cubes used in ROI-C2 and ROI-C3 are shown in Table A2.
+3.2.4
+Training and evaluation setup
+We train all supervised and self-supervised models using a 0.03 learning rate, a batch size of 256
+(chosen for a stable linear layer evaluation) and 5 different random seed values: 1, 100, 350, 631
+and 872. We compile these 5 results in order to calculate the mean and standard deviation of the
+classification accuracy for each considered approach. For the Resnet18-Mixup setting, we evaluate
+5 different probabilities of applying the augmentation during training: 0.01, 0.1, 0.3, 0.5 and 0.7;
+and report the setting with the best performing mean accuracy.
+3.3
+Additional experiments
+We report additional tests in Table A3, where we evaluate all considered models under the ROI-
+C1 training setting, but using different downsampling factors on the image samples: 4x, 2x and
+full resolution (no downsampling). We observe that performance increases with the resolution (as
+more information is available to the models). Surprisingly, we see a significantly larger performance
+jump in the SSL methods when moving from 2x downsampling to full resolution tests with respect
+to the supervised methods (which increase their performance only by a slight margin). This further
+20
+
+Table A3: Results on image classification for brain area prediction (Task 1).
+ROI-C1
+Downsampling =4
+Downsampling =2
+Full-res
+Resnet18
+0.83 ± 0.06
+0.88 ± 0.03
+0.87 ± 0.1
+Resnet18-Mixup
+0.85 ± 0.06
+0.90 ± 0.04
+0.91 ± 0.03
+BYOL
+0.83 ± 0.04
+0.88 ± 0.02
+0.98 ± 0.01
+MYOW
+0.84 ± 0.04
+0.90 ± 0.02
+0.96 ± 0.03
+MYOW-m
+0.84 ± 0.05
+0.94 ± 0.02
+0.99 ± 0.01
+PCA
+0.59
+0.59
+0.59
+NMF
+0.55
+0.62
+0.54
+supports our observation that SSL models benefit more from exposure to additional data than their
+supervised counterparts. Each SSL training instance took on average 30 minutes to train under 4x
+downsampling, 1.5 hours under 2x downsampling, and 4 hours under full resolution setting (runtime
+for a single random seed, trained on an RTX 3090 GPU node). We chose to use 2x downsampling
+for the rest of our experiments in task 1, since it provided a good compromise between performance
+and runtime.
+4
+Further Details on Task 2
+4.1
+Baselines
+4.1.1
+2D Models
+The models we use for the 2D segmentation task are:
+• A standard 2D U-Net model (48; 49)
+• Selected models from the ’segmentation models.pytorch’ library (50):
+– MAnet - a model utilizing a multi-scale attention mechanism, originally design for
+liver and liver tumor segmentation (51),
+– FPN - the Feature Pyramid Network architecture for object detection modified for
+image segmentation (52),
+– U-Net++ - a nested U-Net architecture developed for medical image segmentation
+(53),
+– PAN (Pyramid Attention Network) - a model incorporating spatial pyramid attention
+structure, designed to utilize global contextual information in semantic segmentation
+(54),
+– PSPNet (Pyramid Scene Parsing Network) - a model utilizing pyramid pooling and a
+scene parsing network to learn global context information better (55).
+4.1.2
+3D Models
+The models we use for the 3D segmentation task are:
+• A standard 3D U-Net model (48),
+• Selected models from ‘MedicalZooPytorch’ (56):
+– VNetLight - a lighter version of the V-Net convolutional network architecture devel-
+oped to perform volumetric segmentation (57; 56),
+– HighResNet - a compact, high-resolution convolutional network for volumetric seg-
+mentation, originally demonstrated on brain parcellation pretext task on brain MR
+images (58).
+4.2
+More Details on Training
+4.2.1
+Configuration Files
+Two types configuration files are provided as input to training:
+21
+
+• Network Configuration - a ’.json’ file containing settings for the optimizer, augmentation
+setting, the model settings, evaluation settings, settings for saving the output and the seed
+to use for the training.
+• Task Configuration - a ’.json’ file specifying information regarding the data slices to ac-
+cess for the training. This includes the database channel to access for the data and an-
+notations, the x and y ranges for the slices from each region, the size of the slices, the
+train-val-test split, the size of the volume slice (3D) and the whether the training is for 3-
+Class or 4-Class setting. The task configuration used for the results in Table 2 are listed in
+Table A1.
+Together, these two files completely specify the configurations for the training run.
+4.2.2
+Model Settings
+The basic model settings are:
+• encoder name - the encoder to use with the model. This is applicable only for the 2D mod-
+els and the ’UNet 3D’ model. For more information visit ’segmentation models.pytorch‘
+library ((50)). For the training results in Table 2, ’efficientnet-b7’ was used as the encoder
+as it gave the best performance among several other encoders that were tried.
+• encoder weights - the pre-trained weights to use for the model. For the 2D model and
+’UNet 3D’ results in Table 2, weights trained on ImageNet were used.
+• in channels - the number of input channels.
+• classes - the number of output classes.
+4.2.3
+Details on Class Proportions
+At the microstructure level/scale, the frequency of each class across the brain regions are as follows:
+• Cortex - Background: 93%; Blood Vessel: 2.33%; Cell: 4.64%; Axons: 0%
+• Striatum - Background: 72.63%; Blood Vessel: 2.5%; Cell: 5.66%; Axons: 19.22%
+• VP - Background: 22.75%; Blood Vessel: 4.73%; Cell: 5.73%; Axons: 66.8%
+• ZI - Background: 32.15%; Blood Vessel: 5.4%; Cell: 9.41%; Axons: 53.04%
+• Total - Background: 55.13%; Blood Vessel: 3.74%; Cell: 6.36%; Axons: 34.77%
+4.2.4
+Hyper-Parameters and Random Seeds
+For the selection of optimal hyper parameters for each model a hyper parameter search is performed
+by training the models for the following learning rates: 0.1, 0.05, 0.01, 0.005, 0.001; and the follow-
+ing batch sizes: 2,4,6,8,10. The best performing learning rate and batch size is chosen and 5 separate
+instances of each model are trained with this optimal learning rate and batch size, with seeds: 1, 100,
+350, 631 and 872 respectively. These 5 results (for each model) are used to calculate the mean and
+SD (Mean ± SD) of the performance metrics (which is reported in Table 2). Also across several
+models it was seen that Adam optimizer was yielding a better result than SGD, so Adam optimizer
+was used for all the training runs. The optimal hyperparameters found for each model in each setting
+are listed in Table A4.
+5
+Further Details on Task 3
+In order to extract the semantic features from the microstructure annotations in the 4 densely con-
+nected cubes, we first isolate the relevant corresponding pixel-level labels (either cells, blood vessels
+or axons). Once we have extracted the desired class and labeled everything else as background, we
+can perform a connected component analysis to compute the desired semantic features (cell count,
+size, and distance, as well as axon and vessel density) for each image sample.
+Once the semantic features for different tasks are calculated, we extract the representations of all
+samples across the 4 cubes using the models trained in task 1 ROI-C1 (Table 3, top) and ROI-C3
+22
+
+Table A4: Optimal Hyperparameters used for Task 2 models in Table 2
+I. 2D Pixel-level Segmentation
+3-Class
+4-Class without ZI
+Model
+Learning Rate Batch Size Learning Rate Batch Size
+2D U-Net
+0.1
+8
+0.01
+4
+MA-Net
+0.01
+2
+0.001
+8
+FPN
+0.01
+2
+0.01
+4
+U-Net++
+0.001
+8
+0.01
+8
+PAN
+0.01
+10
+0.001
+8
+PSPNet
+0.01
+4
+0.1
+2
+II. 3D Pixel-level Segmentation
+3-Class
+4-Class without ZI
+Model
+Learning Rate Batch Size Learning Rate Batch Size
+3D U-Net
+0.005
+1
+0.01
+1
+VNetLight
+0.01
+1
+0.01
+1
+HighResNet
+0.005
+1
+0.001
+1
+(Table 3, bottom), and use scikit-learn to fit a linear regression in order to predict the semantic fea-
+tures and report the R2 on the entire subvolume of interest. Furthermore, we provide visualizations
+of the representations and the corresponding semantic features of the 4 interest cubes using a BYOL
+encoder trained under two different dataset conditions (ROI-C1 and ROI-C3) in Figure A4.
+Figure A4: Visualizations of semantic features overlaid on two dimensional learned representations.
+Here,
+the first row shows 2D U-map projections of learned BYOL (ROI-C1) embeddings and overlay the different
+semantic attributes on the latents . From left to right, we color the embeddings by brain area (class), % blood
+vessels, % axons, cell count, and cell size. The second row shows the 2D projections of learned BYOL (ROI-
+C3) embedding which too are overlaid with different semantic attributes.
+6
+Maintenance, Licensing, and Ethical Concerns
+The dataset, data access API, and dataloaders are maintained by the BossDB team (bossdb.org),
+which is tasked as the BRAIN Initiative archive of record for nanoscale connectomics datasets.
+The team is developing standards in accordance with FAIR principles (https://www.go-fair.
+org/fair-principles/) to ensure permanent identifiers and broad accessibility. The data are
+available under the CC-BY-4.0 license. The baseline code is available open source hosted in a
+github organization, and community forks and pull requests will be welcome, to be reviewed by the
+repository maintainers. As improvements are made to the baseline codebase for future challenges,
+changes will be pushed to the MTNeuro repository as a new versioned release.
+The dataset used includes animal data which was collected under an approved IACUC protocol, as
+detailed in the original paper. There is no human derived data in this dataset. We emphasize this
+23
+
+Axon Pixels (%)
+10
+10
+20
+30
+40
+50
+60
+70
+80
+90BloodVesselpixels(%)
+18
+10
+16
+14
+12Cell Count
+4
+10
+8
+12
+16
+20
+24
+28
+32
+36Cell Size
+0
+15
+10
+30
+45
+60
+75
+90
+105
+12010
+Cortex
+Striatum
+VP
+ZIBloodVesselpixels(%)
+0
+24
+10
+6
+10
+12Axon Pixels (%
+10
+20
+.8
+30
+40
+50
+60
+70
+80
+90
+14
+12Cell Count
+4
+8
+.8
+12
+16
+20
+24
+16
+28
+32
+36
+14
+12
+10Cell Size
+0
+15
+30
+.8
+45
+60
+75
+90
+105
+120
+14
+.2
+018
+16-
+14
+12
+Cortex
+Striatum
+10
+VP
+ZIdataset is for development of algorithms for fundamental analysis of structural neuroscience data,
+but it is possible future efforts could use these data inappropriately for the development of clinically
+relevant algorithms which could result in negative societal impacts. This use is discouraged due to
+the unknown generalizations of high-resolution X-ray Microtomography to clinical modalities.
+24
+

ItE3T4oBgHgl3EQfXAqD/content/tmp_files/2301.04475v1.pdf.txt

@@ -0,0 +1,3162 @@
+arXiv:2301.04475v1  [math-ph]  11 Jan 2023
+MIURA-RECIPROCAL TRANSFORMATIONS AND LOCALIZABLE
+POISSON PENCILS
+P. LORENZONI, S. SHADRIN, AND R. VITOLO
+Abstract. We show that the equivalence classes of deformations of localizable semisimple
+Poisson pencils of hydrodynamic type with respect to the action of the Miura-reciprocal group
+contain a local representative and are in one-to-one correspondence with the equivalence classes
+of deformations of local semisimple Poisson pencils of hydrodynamic type with respect to the
+action of the Miura group.
+Contents
+1.
+Introduction
+1
+1.1.
+A variety of jet space transformation groups
+2
+1.2.
+Action of the transformation groups
+4
+1.3.
+Weakly non-local Poisson bi-vectors of localizable shape
+5
+1.4.
+Localizability
+6
+1.5.
+Projective group and Doyle–Pot¨emin form
+7
+1.6.
+Acknowledgments
+8
+2.
+Formulae for the action
+8
+2.1.
+The Ferapontov–Pavlov formula
+10
+2.2.
+Weakly non-local bi-vectors of localizable shape
+11
+3.
+Schouten bracket for weakly non-local operators of localizable shape
+13
+3.1.
+The two approaches
+13
+3.2.
+Identification of the two approaches
+14
+4.
+Pencils of weakly non-local bi-vectors of localizable shape
+16
+4.1.
+Bi-Hamiltonian cohomology
+16
+4.2.
+Comparison with the purely local deformations
+20
+4.3.
+Roots of the characteristic polynomial of the symbol
+21
+5.
+Projective-reciprocal invariance of the Doyle–Pot¨emin form
+23
+References
+24
+1. Introduction
+In 2001, Dubrovin and Zhang initiated a classification programme of bi-Hamiltonian inte-
+grable PDEs in two independent variables [DZ01].
+The group action that they considered
+was that of Miura transformations, i.e., transformations depending on the field variables and,
+polynomially, by their derivatives of higher order through a perturbative series.
+Among the questions that the above approach raises there is the issue of extending the group
+action to include (possibly non-local) changes of variables in one of the independent variables.
+Indeed, an important class of such transformations is that of reciprocal transformations, which
+play an important role in Mathematical Physics (see e.g. [Rog69; Rog68; Fer89; Fer91; FP03;
+XZ06; AG07; Abe09; BS09; LZ11; AL13]).
+2020 Mathematics Subject Classification. 37K05, 37K10, 37K20, 37K25.
+Key words and phrases. bi-Hamiltonian PDE, Hamiltonian operator, Miura transformation, reciprocal trans-
+formation, integrable systems.
+1
+
+2
+P. LORENZONI, S. SHADRIN, AND R. VITOLO
+This paper is concerned with the action of the group of Miura-reciprocal transformations,
+that is a natural group of simultaneous transformations of the independent and the dependent
+variables of a (bi-)Hamiltonian system through a perturbative series of derivatives of the field
+variables. Among other things, we consider (1) the Miura-reciprocal transformations of the
+1st kind and rederive from the scratch the Ferapontov–Pavlov formula for the transformation
+of a hydrodynamic bivector; (2) Miura-reciprocal transformations of the 2nd kind (close to
+identity) and classify the orbits of their action on Poisson pencils of weakly non-local bi-vectors
+of localizable shape with localizable semi-simple hydrodynamic leading term; (3) a smaller
+group of projective-reciprocal transformations and prove that they preserve the Doyle–Pot¨emin
+canonical form of the bi-vectors.
+A detailed comparison between previous results and our results can be read in the following
+Subsections; we just stress that our result on the classification of bi-Hamiltonian integrable
+structures provides a natural extension for the classification program in [DZ01] (that also in-
+corporates and explains some of the results in [LZ11]). To the best of our knowledge it is the
+first result in the literature that provides a systematic classification of orbits of the action of the
+group of Miura-reciprocal transformations in the bi-Hamiltonian context (for a single Poisson
+structure this type of result is established in [FP03; LZ11]).
+1.1. A variety of jet space transformation groups. We consider a jet space Jr(1, N),
+r ≥ 0, with independent variable x and dependent variables ui, i = 1, . . . , N, considered as
+coordinates on some open domain U ⊂ RN. Let ui,σ denote the x-derivative of ui taken σ times.
+Consider the transformations (i.e. diffeomorphisms) of the jet space Jr(1, n).
+We begin
+from the most general type of transformation: a reciprocal transformation coupled with a
+differential substitution. Reciprocal transformations in a modern setting were introduced in
+[Rog69; Rog68] in the study of gas dynamics, and later analyzed under a geometric viewpoint
+in [Fer89; Fer91] and many other authors (see e. g. [FP03; XZ06; AG07; Abe09; BS09; LZ11;
+AL13] and references therein). The class of differential substitutions was introduced in [Ibr85],
+although many particular differential substitutions were already present in the literature (in
+particular, the Miura transformations).
+Definition 1.1. A reciprocal differential substitution is a nonlocal transformation of the inde-
+pendent variable x into the independent variable y of the type
+(1)
+dy = Bdx,
+B = B(x, ui, ui,σ)
+coupled with a differential substitution of the dependent variables of the form
+(2)
+wi = Qi(x, uj, uj,σ).
+By the fact that dx(∂x) = 1 = dy(∂y) we obtain that total derivatives are related by the
+formula ∂x = B∂y.
+Note that, in general, the inversion of a differential substitution is a
+nonlocal operation. We will soon focus on a more restrictive class of transformations.
+Reciprocal differential substitutions admit several interesting subclasses:
+• A reciprocal transformation is a nonlocal transformation of the independent variables x
+into the independent variable y of the type
+(3)
+dy = Bdx,
+B = B(x, ui, ui,σ)
+coupled with the identical transformation of the dependent variables. In practical ap-
+plications the functions ui depend also on an additional parameter that plays the role
+of “time” of the system of evolutionary PDEs
+ui
+t = F i(x, ui, ui,σ),
+i = 1, ..., n
+governing their evolution. Taking into account this additional variable reciprocal trans-
+formations are often defined as
+dy = Adt + Bdx,
+
+MIURA-RECIPROCAL TRANSFORMATIONS AND LOCALIZABLE POISSON PENCILS
+3
+where the function A, B are submitted to the closure condition Bt = Ax, that is, dy
+is a conservation low for the equation. Note that the coefficient A doesn’t enter the
+transformation law for ∂x, and thus can be disregarded throughout this paper.
+• A reciprocal differential substitution is said to be holonomic if there exists a differential
+function P such that B = ∂xP.
+• A general differential substitution of (x, ui) into (y, wj):
+(4)
+y = P(x, uj, uj,σ),
+wi = Qi(x, uj, uj,σ),
+yields a holonomic reciprocal differential substitution dy = ∂xPdx, wi = Qi by differen-
+tiation (in this sense the two classes of transformations coincide).
+The above two categories of transformations, basically local and nonlocal differential substi-
+tutions, are, on the one hand, too wide to be used in the context of the classification programs
+for evolutionary PDEs and related geometric structures as, for instance, the one initiated by
+Dubrovin and Zhang in [DZ01], and on the other hand too restrictive since we are limited by
+fixing the parameter r ≥ 0 that controls the maximal order of jets.
+For this reason, we introduce the space of differential polynomials A, and the following group
+of transformations, which is a subclass of the reciprocal differential substitutions. Consider a
+jet space J∞(1, N) (considered as an inductive limit of the jet spaces Jr(1, N), r → ∞) with
+independent variable x and dependent variables ui, i = 1, . . . , N. Denote ui,σ the x-derivative
+of ui taken σ times. We associate with this space the algebra of functions A := C∞(U)[[ui,σ, i =
+1, . . . , N, σ ≥ 1]], where C∞(U) is the space of smooth functions on a domain U ⊂ RN in the
+coordinates u1, . . . , uN. There is a natural gradation on the algebra of densities A given by
+deg∂x ui,σ. Let Ad denote the deg∂x-degree d part of A, which is a finite dimensional module
+over C∞(U).
+Definition 1.2. A Miura-type reciprocal differential substitution, or Miura-reciprocal trans-
+formation for short, is a transformation of the type
+dy =
+� ∞
+�
+k=0
+ǫkHk(uj, uj,1, . . . , uj,k)
+�
+dx,
+(5)
+wi =
+∞
+�
+k=0
+ǫkKi
+k(uj, uj,1, . . . , uj,k),
+i = 1, . . . , N,
+with Hk, Ki
+k ∈ Ak and
+H0 ̸= 0,
+det
+�∂Ki
+0(uj)
+∂uk
+�
+̸= 0.
+(6)
+The formal dispersive parameter ǫ that we introduce here to control the deg∂x-degree is,
+in principle, not strictly necessary but it is very convenient in particular computations and
+applications.
+The set of all Miura-reciprocal transformations is denoted by R. It is a group with respect
+to the composition, and it has some distinguished subgroups:
+• the subgroup RDS of Miura differential substitutions, that are Miura-type reciprocal
+differential substitutions which are also holonomic differential substitutions of the fol-
+lowing type:
+There exists P = x + P0, with P0 = �∞
+k=0 ǫkFk and Fk ∈ Ak, such that
+(7)
+∂xP =
+∞
+�
+k=0
+ǫkHk(uj, uj
+x, . . . , uj
+σ);
+• the subgroup of Miura transformations characterized by H0 = 1 and Hk = 0 for all
+k ≥ 1. This subgroup is called the Miura group G ⊂ R [DZ01] and bears his name
+from the transformation relating KdV and modified KdV equations introduced by Miura
+
+4
+P. LORENZONI, S. SHADRIN, AND R. VITOLO
+[Miu68]. Note that the Miura group is also a subgroup of the group of Miura differential
+substitutions: G ⊂ RDS.
+Definition 1.3. By analogy with the way the standard Miura group is typically presented, we
+introduce the following two subgroups.
+• We define Miura-reciprocal transformations of the 1st kind to be the Miura-reciprocal
+transformations of the form
+dy = H0(uj)dx,
+(8)
+wi = Ki
+0(uj),
+i = 1, . . . , N.
+The group of all Miura RDS of the first type is denoted by RI. This group contains as
+a subgroup the group of Miura transformations of the 1st kind, GI ⊂ RI, characterized
+by H0 = 1.
+• We define Miura-reciprocal transformations of the 2nd kind to be the Miura-reciprocal
+transformations of the form
+dy =
+�
+1 +
+∞
+�
+k=1
+ǫkHk(uj, uj,1, . . . , uj,σ)
+�
+dx,
+(9)
+wi = ui +
+∞
+�
+k=1
+ǫkKi
+k(uj, uj,x, . . . , uj,σ),
+i = 1, . . . , N.
+The group of all Miura-reciprocal transformations of the second type is denoted by
+RII. It contains as a subgroup the group of Miura transformations of the 2nd kind,
+GII ⊂ RII, characterized by Hk = 0 for all k ≥ 1.
+Definition 1.4. A distinguished subgroup of RI is the group of projective reciprocal transfor-
+mations P. Such transformations are characterized by the requirements that Ki in Equation (8)
+is a projective transformation (in an affine chart) and H0 is the common denominator of the
+projective transformation. More explicitly,
+dy = (a0
+juj + a0
+0)dx,
+(10)
+wi = ai
+juj + ai
+0
+a0
+juj + a0
+0
+,
+i = 1, . . . , N.
+The goal of this paper is to discuss some aspects of the actions of these groups on the natural
+suitable geometric structures that emerge in the study of integrable systems of evolutionary
+PDEs. In order to describe our results we have to recall some of these structures, which we do
+in the rest of the introduction.
+1.2. Action of the transformation groups. The above group of Miura reciprocal differential
+substitutions act on spaces of geometric entities that play important roles in the geometric
+theory of integrability. In particular, it acts on:
+• densities, that have the form
+(11)
+F =
+�
+f(uj, uj,σ) dx ∈ F := A/∂xA,
+with
+f ∈ A;
+• variational vector fields, that include symmetries of partial differential equations, and
+have the form
+(12)
+ϕ = ϕi(uj, uj,σ)δui,
+ϕi ∈ A;
+• covector-valued densities, that include characteristic vectors of conserved quantities of
+differential equations, and have the form
+(13)
+ψ = ψi(uj, uj,σ)dui ⊗ dx,
+ψi ∈ A;
+
+MIURA-RECIPROCAL TRANSFORMATIONS AND LOCALIZABLE POISSON PENCILS
+5
+• the Euler–Lagrange operator, which sends densities into covector-valued densities,
+(14)
+E(F) = δuiFdui ⊗ dx;
+• variational multivectors of degree p, that include Hamiltonian operators of partial dif-
+ferential equations as particular bivectors. They can be regarded as maps from (p − 1)-
+covector-valued densities to variational vector fields.
+In Section 2 we will prove our change of coordinate formulae for reciprocal differential substi-
+tutions for these geometric objects. As an example, we re-derive in Section 2.1 the Ferapontov–
+Pavlov formula for the reciprocal transformation of a Poisson bi-vector of the differential order
+1 [FP03], and this brings us to the realm of weakly non-local Poisson structures of localizable
+shape.
+1.3. Weakly non-local Poisson bi-vectors of localizable shape. Let dependent variables
+ui also dependent on one external parameter, denoted by t. The most studied structures in
+geometric theory of integrability are the local Poisson structures needed for representation of
+equations of the form
+ui
+t = f i(uj, uj,σ)
+(15)
+in Hamiltonian form (note that we don’t allow possible explicit dependence of f i’s on x), that
+is, in the form
+ui
+t =
+d
+�
+s=0
+P ij
+s ∂s
+δ
+δuj
+�
+h(uk, uk,σ)dx,
+(16)
+where H =
+�
+h(uk, uk,σ)dx is the Hamiltonian functional and P = �d
+s=0 P ij
+s ∂s, P ij
+s ∈ A defines
+a bi-vector which in the language of densities can be written as
+{ui(x), uj(y)}P =
+d
+�
+s=0
+P ij
+s ∂s
+xδ(x − y)
+(17)
+(in this paper bi-vectors and, more generally, multivectors are assumed to be skew-symmetric
+by default).
+In addition to the language of densities, there is a very convenient formalism, the so-called
+θ-formalism, to encode the variational multivectors [Get02], see also [IVV02]. Namely, extend
+the space A to a space ˆ
+A := A[[θσ
+i , i = 1, . . . , N, σ ≥ 0]], where θσ
+i are formal odd variables. We
+often denote θ0
+i by θi, and we extend the ∂x operator to ˆ
+A as ∂x := �∞
+s=0 ui,s+1∂ui,s + θs+1
+i
+∂θs
+i .
+The deg∂x-gradation is extended to ˆ
+A by deg∂x θσ
+i = σ, and there is a natural θ-degree given by
+degθ ui,σ = 0 and degθ θσ
+i = 1. Let ˆ
+Ap denote the subspace ˆ
+A of θ-degree p. We can consider it
+as a space of densities of variational p-vectors. Let ˆ
+Ap
+d := ˆ
+Ad ∩ ˆ
+Ap.
+The space ˆF := ˆ
+A/∂x ˆ
+A can be considered as the space of variational multivectors. It inherits
+under the projection
+�
+: ˆ
+A → ˆF both gradations, deg∂x and degθ, and F p
+d denotes its subspace
+of p-vectors of differential degree deg∂x = d. The Schouten bracket is defined as
+(18)
+� �
+P,
+�
+Q
+�
+=
+�
+(−1)degθ PδuiPδθiQ + δθiPδuiQ
+for P, Q ∈ ˆ
+A, where δui := �∞
+s=0(−∂x)s∂ui,s and δθi := �∞
+s=0(−∂x)s∂θs
+i . Various cohomological
+computations in terms of this space and related formalism allow to efficiently control the de-
+formation theory of Poisson bi-vectors and their pencils, see e. g. [LZ05; LZ11; DLZ06; LZ13a;
+CPS18; CKS18; CPS16a; CPS16b; CCS17; CCS18].
+However, studying the action of the group of Miura-reciprocal transformations we can not
+restrict ourselves to the local Poisson bi-vectors. As we have seen, the reciprocal transformations
+
+6
+P. LORENZONI, S. SHADRIN, AND R. VITOLO
+generate non-locality of some very particular shape, and in terms of the operator P we have to
+extend its possible shape to
+P =
+d
+�
+s=0
+P ij
+s ∂s + ui,1∂−1
+x V j + V i∂−1
+x uj,1,
+P ij
+s , V i ∈ A.
+(19)
+Hamiltonian operators of the form above with d = 1, P ij
+1 = gij(u) (det gij ̸= 0), P ij
+0 = −gilΓj
+lkuk
+x
+and V i = V i
+j (u)uj
+x were studied by Ferapontov in [Fer95a]. They belong to the larger class of
+weakly non-local operators, that was introduced in [MN01]. Like in the local case the coefficients
+gij define a metric and the coefficients Γj
+lk the Christoffel symbols of the associated Levi-Civita
+connection but unlike in the local case the metric is no longer flat.
+It turns out that the
+Riemann tensor R and the tensor field V defining the non-local tail satisfy the conditions
+gisV s
+j = gjsV s
+i ,
+∇jV k
+i = ∇iV k
+j ,
+Rij
+kl = V i
+kδj
+l + V j
+l δi
+k − V j
+k δi
+l − V i
+l δj
+k.
+These are a particular instance of Ferapontov’s conditions for weakly non-local Hamiltonian
+operators of hydrodynamic type [Fer95b]. An algorithm to compute such conditions for general
+weakly non-local Hamiltonian operators has been developed in [CLV20] and implemented in
+three different computer algebra systems in [Cas+22].
+A natural question here is how to extend the θ-formalism briefly recalled above to accommo-
+date this type of non-locality. There are two recipes in the literature given in [LZ11] (specific
+for this case) and [LV20] (suitable for general weakly non-local operators). The identification of
+the two approaches should indirectly follow from the uniqueness arguments in [LZ11], but we
+wanted to establish an explicit identification. We do it by an explicit computation in Section 3.
+Remark 1.5. It is important to comment on the action of the operator ∂−1
+x . It can be defined
+on ∂xA by ∂−1
+x (∂x(f)) = f + C for any f ∈ A, here C is some constant. For a more general
+element g ∈ A, g ̸∈ ∂xA, we can represent ∂−1
+x (g), for instance, as an element of a localization
+of A given by A((
+1
+u1,1)), that is, we can perturbatively represent it as a series C + �∞
+i=1
+hi
+(u1,1)i
+with hi ∈ A such that ∂u1,1hi = 0 (this idea is coming from [DLZ06]), here C is also a constant.
+Both approaches that we compare assert that for the analysis of the weakly non-local Poisson
+bi-vectors of localizable shape it is sufficient to formally apply ∂−1
+x
+to just one element −ui,1θi ∈
+ˆ
+A and denote the result by ζ, which has different meaning in these two approaches.
+The
+subsequent usage of ζ in computations implies that the extra constant that might occur by
+inverting ∂x is uniformly set to C = 0.
+1.4. Localizability. Consider a dispersive weakly non-local Poisson structure of localizable
+shape given by
+P =
+∞
+�
+d=1
+ǫd−1
+�
+d
+�
+s=0
+P ij
+d,d−s∂s + ui,1∂−1
+x V j
+d + V i
+d∂−1
+x uj,1
+�
+,
+P ij
+d,k, V i
+k ∈ Ak.
+(20)
+The leading term (d = 1) of this structure is a Poisson structure of hydrodynamic type and
+thus the full Poisson structure can be thought as a deformation of a Poisson structure of
+hydrodynamic type. If det P ij
+1,0 ̸= 0, Liu and Zhang prove in [LZ11] that there is always an
+element in R that turns P into a constant local Poisson structure ηij∂x.
+In the case of a
+purely local structure the same results is established under the action of group G in [Get02]
+(see also [DMS05] and [DZ01]), and in the case ǫ = 0 (that is, a purely degree 1 case) it is
+established under the action of the group RI in [LZ11] for N = 1, 2 and in [FP03] for N ≥ 3.
+Now consider a pencil P − λQ of dispersive weakly non-local Poisson structure of localizable
+shape. Let us fix the leading term (P − λQ)|ǫ=0 of the pencil and assume it is semi-simple. In
+the purely local case (that is, under the additional assumption that both P and Q are purely
+local), it was suggested in [LZ05; DLZ06] (see also [Lor02] for the scalar case) and proved
+in [LZ13a] (N = 1 case) and [CPS18; CKS18] (any N ≥ 1) that the space of orbits of the
+
+MIURA-RECIPROCAL TRANSFORMATIONS AND LOCALIZABLE POISSON PENCILS
+7
+action of GII on such pencils is naturally parametrized by N smooth functions of one variable,
+called the central invariants.
+In Section 4 we generalize these results in the following way. Let us fix the leading term
+(P − λQ)|ǫ=0 of the pencil and assume that P|ǫ=0 and Q|ǫ=0 are simultaneously localizable
+under the action of the group RI. We also still assume that (P − λQ)|ǫ=0 is semi-simple. In
+this case, we prove that the set of orbits of the action of RII on such pencils is also naturally
+parametrized by N smooth functions of one variable. Note that while the statement is literally
+the same as in the purely local case, it is quite different as both the group and the space of
+structures on which the group acts is much bigger. We show that it is still possible to read the
+central invariants from the symbol of the pencil.
+This result is proved by a direct application of various techniques and results proposed
+in [LZ11; LZ13a; CPS18; CKS18]. From the comparison with the computations in the local
+case, we obtain the following extra result: under the assumptions above, each orbit of RII
+contains a purely local representative. In other words, we prove that if the leading term of a
+semi-simple pencil P −λQ of dispersive weakly non-local Poisson structure of localizable shape
+is localizable by the action of the group RI, then the whole pencil is localizable by the action
+of the group R.
+It is worth to mention that this result also generalizes and put in the right context a theorem
+of Liu and Zhang [LZ11, Theorem 1.3] that states that if two local Poisson pencils with the
+leading semi-simple hydrodynamic term are related by a reciprocal transformation, then their
+central invariants are the same.
+1.5. Projective group and Doyle–Pot¨emin form. Finally, we consider the action of the
+group P ⊂ RI. It is a quite small group with transparent structure, and we expect that in
+general the orbits of its action should have a rich geometric structure. In this paper we find a
+surprising connection of this group to a conjecture of Mokhov on the possible form of the local
+Poisson structures of differential degree deg∂x ≥ 2.
+It was independently proved by Doyle [Doy93] and Pot¨emin [Pot91; Pot97] that homogeneous
+local Poisson structures of differential degree d = 2, 3, i.e. of the form
+(21)
+d
+�
+s=0
+P ij
+d−s∂s
+x,
+P ij
+k ∈ Ak,
+can always be transformed by the action of the group GI to an operator of the shape
+(22)
+∂x ◦
+d−2
+�
+s=0
+Qij
+d−2−s∂s
+x ◦ ∂x,
+Qij
+k ∈ Ak.
+Mokhov made the following interesting conjecture:
+Conjecture 1.6 (See [Mok98, Proposition 2.3 and text afterwards]). Let P = �d+2
+e=1 P ij
+e ∂d+2−e
+x
+be a local operator of homogeneous differential order d + 2 (that is, deg∂x P ij
+e
+= e), d ≥ 0.
+Assume that P defines a Poisson bracket. Then there exists a local skew-symmetric operator
+Qij of homogenenous differential order d such that P = ∂x ◦ Qij ◦ ∂x.
+The form (22) is called the Doyle–Pot¨emin form of a local homogeneous bi-vector of differ-
+ential degree deg∂x ≥ 2.
+It was recently proved that in the cases of homogeneous local Poisson structures of degree
+d = 2 [VV] and d = 3 [FPV14] the form (22) is preserved by the action of the group P. In
+Section 5, thanks to our change of coordinates formulae from Subsection 1.2, we generalize the
+above results to local homogeneous bi-vectors (i.e., not necessarily Poisson structures) of degree
+d ≥ 2 and prove that the group P preserves the set of local bi-vectors of Doyle–Pot¨emin form.
+A nice example of application to a Hamiltonian operator for the Dubrovin–Zhang hierarchy is
+pointed out.
+
+8
+P. LORENZONI, S. SHADRIN, AND R. VITOLO
+1.6. Acknowledgments. S. S. and R. V. were supported by the Netherlands Organization
+for Scientific Research. P. L. and R. V. are supported by funds of INFN (Istituto Nazionale di
+Fisica Nucleare) by IS-CSN4 Mathematical Methods of Nonlinear Physics. P. L. is supported
+by funds of H2020-MSCA-RISE-2017 Project No. 778010 IPaDEGAN. P. L. and R. V. are also
+thankful to GNFM (Gruppo Nazionale di Fisica Matematica) for supporting activities that
+contributed to the research reported in this paper.
+2. Formulae for the action
+The goal of this Section is to compute from the scratch the effect of general reciprocal
+differential substitutions on variational (multi)vector fields and related geometric objects. It is
+clear that a reciprocal differential substitution given by dy = Bdx (or y = P in the holonomic
+case), wi = Qi, also yields a coordinate change of the y-derivative variables:
+(23)
+wi,τ = ∂τ
+y Qi =
+� 1
+B ∂x
+�τ
+Qi.
+It is convenient to introduce the Fr´echet derivative1 of a differential function F ∈ A, as
+(24)
+ℓF(X) = (ℓF)i(Xi) =
+∞
+�
+σ=0
+∂F
+∂ui,σ ∂σ
+xXi,
+where X = Xiδui is a variational vector field. The formal adjoint of the above operator is
+(25)
+(ℓ∗
+F)i =
+∞
+�
+σ=0
+(−∂x)σ ◦ ∂F
+∂ui,σ
+acting on covector-valued densities.
+A change of coordinates formula for Hamiltonian operators under the action of differential
+substitutions was already given in [Mok87; Olv88]. We rephrase the arguments of the proof
+in [Olv88] and obtain change of coordinates formulae for the geometric objects that we listed
+in Subsection 1.2 which turn out to be valid in the more general case of reciprocal differential
+substitutions.
+We observe that also in [LZ11] there are formulae for coordinate change, but their validity
+is limited to the action of Miura reciprocal transformations on operators of localizable shape,
+while we do not have this limitation.
+First of all, we provide a formula for the coordinate change of an variational vector field
+under a differential substitution.
+Proposition 2.1. Let Xiδui = Y iδwi be a variational vector field in the coordinate systems
+(x, ui,σ) and (y, wi,σ), respectively, where the latter coordinates systems are related by a holo-
+nomic reciprocal differential substitution y = P, wi = Qi. Then the following change of coordi-
+nate formula holds:
+(26)
+Y j =
+1
+∂xP Dj
+i (Xi)
+where
+(27)
+Dj
+i = ∂xP(ℓQj)i − ∂xQj(ℓP)i.
+Proof. The proof uses arguments that provide a change of coordinates formula for Euler–
+Lagrange operators in [Olv93], Theorem 4.8 and Exercise 5.49. Let
+(28)
+ui = f i(x),
+x ∈ Ω,
+wi = gi(y),
+y ∈ ˜Ω
+be functions that are put in correspondence by a transformation.
+We can consider a one-
+parameter family of such functions defined by the variation field Xi∂ui:
+(29)
+ui
+ǫ = f i(x, ǫ) = f i(x) + ǫXi(x),
+1It should be the Gateaux derivative, but Fr´echet is prevailing in the literature.
+
+MIURA-RECIPROCAL TRANSFORMATIONS AND LOCALIZABLE POISSON PENCILS
+9
+where Xi∂ui has compact support in Ω. Its transformed version
+(30)
+wi
+ǫ = gi(y, ǫ) = gi(y) + ǫY i(y) + O(ǫ2).
+is determined by the formulae
+(31)
+y = P(x, ∂σ
+x(f j(x) + ǫXj(x))),
+wi
+ǫ = Qi(x, ∂σ
+x(f j(x) + ǫXj(x))).
+Since η has compact support on Ω, each gi(y, ǫ) is defined on a common compact domain ˜Ω =
+{x ∈ Ω | y = P(x, ∂σ
+xf j(x))}. The transformed variation field is given by Y i(y) = ∂ǫgi(y, ǫ)
+��
+ǫ=0.
+As variation fields do not depend on ǫ we have
+(32)
+∂ǫy = 0 = ∂xP∂ǫx +
+∞
+�
+σ=0
+∂uj,σP∂σ
+xXj,
+hence
+(33)
+∂ǫx
+��
+ǫ=0 = − 1
+∂xP
+∞
+�
+σ=0
+∂uj,σP∂σ
+xXj.
+We have:
+Y j = ∂ǫgj(y, ǫ)
+��
+ǫ=0 =
+∞
+�
+σ=0
+∂ui,σQj∂σ
+x∂ǫf i(x, ǫ)
+��
+ǫ=0 + ∂xQj∂ǫx
+��
+ǫ=0
+(34)
+=
+1
+∂xP
+�
+∂xP(ℓQj)i − ∂xQj(ℓP)i
+�
+Xi.
+□
+In the non-holonomic case, we have to regard the differential function P as the primitive of
+a differential function B, P = ∂−1
+x B, and we obtain the following Corollary.
+Corollary 2.2. In the non-holonomic case of the reciprocal differential substitution dy = Bdx,
+wi = Qi the following change of coordinate formula holds for an variational vector field Xiδui =
+Y iδwi:
+(35)
+Y j = 1
+B Dj
+i (Xi),
+where
+(36)
+Dj
+i = B(ℓQj)i − ∂xQj ◦ ∂−1
+x
+◦ (ℓB)i.
+Note that we used the property ℓB ◦ ∂−1 = ∂−1 ◦ ℓB, which is very useful in computations.
+Dualizing the computation above, we obtain the formulae for the change of coordinates
+formula for the Euler–Lagrange operator.
+Corollary 2.3. Let the coordinate systems (x, ui,σ) and (y, wi,σ), respectively, where the latter
+coordinates systems are related by a reciprocal differential distribution dy = Bdx, wi = Qi, and
+let Ex
+i , Ey
+i be the Euler–Lagrange operator with respect to the coordinates (x, ui
+σ), (y, wi,σ). Then
+the change of coordinate formula is
+(37)
+Ey
+i = (D∗)k
+i ◦ Ex
+k ,
+with D given by Equation (36).
+In the holonomic case, the formula reduces to the known formula in [Olv93, Exercise 5.49]
+(with D given by Equation (27)). In the particular case of a differential substitution of the
+dependent variable only we have ℓB = 0 and the above formula reduces to the well-known
+formula Ex = (ℓ∗
+Qk)i ◦ Ey
+k.
+
+10
+P. LORENZONI, S. SHADRIN, AND R. VITOLO
+Corollary 2.4. Consider a covector-valued density Ξidui ⊗ dx = Ψidwi ⊗ dy in the coordinate
+systems (x, ui,σ) and (y, wi,σ) related by dy = Bdx, wi = Qi. Then we have the following change
+of coordinates formula:
+(38)
+Ξi = (D∗)k
+i (Ψk),
+where D is as in Equation (36) (or as in Equation (27) in the holonomic case y = P).
+Finally, we obtain the following:
+Proposition 2.5. Consider a reciprocal differential substitution dy = Bdx, wi = Qi and let
+P ij
+x , P ij
+y
+be its coordinate expressions of a (possibly non-local or non-Poisson) bi-vector with
+respect to the coordinates (x, ui
+σ) and (y, wi
+σ). Then we have the change of coordinate formula
+(39)
+P hk
+y
+= 1
+B (D)h
+i P ij
+x (D∗)k
+j,
+where D is as in Equation (36).
+Proof. The proposition has already been proved in [Mok87; Olv88] for the particular case
+of Hamiltonian operators and differential substitutions. In our case, the proof follows from
+the fact that P ij maps covector-valued densities into variational vector fields.
+So, we can
+use the change of coordinates formulae for these two geometric objects (independently of the
+Hamiltonian property) and find the above result, that holds also in the case of (nonlocal)
+reciprocal differential substitutions.
+□
+Remark 2.6. The same argument can be applied to multivector fields considered as maps from
+multicovector-valued densities to variational vector fields. For instance, in the same set-up as
+Theorem 2.5 let T ijk and ˜T ijk be the coordinate expressions of a trivector. Then
+(40)
+˜T ijk = 1
+B (D)i
+mT mnp((D∗)j
+n, (D∗)k
+p).
+2.1. The Ferapontov–Pavlov formula. Let us apply a special case of Theorem 2.5 to a local
+Poisson bi-vector of order deg∂x = 1 and a reciprocal transformation in R that only changes
+the independent variable. This should reproduce the Ferapontov–Pavlov formula first derived
+in [FP03, Section 3] (based on [Fer95a]).
+Consider the change of x given by
+∂x = B∂y,
+∂−1
+y B−1 = ∂−1
+x
+(41)
+as an element of RI, that is, we assume that B = B(uj).
+Let a local Poisson bracket of
+differential degree 1 be given by the operator
+P ij := gij∂x + Γij
+k uk
+x,
+(P ∗)ji = −P ij
+(42)
+Convention 2.7. Throughout the computations in this Section it is important for us to distin-
+guish between ∂xuk and ∂yuk, so we use the notation uk
+x and uk
+y rather than uk,1.
+Proposition 2.8. The action of the reciprocal transformation (41) on the operator (42) pro-
+duces a weakly non-local operator of localizable shape, whose local part is given explicitly as
+(43)
+gijB2∂y + Γij
+k B2uk
+y − 1
+2giℓB2
+�
+gℓm
+∂B−2
+∂uk + gkm
+∂B−2
+∂uℓ − gℓk
+∂B−2
+∂um
+�
+gmjB2uk
+y
+and the non-local part is equal to
+(44)
+�
+P iℓ
+�∂B
+∂uℓ
+�
+− 1
+2ui
+y
+∂B
+∂uk gkℓ ∂B
+∂uℓ
+�
+∂−1
+y uj
+y + ui
+y∂−1
+y
+�
+P jk
+� ∂B
+∂uk
+�
+− 1
+2
+∂B
+∂uk gkℓ ∂B
+∂uℓuj
+y
+�
+.
+Remark 2.9. Note that Γij
+k B2 − 1
+2giℓB2 (gℓm∂ukB−2 + gkm∂uℓB−2 − gℓk∂umB−2) gmjB2 is exactly
+the covariant Christoffel symbol for the metric gijB2, so we indeed reproduce the Ferapontov–
+Pavlov formula in [FP03].
+
+MIURA-RECIPROCAL TRANSFORMATIONS AND LOCALIZABLE POISSON PENCILS
+11
+Proof of Proposition 2.8. By Theorem 2.5 the bi-vector P ij is transformed under the substitu-
+tion (41) to
+B−1
+�
+Bδi
+k − ui
+x∂−1
+x
+∂B
+∂uk
+�
+P kℓ
+�
+Bδj
+ℓ + ∂B
+∂uℓ∂−1
+x uj
+x
+�
+.
+(45)
+Expanding the brackets, we have the following four summands (we intentionally keep derivatives
+in x instead of y as long as possible):
+−B−1ui
+x∂−1
+x
+∂B
+∂uk P kℓ ∂B
+∂uℓ∂−1
+x uj
+x = −1
+2B−1ui
+x∂−1
+x
+�
+∂x
+∂B
+∂uk gkℓ ∂B
+∂uℓ + ∂B
+∂uk gkℓ ∂B
+∂uℓ∂x
+�
+∂−1
+x uj
+x
+(46)
+= −1
+2B−1ui
+x
+∂B
+∂uk gkℓ ∂B
+∂uℓ∂−1
+x uj
+x − 1
+2B−1ui
+x∂−1
+x
+∂B
+∂uk gkℓ ∂B
+∂uℓuj
+x
+= −1
+2ui
+y
+∂B
+∂uk gkℓ ∂B
+∂uℓ∂−1
+y uj
+y − 1
+2ui
+y∂−1
+y
+∂B
+∂uk gkℓ ∂B
+∂uℓuj
+y ;
+B−1Bδi
+kP kℓ ∂B
+∂uℓ∂−1
+x uj
+x = P iℓ
+�∂B
+∂uℓ
+�
+∂−1
+x uj
+x + giℓ ∂B
+∂uℓuj
+x
+(47)
+= P iℓ
+�∂B
+∂uℓ
+�
+∂−1
+y uj
+y + giℓ ∂B
+∂uℓ Buj
+y ;
+−B−1ui
+x∂−1
+x
+∂B
+∂uk P kℓBδj
+ℓ = −B−1ui
+x∂−1
+x (P ∗)kj
+� ∂B
+∂uk
+�
+B − B−1ui
+x
+∂B
+∂uk gkjB
+(48)
+= ui
+y∂−1
+y P jk
+� ∂B
+∂uk
+�
+− ui
+y
+∂B
+∂uk gkjB ;
+B−1Bδi
+kP kℓBδj
+ℓ = P ijB .
+(49)
+Thus the non-local term is given by Equation (44), and the local term is given by
+P ijB + giℓ ∂B
+∂uℓBuj
+y − ui
+y
+∂B
+∂uk gkjB
+(50)
+= gijB2∂y + gijB ∂B
+∂uk uk
+y + Γij
+k B2uk
+y − 1
+2giℓB4∂B−2
+∂uℓ uj
+y + 1
+2ui
+y
+∂B−2
+∂uk gkjB4,
+where the latter expression is equal to (43).
+□
+2.2. Weakly non-local bi-vectors of localizable shape. The goal of this Section is to prove
+that the space of weakly non-local bi-vectors of localizable shape is closed under the action of
+reciprocal differential substitutions. We narrow the scope to the Miura-type substitutions R
+as in Definition 1.2.
+Let us consider the effect of a reciprocal transformation of the form (41) on a general weakly
+nonlocal bi-vector of localizable shape:
+(51)
+P ij =
+∞
+�
+d=1
+ǫd−1
+�
+d
+�
+s=0
+P ij
+d,d−s∂s
+x + ui,1∂−1
+x V j
+d + V i
+d∂−1
+x uj,1
+�
+= P ij
+loc + P ij
+nonloc,
+where P ij
+d,d−s ∈ Ad−s and V i
+d ∈ Ad. Note that both P ij
+loc and P ij
+nonloc define skew-symmetric
+bi-vectors, that is (P ∗
+loc)ij = −P ji
+loc and (P ∗
+nonloc)ij = −P ji
+nonloc.
+Proposition 2.10. Consider a Miura-reciprocal transformation in R given by dy = Bdx,
+wi = Qi. Under this transformation any weakly non-local bi-vector P ij of localizable shape (51)
+is transformed into a weakly non-local bi-vector of localizable shape.
+Remark 2.11. In principle, this proposition follows from the arguments of [LZ11] and [FP03].
+However, it can also be directly obtained using Theorem 2.5.
+
+12
+P. LORENZONI, S. SHADRIN, AND R. VITOLO
+Proof. Repeating mutatis mutandis the proof of Proposition 2.8 one can check that the local
+part P ij
+loc produces a weakly non-local operator of localizable shape (the only thing that matters
+for that computation is skew-symmetry of the bi-vector defined by P ij
+loc).
+So let us focus
+on the non-local part P ij
+nonloc = ui
+x∂−1
+x V j + V i∂−1
+x uj
+x, where V i = �∞
+d=1 ǫd−1V i
+d and we use
+Convention 2.7 here and below in computations. We have:
+B−1
+�
+B ∂Qi
+∂uk,s∂s
+x − Qi
+x∂−1
+x
+∂B
+∂uk,s∂s
+x
+�
+◦
+�
+uk
+x∂−1
+x V l + V k∂−1
+x ul
+x
+�
+(52)
+�
+(−∂x)t ◦ ∂Qj
+∂ul,tB + (−∂x)t ◦ ∂B
+∂ul,t∂−1
+x Qj
+x
+�
+(we omit the summation over s and t for brevity). We compute (52) as follows. First, note that
+∂Qi
+∂uk,s∂s
+x ◦ uk
+x∂−1
+x V l(−∂x)t ◦ ∂Qj
+∂ul,tB = Qi
+x∂−1
+x (ℓQj)l(V l)B + loc;
+(53)
+∂Qi
+∂uk,s∂s
+x ◦ V k∂−1
+x ul
+x(−∂x)t ◦ ∂Qj
+∂ul,tB = (ℓQi)k(V k)∂−1
+x Qj
+xB + loc.
+Here loc are the terms where we collect some purely local operators. Furthermore,
+− 1
+B Qi
+x∂−1
+x
+∂B
+∂uk,s∂s
+x ◦ uk
+x∂−1
+x V l(−∂x)t ◦ ∂Qj
+∂ul,tB = − 1
+B Qi
+x∂−1
+x Bx∂−1
+x (ℓQj)l(V l)B − 1
+B Qi
+x∂−1
+x Oj
+BuV QB;
+(54)
+− 1
+B Qi
+x∂−1
+x
+∂B
+∂uk,s∂s
+x ◦ V k∂−1
+x ul
+x(−∂x)t ◦ ∂Qj
+∂ul,tB = − 1
+B Qi
+x∂−1
+x (ℓB)k(V k)∂−1
+x Qj
+xB − 1
+B Qi
+x∂−1
+x Oj
+BV uQB;
+∂Qi
+∂uk,s∂s
+x ◦ uk
+x∂−1
+x V l(−∂x)t ◦ ∂B
+∂ul,t∂−1
+x Qj
+x = Qi
+x∂−1
+x (ℓB)l(V l)∂−1
+x Qj
+x + Oi
+QuV B∂−1
+x Qj
+x;
+∂Qi
+∂uk,s∂s
+x ◦ V k∂−1
+x ul
+x(−∂x)t ◦ ∂B
+∂ul,t∂−1
+x Qj
+x = (ℓQi)l(V l)∂−1
+x Bx∂−1
+x Qj
+x + Oi
+QV uB∂−1
+x Qj
+x.
+Here Oj
+BuV Q, Oj
+BV uQ, Oi
+QuV B, and Oi
+QV uB are some scalar local operators, whose main property
+is that (Oj
+BuV Q)∗ = −Oj
+QV uB and (Oj
+BV uQ)∗ = −Oj
+QuV B.
+We omit their explicit formulas.
+Finally,
+− 1
+B Qi
+x∂−1
+x
+∂B
+∂uk,s∂s
+x ◦ uk
+x∂−1
+x V l(−∂x)t ◦ ∂B
+∂ul,t∂−1
+x Qj
+x = − 1
+B Qi
+x∂−1
+x Bx∂−1
+x (ℓB)l(V l)∂−1
+x Qj
+x
+(55)
+− 1
+B Qi
+x∂−1
+x OBuV B∂−1
+x Qj
+x;
+− 1
+B Qi
+x∂−1
+x
+∂B
+∂uk,s∂s
+x ◦ V k∂−1
+x ul
+x(−∂x)t ◦ ∂B
+∂ul,t∂−1
+x Qj
+x = − 1
+B Qi
+x∂−1
+x (ℓB)l(V l)∂−1
+x Bx∂−1
+x Qj
+x
+− 1
+B Qi
+x∂−1
+x OBV uB∂−1
+x Qj
+x,
+where OBuV B and OBV uB are scalar local operators such that O∗
+BuV B = −OBV uB. We omit
+their explicit formulas, but we use below that OBuV B +OBV uB = − ˜O∗∂x −∂x ◦ ˜O for some local
+operator ˜O.
+Now we collect the terms together. Firstly, we list all terms with Bx that emerged in (54)
+and (55):
+− 1
+B Qi
+x∂−1
+x Bx∂−1
+x (ℓQj)l(V l)B = −Qi
+x∂−1
+x (ℓQj)l(V l)B + 1
+B Qi
+x∂−1
+x (ℓQj)l(V l)B2
+(56)
+(ℓQi)l(V l)∂−1
+x Bx∂−1
+x Qj
+x = (ℓQi)l(V l)B∂−1
+x Qj
+x − (ℓQi)l(V l)∂−1
+x Qj
+xB
+− 1
+B Qi
+x∂−1
+x Bx∂−1
+x (ℓB)l(V l)∂−1
+x Qj
+x = −Qi
+x∂−1
+x (ℓB)l(V l)∂−1
+x Qj
+x + 1
+B Qi
+x∂−1
+x B(ℓB)l(V l)∂−1
+x Qj
+x
+
+MIURA-RECIPROCAL TRANSFORMATIONS AND LOCALIZABLE POISSON PENCILS
+13
+− 1
+B Qi
+x∂−1
+x (ℓB)l(V l)∂−1
+x Bx∂−1
+x Qj
+x = − 1
+B Qi
+x∂−1
+x (ℓB)l(V l)B∂−1
+x Qj
+x + 1
+B Qi
+x∂−1
+x (ℓB)l(V l)∂−1
+x Qj
+xB
+Note some cancellations: the non-local terms in (53) cancel with the corresponding summands
+in the first and the second line of (56), two non-local terms in the second and third line of (54)
+cancel with the two terms in the third and forth line of (56), are there are two terms in the
+latter lines that cancel each other. So, modulo the purely local terms, (52) is equal to the sum
+of the following four expressions:
+1
+B Qi
+x∂−1
+x (ℓQj)l(V l)B2 + (ℓQi)l(V l)B∂−1
+x Qj
+x = wi
+y∂−1
+y (ℓQj)l(V l)B + (ℓQi)l(V l)B∂−1
+y wj
+y;
+(57)
+1
+B Qi
+x∂−1
+x (Oj
+QV uB)∗B + Oi
+QV uB∂−1
+x Qj
+x = 1
+B Qi
+x∂−1
+x Oj
+QV uB(1)B + Oi
+QV uB(1)∂−1
+x Qj
+x + loc
+= wi
+y∂−1
+y Oj
+QV uB(1) + Oi
+QV uB(1)∂−1
+y wj
+y + loc;
+1
+B Qi
+x∂−1
+x (Oj
+QuV B)∗B + Oi
+QuV B∂−1
+x Qj
+x = 1
+B Qi
+x∂−1
+x Oj
+QuV B(1)B + Oi
+QuV B(1)∂−1
+x Qj
+x + loc
+= wi
+y∂−1
+y Oj
+QuV B(1) + Oi
+QuV B(1)∂−1
+y wj
+y + loc;
+− 1
+B Qi
+x∂−1
+x OBuV B∂−1
+x Qj
+x − 1
+B Qi
+x∂−1
+x OBV uB∂−1
+x Qj
+x = 1
+B Qi
+x∂−1
+x ( ˜O∗∂x + ∂x ◦ ˜O)∂−1
+x Qj
+x
+= 1
+B Qi
+x∂−1
+x
+˜O(1)Qj
+x + 1
+B Qi
+x ˜O(1)∂−1
+x Qj
+x + loc
+= wi
+y∂−1
+y
+1
+B
+˜O(1)Qj
+x + 1
+B Qi
+x ˜O(1)∂−1
+y wj
+y + loc,
+which is manifestly a weakly non-local operator of localizable shape.
+□
+3. Schouten bracket for weakly non-local operators of localizable shape
+The goal of this Section is to compare two ways to encode weakly non-local Poisson structures
+of localizable shape: the one given in [LZ11] (by design only working for the localizable shape
+case) and [LV20] (it is working for general weakly non-local case, but we specialize it for the
+localizable shape). In principle, the identification of these two approaches follows from the
+uniqueness property of the bracket, c.f. [LZ11, Theorem 2.4.1], but we want to present an
+explicit computation for this identification.
+3.1. The two approaches. In both approaches the weakly non-local p-vectors of localizable
+shape are encoded as
+�
+P =
+�
+PL + ζPN,
+(58)
+where PL ∈ ˆ
+Ap, PN ∈ ˆ
+Ap−1, and
+(59)
+∂xζ = −ui,1θi.
+The difference in two approaches is the meaning of ζ. In the approach of [LZ11], ζ is a new
+dependent variable such that deg∂x ζ = 0 and degθ ζ = 1. The new space of multivector densities
+is defined as S := ˆ
+A[ζ], equipped with the operator
+∂x = −ui,1θi∂ζ +
+�
+ui,d+1∂ui,d + θd+1
+i
+∂θd
+i ,
+(60)
+and the space of weakly non-local multivectors of localizable shape is defined as E := S/∂xS.
+In the approach of [LV20], ζ is not a new dependent variable, but rather an expression in
+the existing dependent variables (still of differential degree deg∂x ζ = 0 and multivector degree
+degθ ζ = 1), such that Equation (59) is satisfied for the standard operator
+˜∂x =
+�
+ui,d+1∂ui,d + θd+1
+i
+∂θd
+i .
+(61)
+
+14
+P. LORENZONI, S. SHADRIN, AND R. VITOLO
+For instance, one can find such a function in ˆ
+A((
+1
+u1,1)), cf. [DLZ06]. To this end, one looks for
+a unique solution ˜∂xζ = −ui,1θi of the form ζ = �∞
+i=1
+fi
+(u1,1)i, with fi ∈ ˆ
+A such that ∂u1,1fi = 0.
+Once the objects are defined, we have two different formulae for the Schouten bracket in
+these two approaches:
+• The formula in the approach of [LV20] is
+(62)
+� �
+P,
+�
+Q
+�
+=
+�
+(−1)degθ P ˜δuiP ˜δθiQ + ˜δθiP ˜δuiQ.
+Recall that ζ is regarded as a function of (ui
+σ, θσ
+i ) in the variational derivatives (which
+are denoted by ˜δui and ˜δθi for that reason).
+• The formula in the approach of [LZ11] is
+(63)
+� �
+P,
+�
+Q
+�
+=
+�
+(−1)degθ PδuiPδθiQ + δθiPδuiQ + (−1)degθ P ˆE(P)∂ζQ + ∂ζP ˆE(Q)
+Here ζ is regarded as an extra dependent variable, and the operator ˆE is defined as
+ˆE =
+�
+s≥1
+t≥0
+�
+ui,s(−∂x)t∂ui,s+t + θs
+i (−∂x)t∂θs+t
+i
+�
+− 1 + θiδθi.
+(64)
+3.2. Identification of the two approaches. We prove the following:
+Theorem 3.1. The identity map ˆ
+A[ζ] → ˆ
+A[ζ] induces the isomorphism of the Lie algebras of
+local multivector fields defined by the Schouten brackets in these two approaches.
+Proof. We represent any density P ∈ ˆ
+A[ζ] as P = PL + ζPN and consider ζ to be a nonlocal
+function. Note that
+˜δuiP = δuiP + (−∂x)σ (∂ui,σζPN)
+(65)
+= δuiPL + (−∂x)σ (ζ∂ui,σPN) + (−∂x)σ (∂ui,σζPN) ,
+˜δθiP = δθiP + (−∂x)σ �
+∂θσ
+i ζPN
+�
+(66)
+= δθiPL − (−∂x)σ �
+ζ∂θσ
+i PN
+�
++ (−∂x)σ �
+∂θσ
+i ζPN
+�
+,
+where we used that
+δuiP =δuiPL + (−∂x)σ (ζ∂ui,σPN) ,
+(67)
+δθiP =δθiPL − (−∂x)σ �
+ζ∂θσ
+i PN
+�
+.
+(68)
+Using these formulas, we obtain
+˜δuiP ˜δθiQ = (δuiP + (−∂x)σ (∂ui,σζPN))
+�
+δθiQ + (−∂x)σ �
+∂θσ
+i ζQN
+��
+(69)
+= δuiPδθiQ + δuiP(−∂x)σ �
+∂θσ
+i ζQN
+�
++ (−∂x)σ (∂ui,σζPN) δθiQ
++ (−∂x)σ (∂ui,σζPN) (−∂x)σ �
+∂θσ
+i ζQN
+�
+;
+˜δθiP ˜δuiQ =
+�
+δθiP + (−∂x)σ �
+∂θσ
+i ζPN
+��
+(δuiQ + (−∂x)σ (∂ui,σζQN))
+(70)
+= δθiPδuiQ + δθiP(−∂x)σ (∂ui,σζQN) + (−∂x)σ �
+∂θσ
+i ζPN
+�
+δuiQ
++ (−∂x)σ �
+∂θσ
+i ζPN
+�
+(−∂x)σ (∂ui,σζQN) .
+
+MIURA-RECIPROCAL TRANSFORMATIONS AND LOCALIZABLE POISSON PENCILS
+15
+If we want to treat ζ as a new dependent variable, we have ˆE(P)∂ζQ = ˆE(P)QN (and
+similarly for the other summand in the formula), so we have to prove that
+�
+(−1)degθ P ˆE(P)QN + PN ˆE(Q)
+(71)
+=
+�
+(−1)degθ P�
+δuiP(−∂x)σ �
+∂θσ
+i ζQN
+�
++ (−∂x)σ (∂ui,σζPN) δθiQ
++ (−∂x)σ (∂ui,σζPN) (−∂x)σ �
+∂θσ
+i ζQN
+� �
++
+�
+δθiP(−∂x)σ (∂ui,σζQN) + (−∂x)σ �
+∂θσ
+i ζPN
+�
+δuiQ
++ (−∂x)σ �
+∂θσ
+i ζPN
+�
+(−∂x)σ (∂ui,σζQN)
+�
+.
+Let us use the following property of the operator ˆE:
+∂x ˆE = −ui,1δui + θi∂xδθi + ui,1θiδζ;
+(72)
+So, we obtain
+�
+ˆE(P)QN =
+�
+∂−1
+x
+�
+−ui,1δuiP + θi∂xδθiP + ui,1θiPN
+�
+QN
+(73)
+=
+�
+−
+�
+−ui,1δuiP + θi∂xδθiP + ui,1θiPN
+�
+∂−1
+x (QN),
+�
+PN ˆE(Q) =
+�
+PN∂−1
+x
+�
+−ui,1δuiQ + θi∂xδθiQ + ui,1θiQN
+�
+(74)
+=
+�
+−∂−1
+x (PN)
+�
+−ui,1δuiQ + θi∂xδθiQ + ui,1θiQN
+�
+.
+Substituting Equations (73) and (74) into (71), we see that the statement of the theorem reduces
+to the following equality:
+�
+(−1)degθ P +1 �
+−ui,1δuiP + θi∂xδθiP + ui,1θiPN
+�
+∂−1
+x (QN)
+(75)
+− ∂−1
+x (PN)
+�
+−ui,1δuiQ + θi∂xδθiQ + ui,1θiQN
+�
+=
+�
+(−1)degθ P�
+δuiP(−∂x)σ �
+∂θσ
+i ζQN
+�
++ (−∂x)σ (∂ui,σζPN) δθiQ
++ (−∂x)σ (∂ui,σζPN) (���∂x)σ �
+∂θσ
+i ζQN
+� �
++
+�
+δθiP(−∂x)σ (∂ui,σζQN) + (−∂x)σ �
+∂θσ
+i ζPN
+�
+δuiQ
++ (−∂x)σ �
+∂θσ
+i ζPN
+�
+(−∂x)σ (∂ui,σζQN)
+�
+.
+In order to prove this equality, our strategy is move ∂−1
+x
+in ζ = ∂−1
+x (−ui,1θi) to the other
+factor (PN or QN) using integration by parts. We have:
+δuiP(−∂x)σ �
+∂θσ
+i ζQN
+�
+= δuiPui,1∂−1
+x (QN),
+(76)
+(−∂x)σ �
+∂uiσζPN
+�
+δθiQ = −∂x
+�
+θi∂−1
+x (PN)
+�
+δθiQ
+(77)
+(−∂x)σ �
+∂uiσζPN
+�
+(−∂x)σ �
+∂θσ
+i ζQN
+�
+= −∂x
+�
+θi∂−1
+x (PN)
+�
+ui,1∂−1
+x (QN)
+(78)
+(−∂x)σ �
+∂θσ
+i ζPN
+�
+δuiQ = ui,1∂−1
+x (PN) δuiQ
+(79)
+δθiP(−∂x)σ �
+∂uiσζQN
+�
+= −δθiP∂x
+�
+θi∂−1
+x (QN)
+�
+(80)
+(−∂x)σ �
+∂θσ
+i ζPN
+�
+(−∂x)σ �
+∂uiσζQN
+�
+= −
+�
+ui,1∂−1
+x (PN)
+�
+∂x
+�
+θi∂−1
+x (QN)
+�
+(81)
+
+16
+P. LORENZONI, S. SHADRIN, AND R. VITOLO
+Substituting the above expressions into the equality (75) that we shall prove, we are led to the
+simplified equality:
+�
+(−1)degθ P +1 �
+θi∂xδθiP + ui,1θiPN
+�
+∂−1
+x (QN) − ∂−1
+x (PN)
+�
+θi∂xδθiQ + ui,1θiQN
+�
+(82)
+=
+�
+(−1)degθ P +1�
+∂x(θi∂−1
+x (PN))δθiQ + ∂x(θi∂−1
+x (PN))ui,1∂−1
+x (QN)
+�
+−
+�
+δθiP∂x(θi∂−1
+x (QN)) + ui,1∂−1
+x (PN)∂x(θi∂−1
+x (QN))
+�
+.
+Integrating by parts the summands containing ∂xδθiP, ∂xδθiQ we obtain the further simplifica-
+tion of the equality (75) (note that degθ PN = degθ P − 1):
+�
+(−1)degθ P +1ui,1θiPN∂−1
+x (QN) − ∂−1
+x (PN)ui,1θiQN
+(83)
+=
+�
+(−1)degθ P +1∂x(θi∂−1
+x (PN))ui,1∂−1
+x (QN) − ui,1∂−1
+x (PN)∂x(θi∂−1
+x (QN))
+Expanding the total derivatives on the right-hand side we easily see that the above equality is
+an identity. This completes the proof of the theorem.
+□
+4. Pencils of weakly non-local bi-vectors of localizable shape
+In this Section we compute the bi-Hamiltonian cohomology for a semi-simple pencil of weakly
+non-local Poisson bi-vectors of localizable shape of differential order deg∂x = 1 satisfying the
+extra condition: the pencil of these bi-vectors should be localizable (or, equivalently, they
+should be simultaneously localizable) with respect to the Miura-reciprocal group. As a result
+of this computation and some further arguments we prove the following theorem:
+Theorem 4.1. Let P1 and P2 be two of commuting non-local Poisson bi-vectors of localizable
+shape.
+We assume that P1 and P2 have dispersive expansion given by Pa = �∞
+i=1 ǫi−1Pa,i,
+deg∂x Pa,i = i, a = 1, 2, i = 1, 2, . . . .
+If the leading terms of degree deg∂x = 1, P1,1 and P2,1, are simultaneously localizable under
+the action of the Miura-reciprocal group and form a semi-simple Poisson pencil, then the full
+dispersive brackets P1 and P2 are simultaneously localizable under the action of the Miura-
+reciprocal group.
+In order to prove this theorem, we have to make a few preliminary computations with bi-
+Hamiltonian cohomology, following the ideas in [LZ11] subsequent steps in [CPS18; CKS18].
+4.1. Bi-Hamiltonian cohomology.
+4.1.1. Setup for a deformation problem. Recall that following Liu and Zhang [LZ11] we denote
+S := ˆ
+A[ζ], with ∂x : S → S given by ∂x = −ui,1θi∂ζ + � ui,d+1∂ui,d + θd+1
+i
+∂θd
+i , and E := S/∂xS.
+Let P1, P2 ∈ S2
+1 such that
+�
+P1 and
+�
+P2 form a pencil of Poisson structures (possibly non-
+local, but then they are automatically weakly non-local of localizable shape, since it is the only
+type of non-locality accommodated in the space E), that is, we assume that
+� �
+P2 − λP1,
+�
+P2 − λP1
+�
+= 0
+(84)
+Recall that there is a group RI of the Miura-reciprocal transformations of the 1st kind acting
+on them, see Equation (8). We assume that the pencil
+�
+P2−λP1 is localizable under the action
+of RI. We also assume that the pencil formed by P1 and P2 is semi-simple, which together with
+the assumption of localizability implies that the we can choose the coordinates x, u1, . . . , uN
+
+MIURA-RECIPROCAL TRANSFORMATIONS AND LOCALIZABLE POISSON PENCILS
+17
+such that the densities P1 and P2 of the bivectors
+�
+P1 and
+�
+P2 take the form
+P1 =
+�
+N
+�
+i=1
+f iθiθ1
+i
+�
++ Γij
+1,kuk,1θiθj;
+(85)
+P2 =
+�
+N
+�
+i=1
+uif iθiθ1
+i
+�
++ Γij
+2,kuk,1θiθj.
+(86)
+We are interested to classify the equivalence classes of the higher order dispersive deforma-
+tions of the Poisson pencil
+�
+P2−λP1 in E with respect to the Miura-reciprocal transformations
+of the 2nd kind, RII. Let di := adPi : E → E, i = 1, 2. Then the deformation problem is con-
+trolled by the bi-Hamiltonian cohomology BHp
+d(E, d1, d2) of cohomological degree p = 2 and
+p = 3 and of differential degrees d ≥ 2 and d ≥ 4, respectively. It is a rather standard argument,
+see e. g. [LZ11, Proposition 3.3.5]. The only extra bit that one needs in our case, that is, the
+space E and the group RII of Miura-reciprocal transformations of the 2nd kind, in comparison
+with the usual local case, that is, the space ˆF and the group GII of Miura transformations of
+the 2nd kind, is the identification of the action of the Lie algebra of RII on weakly non-local
+bi-vectors (or, more generally, multivectors) of localizable shape with the adjoint action of E1
+on E2 (resp., E). This is established in [LZ11, Theorems 2.5.7 and 2.6.5]
+4.1.2. Bi-Hamiltonian cohomology computation. We prove the following
+Theorem 4.2. We have:
+BH2
+d(E, d1, d2) ∼=
+�
+0,
+d = 2 and d ≥ 4;
+�N
+i=1 C∞(R, ui),
+d = 3.
+(87)
+BH3
+d(E, d1, d2) ∼= 0,
+d ≥ 4.
+(88)
+Proof. For the proof we use that for d ≥ 2 we have [LZ13b, Lemma 4.4]:
+BHp
+d(E, d1, d2) ∼= Hp
+d(E[λ], d2 − λd1).
+(89)
+In order to compute Hp
+d(E[λ], d2−λd1), we recall the definition of Di := DPi : S → S from [LZ11]:
+DPi := ˆE(Pi)∂ζ +
+∞
+�
+s=0
+∂s
+x
+�
+δujPi
+�
+∂θs
+j + ∂s
+x
+�
+δθjPi
+�
+∂uj,s,
+i = 1, 2.
+(90)
+Note that [∂x, Di] = 0 (by direct computation). We prove that it is a homological vector field
+(which is not true in general, for a non-local bi-vector Pi):
+Lemma 4.3. For a purely local bivector
+�
+P the operator DP does not depend on the choice of
+a purely local density P. Moreover, for purely local densities of the bivectors P, Q ∈ ˆ
+A2 and for
+any T ∈ S we have:
+�
+DP(T) =
+� �
+P,
+�
+T
+�
+(91)
+and
+[DP, DQ] = D[P,Q],
+(92)
+where [P, Q] = δθiPδuiQ + δuiPδθiQ.
+In particular, for a purely local density P of a Poisson bivector
+�
+P we have D2
+P = 0 on S.
+Remark 4.4. The statements of Lemma 4.3 do not hold for not purely local densities.
+
+18
+P. LORENZONI, S. SHADRIN, AND R. VITOLO
+Proof of Lemma 4.3. Firstly, we check the DP does not depend on the choice of a local density
+P. To this end, we remind the definitions and basic properties of ˆE and ∂x. We have:
+∂x = −ui,1θi∂ζ +
+�
+ui,d+1∂ui,d + θd+1
+i
+∂θd
+i ;
+(93)
+ˆE =
+�
+s≥1
+t≥0
+�
+ui,s(−∂x)t∂ui,s+t + θs
+i (−∂x)t∂θs+t
+i
+�
+− 1 + θiδθi;
+(94)
+∂x ˆE = −ui,1δui + θi∂xδθi + ui,1θiδζ;
+(95)
+ˆE∂x = −ui,1θi∂ζ;
+(96)
+δui∂x = ∂xθi∂ζ,
+i = 1, . . . , N;
+(97)
+δθi∂x = −ui,1∂ζ,
+i = 1, . . . , N.
+(98)
+With the last three equations we immediately see that for any local X ∈ ˆ
+A
+ˆE(∂xX)∂ζ +
+∞
+�
+s=0
+�
+∂s
+x
+�
+δuj∂xX
+�
+∂θs
+j + ∂s
+x
+�
+δθj∂xX
+�
+∂uj,s
+�
+=
+(99)
+− ui,1θi∂ζX∂ζ +
+∞
+�
+s=0
+�
+∂s
+x
+�
+∂x(θi∂ζX)
+�
+∂θs
+j + ∂s
+x
+�
+− ui,1∂ζX
+�
+∂uj,s
+�
+= 0,
+since ∂ζ = 0, which implies the first assertion of the lemma.
+Now, Equation (91) is obvious from the definition of the Schouten bracket. So we focus on
+Equation (92). Let us compute the coefficient of ∂ζ on the left hand side. Using the vanishing
+of ∂ζ derivatives, we have:
+∞
+�
+s=0
+�
+∂s
+x
+�
+δujP
+�
+∂θs
+j + ∂s
+x
+�
+δθjP
+�
+∂uj,s
+�
+ˆE(Q) =
+(100)
+∂−1
+x
+∞
+�
+s=0
+�
+∂s
+x
+�
+δujP
+�
+∂θs
+j + ∂s
+x
+�
+δθjP
+�
+∂uj,s
+�
+(−ui,1δui + θi∂xδθi)(Q) =
+∂−1
+x
+�
+δujP∂x(δθjQ) − ∂x(δθjP)δujQ
+�
++ ∂−1
+x (−ui,1)
+∞
+�
+s=0
+�
+∂s
+x
+�
+δujP
+�
+∂θs
+j + ∂s
+x
+�
+δθjP
+�
+∂uj,s
+�
+δuiQ
++ ∂−1
+x (−θi∂x)
+∞
+�
+s=0
+�
+∂s
+x
+�
+δujP
+�
+∂θs
+j + ∂s
+x
+�
+δθjP
+�
+∂uj,s
+�
+δθiQ
+Adding to the latter expression the same one with interchanged P and Q and using that for
+purely local densities
+∞
+�
+s=0
+�
+∂s
+x
+�
+δujP
+�
+∂θs
+j + ∂s
+x
+�
+δθjP
+�
+∂uj,s
+�
+δuiQ +
+∞
+�
+s=0
+�
+∂s
+x
+�
+δujQ
+�
+∂θs
+j + ∂s
+x
+�
+δθjQ
+�
+∂uj,s
+�
+δuiP
+(101)
+= δui
+∞
+�
+s=0
+�
+∂s
+x
+�
+δujP
+�
+∂θs
+jQ + ∂s
+x
+�
+δθjP
+�
+∂uj,sQ
+�
+= δui
+∞
+�
+s=0
+�
+δujPδθjQ + δθjPδujQ
+�
+and
+∞
+�
+s=0
+�
+∂s
+x
+�
+δujP
+�
+∂θs
+j + ∂s
+x
+�
+δθjP
+�
+∂uj,s
+�
+δθiQ +
+∞
+�
+s=0
+�
+∂s
+x
+�
+δujQ
+�
+∂θs
+j + ∂s
+x
+�
+δθjQ
+�
+∂uj,s
+�
+δθiP
+(102)
+= −δθi
+∞
+�
+s=0
+�
+∂s
+x
+�
+δujP
+�
+∂θs
+jQ + ∂s
+x
+�
+δθjP
+�
+∂uj,sQ
+�
+= −δθi
+∞
+�
+s=0
+�
+δujPδθjQ + δθjPδujQ
+�
+,
+
+MIURA-RECIPROCAL TRANSFORMATIONS AND LOCALIZABLE POISSON PENCILS
+19
+we obtain that the coefficient of ∂ζ on the left hand side of Equation (92) is equal to
+∂−1
+x (−ui,1δui + θi∂xδθi)[P, Q] = ˆE
+�
+[P, Q]
+�
+,
+(103)
+which is the coefficient of ∂ζ on the right hand side of Equation (92). The coefficients of all
+other components of the vector fields on the left hand side of Equation (92) are computed in a
+very similar way.
+□
+Lemma 4.3 implies that D2−λD1 is a differential on S[λ], and we have a short exact sequence
+0
+� S[λ]
+R
+∂x
+�
+D2−λD1
+�
+S[λ]
+�
+�
+D2−λD1
+�
+E[λ]
+�
+d2−λd1
+�
+0
+(104)
+and it implies a long exact sequence in the cohomology which reads
+Hp
+d−1(S[λ]/R, D2 − λD1)
+� Hp
+d(S[λ], D2 − λD1)
+� Hp
+d(E[λ], d2 − λd1)
+�
+Hp+1
+d
+(S[λ]/R, D2 − λD1)
+� Hp+1
+d+1(S[λ], D2 − λD1)
+(105)
+Lemma 4.5. We have
+Hp
+d(S[λ], D2 − λD1) ∼=
+
+
+
+
+
+0
+p ≤ d and (p, d) ̸= (3, 3), (0, 0)
+R[λ]
+p = 0, d = 0;
+�N
+i=1 C∞(R, ui)
+p = 3, d = 3.
+(106)
+Also, H3
+2(S[λ], D2 − λD1) ∼= 0.
+Proof. This lemma can be derived from [CKS18, Theorems 2.12 and 2.13]. Indeed, Lemma 4.3
+in particular implies that we have a bicomplex (S[λ], Dloc, Dζ) with the differentials given by
+Dζ := ( ˆE(P2)−λ ˆE(P1))∂ζ and Dloc := D2 −λD1 −Dζ. We start a spectral sequence associated
+with this bicomplex. Obviously, it converges on the second page. The computation of the first
+page splits as
+Hp
+d(S[λ], Dloc) ∼= Hp
+d(A[λ], Dloc) ⊕ Hp
+d(A[λ]ζ, Dloc)
+(107)
+∼= Hp
+d(A[λ], Dloc) ⊕ Hp−1
+d
+(A[λ], Dloc),
+which implies all desired vanishings (for p ≤ d the only non-trivial cohomology groups are
+H0
+0(A[λ], Dloc) ∼= R[λ] and H3
+3(A[λ], Dloc) ∼= �N
+i=1 C∞(R, ui) [CKS18, Theorems 2.12 and
+2.13]).
+Since both Hi
+i(S[λ], Dloc) = 0 for i = 1, 2, 4, and the induced differential on the first page
+has the (p, d)-degree (1, 1), we conclude that
+H0
+0(S[λ], D2 − λD1) ∼= H0
+0(S[λ], Dloc) ∼= R[λ];
+(108)
+H3
+3(S[λ], D2 − λD1) ∼= H3
+3(S[λ], Dloc) ∼=
+N
+�
+i=1
+C∞(R, ui).
+(109)
+□
+Remark 4.6. Almost the same statement holds for the cohomology of S[λ]/R, the only difference
+is H0
+0(S[λ]/R, D2 − λD1) ∼= 0.
+Now we can complete the computation of the cohomology Hp
+d(E[λ], d2 − λd1) for p < d and
+p = 2, d = 2 using the long exact sequence (105). The relevant pieces of this long exact sequence
+
+20
+P. LORENZONI, S. SHADRIN, AND R. VITOLO
+are
+0 = Hp
+d(S[λ], D2 − λD1)
+� Hp
+d(E[λ], dloc
+2 − λdloc
+1 )
+�
+Hp+1
+d
+(S[λ]/R, D2 − λD1) = 0
+,
+(110)
+for p < d and (p + 1, d) ̸= (3, 3), which implies the vanishing for p < d, (p, d) ̸= (2, 3), and
+p = 2, d = 2. Moreover, we have
+0 = H2
+3(S[λ], D2 − λD1)
+� H2
+3(E[λ], dloc
+2 − λdloc
+1 )
+�
+H3
+3(S[λ]/R, D2 − λD1) ∼= �N
+i=1 C∞(R, ui)
+� H3
+4(S[λ], D2 − λD1) = 0
+,
+(111)
+which gives the answer for (p, d) = (2, 3). Now, the special cases of these computations for
+p = 2, d ≥ 2 and p = 3, d ≥ 4 imply all statements of Theorem 4.2.
+□
+An immediate corollary of Theorem 4.2 is the following:
+Corollary 4.7. Let
+�
+P2 − λP1 be a semi-simple pencil of local Poisson bivectors of differen-
+tial order 1. We consider the higher order dispersive extensions of
+�
+P2 − λP1 in the realm
+of weakly non-local Poisson pencils of localizable shape, that is, we consider Poisson pencils
+� �∞
+d=1 ǫd−1(P2,d −λP1,d) ∈ E such that deg∂x(P2,d −λP1,d) = d and
+�
+P2,1 −λP1,1 =
+�
+P2 −λP1.
+The space of orbits of the action of the group RII (the group of Miura-reciprocal transfor-
+mation of the 2nd kind) onto the set of these dispersive extensions is isomorphic to the space
+�N
+i=1 C∞(R, ui).
+This result is strikingly similar to the corresponding statement in the local case, cf. [CPS18,
+Theorem 1], see also [LZ05; LZ13a; DLZ06]. However, in the local case both the space ˆF where
+the deformations of
+�
+P2 − λP1 are allowed as well as the group GII acting on them are much
+smaller than in Corollary 4.7. Our next goal is to compare these two situations.
+4.2. Comparison with the purely local deformations. Within this section it is important
+to have a notation that distinguishes between the operator ∂x as given by Equation (93) on the
+space S = ˆ
+A[ζ] and its purely local version ˜∂x := ∂x + ui,1θi∂ζ defined both on S and on A.
+Note that on S the operator ˜∂x commutes with multiplication by ζ.
+Let T nl denote the space of dispersive weakly non-local Poisson pencils of localizable shape
+� �∞
+d=1 ǫd−1(P2,d − λP1,d) ∈ E with the fixed leading term
+�
+P2,1 − λP1,1 =
+�
+P2 − λP1 that
+is purely local and semi-simple. Let T loc denote the space of dispersive local Poisson pencils
+� �∞
+d=1 ǫd−1(P2,d − λP1,d) ∈ ˆF with the same fixed leading term
+�
+P2,1 − λP1,1 =
+�
+P2 − λP1.
+The group RII acts on T nl and the group GII acts on T loc. Moreover, there is a natural
+embedding I : T loc → T nl that is GII-equivariant (GII acts on T nl as a subgroup of RII). The
+map I induces a map of the sets of orbits ι: T loc/GII → T nl/RII.
+Proposition 4.8. The map ι is injective.
+Proof. This proposition immediately follows from [LZ11, Theorem 1.3] and [CPS18, Theorem
+2].
+By the latter result in the local case, we have an isomorphism of sets ˜c: T loc/GII →
+�N
+i=1 C∞(R, ui) (these are the so-called central invariants in the local case).
+On the other
+hand, [LZ11, Theorem 1.3] states that for any x, y ∈ T loc/GII such that ι(x) = ι(y) we have
+˜c(x) = ˜c(y). Hence, x = y, and ι is surjective.
+□
+Corollary 4.7 implies that there is a RII invariant map C : T nl → �N
+i=1 C∞(R, ui) that
+descends to a bijection c: T nl/RII → �N
+i=1 C∞(R, ui). We have the following
+Proposition 4.9. The composition c ◦ ι: T loc/GII → �N
+i=1 C∞(R, ui) is surjective.
+
+MIURA-RECIPROCAL TRANSFORMATIONS AND LOCALIZABLE POISSON PENCILS
+21
+Proof. Basically, we want to show that any cohomology class in H2
+3(E) has a representative with
+a purely local density. Let
+�
+a2 + a1ζ represent a class in H2
+3(E), a2 ∈ ˆ
+A2
+3[λ] and a1 ∈ ˆ
+A1
+3[λ].
+This means that
+(D2 − λD1)(a2 + a1ζ) ∈ ∂x(S3
+3[λ]),
+(112)
+or, in other words, that there exist b3 ∈ ˆ
+A3
+3[λ] and b2 ∈ ˆ
+A2
+3[λ] such that
+Dloc(a2) − ( ˆE(P2) − λ ˆE(P1))(a1) + Dloc(a1)ζ = ˜∂x(b3) + ui,1θib2 + ˜∂x(b2)ζ.
+(113)
+Since H1
+3( ˆ
+A[λ], Dloc) = 0 [CKS18, Theorem 2.13], there exist e0 ∈ ˆ
+A0
+2[λ] and f 1 ∈ ˆ
+A1
+2[λ] such
+that Dloce0 = a1 + ˜∂x(f 1). Then,
+(D2 − λD1)(e0ζ) = a1ζ + ˜∂x(f 1)ζ − ( ˆE(P2) − λ ˆE(P1))(e0)
+(114)
+= a1ζ − ( ˆE(P2) − λ ˆE(P1))(e0) − ui,1θif 1 + ∂x(f 1ζ),
+which implies that
+(d2 − λd1)
+�
+e0ζ =
+�
+a1ζ − ( ˆE(P2) − λ ˆE(P1))(e0) − ui,1θif 1
+(115)
+Thus, the cocycle
+�
+a2 + a1ζ is cohomologous to
+�
+a2 + ( ˆE(P2) − λ ˆE(P1))(e0) + ui,1θif 1, which
+gives a pure local deformation for
+�
+P2 − λP1.
+□
+Taking into account that that c is a bijection, an immediate corollary of Proposition 4.9 is
+the following:
+Corollary 4.10. The map ι is surjective (and hence a bijection). In particular, every orbit of
+the action of RII on T nl contains a purely local representative.
+It is just a different way to state Theorem 4.1, so this corollary also completes the proof of
+Theorem 4.1
+4.3. Roots of the characteristic polynomial of the symbol. In the purely local case the
+central invariants, besides a purely cohomological definition, can be computed directly from a
+representative of a deformation (see [DLZ06] for details). More precisely, one has to compute
+the eigenvalues of the symbol of a representative of a deformation, which behave as scalars with
+respect to the Miura group action. In this section we extend this viewpoint to the invariants
+of the Miura-reciprocal group.
+First, we recall the construction from [DLZ06]. Let
+� �∞
+d=1 ǫd−1(P2,d −λP1,d) ∈ T loc, and the
+densities are expanded as �d
+s=0(P2,d,s − λP1,d,s)ijθiθd−s
+j
+, d ≥ 1, such that
+d
+�
+s=0
+(P2,d,s − λP1,d,s)ij∂d−s
+x
+= −
+d
+�
+s=0
+(−∂x)d−s ◦ (P2,d,s − λP1,d,s)ji
+(116)
+Consider the symbol of the densities of the bi-vector
+� �∞
+d=1 ǫd−1(P2,d −λP1,d), that is, the sum
+�∞
+d=1 ǫd−1(P2,d,0 − λP1,d,0)ij = �∞
+d=1(−ǫ)d−1(P2,d,0 − λP1,d,0)ji. The construction of the Miura
+group invariants from the eigenvalues of the symbol is based on the following lemma:
+Lemma 4.11. Under the group of Miura transformations G the symbol transforms linearly as
+a pencil of bi-linear forms:
+∞
+�
+d=1
+ǫd−1(P2,d,0 − λP1,d,0)ij �→
+∞
+�
+d=0
+ǫd ∂wi
+d
+∂uk,d
+∞
+�
+d=1
+ǫd−1(P2,d,0 − λP1,d,0)kℓ
+∞
+�
+d=1
+(−ǫ)d ∂wj
+d
+∂uℓ,d
+(117)
+(here wi = �∞
+d=0 ǫdwi
+d, wi
+d ∈ Ad, i = 1, . . . , N, are the new coordinates). Hence, the eigenvalues
+of this pencil behave as scalar with respect to the action of the Miura group.
+
+22
+P. LORENZONI, S. SHADRIN, AND R. VITOLO
+There are N roots λi, i = 1, . . . , N, of the λ-polynomial
+(118)
+det
+� ∞
+�
+d=1
+ǫd−1(P2,d,0 − λP1,d,0)
+�
+which are the formal power series in ǫ with the coefficients given by smooth functions in
+u1, . . . , uN, with the leading terms in ǫ given by mi = ui + O(ǫ). These eigenvalues are fur-
+ther used to derive the closed formulas for the central invariants of a pencil
+� �∞
+d=1 ǫd−1(P2,d −
+λP1,d) ∈ T loc.
+In the weakly non-local case of localizable shape, the densities of
+� �∞
+d=1 ǫd−1(P2,d −λP1,d) ∈
+T nl can be uniquely expanded as �d
+s=0(P2,d,s − λP1,d,s)ijθiθd−s
+j
++ (Q2,d − λQ1,d)iθiζ, d ≥ 1, such
+that
+d
+�
+s=0
+(P2,d,s − λP1,d,s)ij∂d−s
+x
+= −
+d
+�
+s=0
+(−∂x)d−s ◦ (P2,d,s − λP1,d,s)ji.
+(119)
+(this expansion we call the “normal form” below).
+Proposition 4.12. Let
+� �∞
+d=1 ǫd−1(P2,d − λP1,d) ∈ T nl and let
+λi = ri + ǫ2λi
+2 + ǫ4λi
+4 + · · ·
+be the λ-roots of the characteristic polynomial (118). The quantities
+ci
+2k = λi
+2k
+(f i)k ,
+k = 0, 1, 2, ...
+(where f i are the diagonal entries of the first metric in canonical coordinates) are invariant
+under the action of R.
+Proof. Taking into account the above lemma we focus on pure reciprocal transformations. The
+action of reciprocal transformations of 1st kind on the coefficients of the symbols can be easily
+obtained using the same arguments used in the proof of Proposition (2.8). Indeed
+˜P ij
+λ
+=
+B−1
+�
+Bδi
+k − ui
+x∂−1
+x
+∂B
+∂uk
+�
+P kℓ
+λ
+�
+Bδj
+ℓ + ∂B
+∂uℓ∂−1
+x uj
+x
+�
+=
+BP ij
+λ + P il
+λ
+∂B
+∂uℓ∂−1
+x uj
+x − 1
+B ui
+x∂−1
+x
+∂B
+∂uk P kj
+λ − ui
+x∂−1
+x
+∂B
+∂uk P kℓ
+λ
+∂B
+∂uℓ ∂−1
+x uj
+x.
+The second, the third and the fourth terms cannot contribute to the symbol of ˜P ij
+λ , while in
+the first term the only contributions come from
+B
+∞
+�
+d=1
+ǫd−1(P2,d,0−λP1,d,0)∂d
+x = B
+∞
+�
+d=1
+ǫd−1(P2,d,0−λP1,d,0)Bd∂d
+y = B2
+∞
+�
+d=1
+(Bǫ)d−1(P2,d,0−λP1,d,0)∂d
+y
+that implies
+∞
+�
+d=1
+ǫd−1(P2,d,0 − λP1,d,0)ij → B2
+∞
+�
+d=1
+(Bǫ)d−1(P2,d,0 − λP1,d,0)ij.
+This means that
+λi → ri + (Bǫ)2λi
+2 + (Bǫ)4λi
+4 + · · ·
+or, equivalently, that
+λi
+2k → B2kλi
+2k.
+The result then follows from the transformation rule for the contravariant metric (see (43)):
+f i → B2f i. In the case of reciprocal transformation of 2nd kind we observe that they do not
+affect the symbol of the pencil. Indeed a bivector transforms according to the following rule
+˜P ij := B−1
+�
+Bδi
+k − ui
+x∂−1
+x
+∂B
+∂uk
+σ
+∂σ
+x
+�
+P kℓ
+�
+Bδj
+ℓ + (−∂x)τ ∂B
+∂uℓ
+τ
+∂−1
+x uj
+x
+�
+.
+(120)
+
+MIURA-RECIPROCAL TRANSFORMATIONS AND LOCALIZABLE POISSON PENCILS
+23
+where
+B = 1 + H = 1 +
+∞
+�
+k=1
+ǫkHk(uj, uj
+x, . . . , uj
+σ),
+Hk ∈ Ak.
+Thus we have
+˜P ij := P ij − 1
+B ui
+x∂−1
+x
+∂H
+∂uk
+σ
+∂σ
+xP kj +
+�
+δi
+k − 1
+B ui
+x∂−1
+x
+∂H
+∂uk
+σ
+∂σ
+x
+�
+P kℓ
+�
+Hδj
+ℓ + (−∂x)τ ∂H
+∂uℓ
+τ
+∂−1
+x uj
+x
+�
+.
+(121)
+Since the symbol of the bivector contains only the subset of the coefficients which depend only
+on the u’s but not on their x-derivatives the second term and the third terms above cannot
+contribute to it. This implies that the symbol of each bivector defining the pencil is unaffected
+by these transformations. For Miura reciprocal transformations (5) the transformation rule for
+the symbol of the pencil is obtained combining the Lemma 4.11 with the above rule. It turns
+out that the symbol of the pencil transforms in the following way
+∞
+�
+d=1
+ǫd−1(P2,d,0 − λP1,d,0)ij �→ B2
+∞
+�
+d=0
+ǫd ∂wi
+d
+∂uk,d
+∞
+�
+d=1
+(Bǫ)d−1(P2,d,0 − λP1,d,0)kℓ
+∞
+�
+d=1
+(−ǫ)d ∂wj
+d
+∂uℓ,d.
+(122)
+□
+5. Projective-reciprocal invariance of the Doyle–Pot¨emin form
+In this Section we make a first step towards the study of the projective-reciprocal group
+action. Consider a local operator of homogeneous differential order d + 2, d ≥ 2 of the form
+P ij = ∂x ◦ Qij ◦ ∂x. We call this presentation of an operator the Doyle–Pot¨emin form (see
+Subsection 1.5).
+We prove the following theorem:
+Theorem 5.1. The projective group preserves the Doyle–Pot¨emin form of an operator. More
+precisely, the image of a homogeneous skew-symmetric operator of the form ∂x ◦ Qij ◦ ∂x,
+deg∂x Qij = d ≥ 0 under the action of an element of P is a homogeneous skew-symmetric
+operator of the form ∂x ◦ ˜Qij ◦ ∂x, deg∂x ˜Qij = d ≥ 0
+Proof. Consider an element of the group P given by
+dy = A0dx,
+(123)
+wi = Ai/A0,
+i = 1, . . . , N,
+where Ai := ai
+juj + ai
+0, i = 0, 1, . . . , N. Since the functions Ai and A0 do not depend on the
+higher jet variables, Theorem 2.5 implies that the operator P ij = ∂x◦Qij ◦∂x in the coordinates
+y, w1, . . . , wN is represented as
+˜P ij = 1
+A0
+�
+A0∂uk
+� Ai
+A0
+�
+− ∂x
+� Ai
+A0
+�
+∂−1
+x
+◦ ∂ukA0
+�
+∂x ◦ Qkl ◦ ∂x◦
+(124)
+�
+∂ul
+�Aj
+A0
+�
+A0 + ∂ulA0∂−1
+x
+◦ ∂x
+�Aj
+A0
+��
+Now we see that
+∂x ◦
+�
+∂ul
+�Aj
+A0
+�
+A0 + ∂ulA0∂−1
+x
+◦ ∂x
+�Aj
+A0
+��
+= ∂x ◦
+�
+aj
+l − a0
+l
+�Aj
+A0
+�
++ a0
+l ∂−1
+x
+◦ ∂x
+�Aj
+A0
+��
+(125)
+=
+�
+aj
+l − a0
+l
+�Aj
+A0
+��
+∂x = (A0)2∂ulwj∂y
+
+24
+P. LORENZONI, S. SHADRIN, AND R. VITOLO
+and analogously
+1
+A0
+�
+A0∂uk
+� Ai
+A0
+�
+− ∂x
+� Ai
+A0
+�
+∂−1
+x
+◦ ∂ukA0
+�
+∂x = 1
+A0
+�
+ai
+k −
+� Ai
+A0
+�
+a0
+k − ∂x
+� Ai
+A0
+�
+∂−1
+x
+◦ a0
+k
+�
+∂x
+(126)
+= 1
+A0∂x ◦
+�
+ai
+k −
+� Ai
+A0
+�
+a0
+k
+�
+= ∂y ◦ 1
+A0∂ukwi(A0)2.
+Thus we see that ˜P ij takes the form ∂y ◦ ˜Qij ◦ ∂y, where the operator ˜Qij is equal to
+˜Qij = 1
+A0∂ukwi(A0)2Qkl(A0)2∂ulwj
+(127)
+after the substitution wi(u1, . . . , uN) = Ai/A0 and ∂y = (A0)−1∂x, which makes it manifestly
+skew-symmetric and homogeneous of the same degree.
+□
+Remark 5.2. Note that we don’t use the Poisson property in the proof (and we don’t have it in
+the statement of the theorem). This allows us to apply the projective-reciprocal transformation
+to any homogeneous skew-symmetric operators of the Doyle–Pot¨emin form, and the action
+would preserve the form.
+Remark 5.3. Interesting examples of skew-symmetric operators in the Doyle–Pot¨emin form
+are coming from the theory of Dubrovin–Zhang hierarchies [DZ01]. Is it proved in [BPS12b;
+BPS12a] that Dubrovin–Zhang hierarchies posses a Poisson bracket given by an operator of the
+shape �∞
+p=0 ǫ2pP ij
+2p+1, where P ij
+1 = ηij∂x for some constant inner product ηij, and for p ≥ 1 the
+operators P ij
+2p+1 are homogeneous skew-symmetric operators of the shape �2p+1
+e=0 P ij
+2p+1,e∂2p+1−e
+x
+,
+where deg∂x P ij
+2p+1,e = e, such that P ij
+2p+1,0 = 0. Using that the operators P ij
+2p+1, p ≥ 1, are
+skew-symmetric, it is easy to show that each of them is of the Doyle–Pot¨emin form.
+References
+[Abe09]
+S. Abenda. “Reciprocal transformations and local Hamiltonian structures of hydro-
+dynamic-type systems”. In: J. Phys. A 42.9 (2009), pp. 095208, 20. url: https://doi.org/10.1088/1751-8113/42/9/095208.
+[AG07]
+S. Abenda and T. Grava. “Reciprocal transformations and flat metrics on Hurwitz
+spaces”. In: J. Phys. A 40.35 (2007), pp. 10769–10790. url: https://doi.org/10.1088/1751-8113/40/35/004.
+[AL13]
+A. Arsie and P. Lorenzoni. “Reciprocal F-manifolds”. In: J. Geom. Phys. 70 (2013),
+pp. 185–204. url: https://doi.org/10.1016/j.geomphys.2013.03.029.
+[BPS12a]
+A. Buryak, H. Posthuma, and S. Shadrin. “On deformations of quasi-Miura trans-
+formations and the Dubrovin-Zhang bracket”. In: J. Geom. Phys. 62.7 (2012),
+pp. 1639–1651. url: https://doi.org/10.1016/j.geomphys.2012.03.006.
+[BPS12b]
+A. Buryak, H. Posthuma, and S. Shadrin. “A polynomial bracket for the Dubrovin-
+Zhang hierarchies”. In: J. Differential Geom. 92.1 (2012), pp. 153–185. url: http://projecteuclid.org/euclid.jdg/1352211225.
+[BS09]
+M. B�laszak and A. Sergyeyev. “A coordinate-free construction of conservation laws
+and reciprocal transformations for a class of integrable hydrodynamic-type systems”.
+In: Rep. Math. Phys. 64.1-2 (2009), pp. 341–354. url: https://doi.org/10.1016/S0034-4877(09)90038-6.
+[Cas+22]
+M. Casati, P. Lorenzoni, D. Valeri, and R. Vitolo. “Weakly nonlocal Poisson brack-
+ets: tools, examples, computations”. In: Comput. Phys. Commun. 274 (2022), Paper
+No. 108284, 18. url: https://doi.org/10.1016/j.cpc.2022.108284.
+[CCS17]
+G. Carlet, M. Casati, and S. Shadrin. “Poisson cohomology of scalar multidimen-
+sional Dubrovin-Novikov brackets”. In: J. Geom. Phys. 114 (2017), pp. 404–419.
+url: https://doi.org/10.1016/j.geomphys.2016.12.008.
+[CCS18]
+G. Carlet, M. Casati, and S. Shadrin. “Normal forms of dispersive scalar Pois-
+son brackets with two independent variables”. In: Lett. Math. Phys. 108.10 (2018),
+pp. 2229–2253. url: https://doi.org/10.1007/s11005-018-1076-x.
+
+REFERENCES
+25
+[CKS18]
+G. Carlet, R. Kramer, and S. Shadrin. “Central invariants revisited”. In: J. ´Ec.
+polytech. Math. 5 (2018), pp. 149–175. url: https://doi.org/10.5802/jep.66.
+[CLV20]
+M. Casati, P. Lorenzoni, and R. Vitolo. “Three computational approaches to weakly
+nonlocal Poisson brackets”. In: Stud. Appl. Math. 144.4 (2020), pp. 412–448. url:
+https://doi.org/10.1111/sapm.12302.
+[CPS16a]
+G. Carlet, H. Posthuma, and S. Shadrin. “Bihamiltonian cohomology of KdV brack-
+ets”. In: Comm. Math. Phys. 341.3 (2016), pp. 805–819. url: https://doi.org/10.1007/s00220-015-2540-4.
+[CPS16b]
+G. Carlet, H. Posthuma, and S. Shadrin. “The bi-Hamiltonian cohomology of a
+scalar Poisson pencil”. In: Bull. Lond. Math. Soc. 48.4 (2016), pp. 617–627. url:
+https://doi.org/10.1112/blms/bdw017.
+[CPS18]
+G. Carlet, H. Posthuma, and S. Shadrin. “Deformations of semisimple Poisson pen-
+cils of hydrodynamic type are unobstructed”. In: J. Differential Geom. 108.1 (2018),
+pp. 63–89. url: https://doi.org/10.4310/jdg/1513998030.
+[DLZ06]
+B. Dubrovin, S.-Q. Liu, and Y. Zhang. “On Hamiltonian perturbations of hyperbolic
+systems of conservation laws. I. Quasi-triviality of bi-Hamiltonian perturbations”.
+In: Comm. Pure Appl. Math. 59.4 (2006), pp. 559–615. url: https://doi.org/10.1002/cpa.20111.
+[DMS05]
+L. Degiovanni, F. Magri, and V. Sciacca. “On deformation of Poisson manifolds of
+hydrodynamic type”. In: Comm. Math. Phys. 253.1 (2005), pp. 1–24. url: https://doi.org/10.1007/s00220-004-1190-8.
+[Doy93]
+P. W. Doyle. “Differential geometric Poisson bivectors in one space variable”. In: J.
+Math. Phys. 34.4 (1993), pp. 1314–1338. url: https://doi.org/10.1063/1.530213.
+[DZ01]
+B. Dubrovin and Y. Zhang. Normal forms of hierarchies of integrable PDEs, Frobe-
+nius manifolds and Gromov - Witten invariants. 2001. url: https://arxiv.org/abs/math/0108160.
+[Fer89]
+E. V. Ferapontov. “Reciprocal transformations and their invariants”. In: Differ-
+entsial ′nye Uravneniya 25.7 (1989), pp. 1256–1265, 1286.
+[Fer91]
+E. V. Ferapontov. “Autotransformations with respect to the solution, and hydro-
+dynamic symmetries”. In: Differentsial ′nye Uravneniya 27.7 (1991), pp. 1250–1263,
+1287.
+[Fer95a]
+E. V. Ferapontov. “Conformally flat metrics, systems of hydrodynamic type and
+nonlocal Hamiltonian operators”. In: Uspekhi Mat. Nauk 50.4(304) (1995), pp. 175–
+176. url: https://doi.org/10.1070/RM1995v050n04ABEH002582.
+[Fer95b]
+E. V. Ferapontov. “Nonlocal Hamiltonian operators of hydrodynamic type: differ-
+ential geometry and applications”. In: Topics in topology and mathematical physics.
+Vol. 170. Amer. Math. Soc. Transl. Ser. 2. Amer. Math. Soc., Providence, RI, 1995,
+pp. 33–58. url: https://doi.org/10.1090/trans2/170/03.
+[FP03]
+E. V. Ferapontov and M. V. Pavlov. “Reciprocal transformations of Hamiltonian op-
+erators of hydrodynamic type: nonlocal Hamiltonian formalism for linearly degener-
+ate systems”. In: J. Math. Phys. 44.3 (2003), pp. 1150–1172. url: https://doi.org/10.1063/1.1542921.
+[FPV14]
+E. V. Ferapontov, M. V. Pavlov, and R. F. Vitolo. “Projective-geometric aspects
+of homogeneous third-order Hamiltonian operators”. In: J. Geom. Phys. 85 (2014),
+pp. 16–28. url: https://doi.org/10.1016/j.geomphys.2014.05.027.
+[Get02]
+E. Getzler. “A Darboux theorem for Hamiltonian operators in the formal calculus of
+variations”. In: Duke Math. J. 111.3 (2002), pp. 535–560. url: https://doi.org/10.1215/S0012-7094-02-11136-3.
+[Ibr85]
+N. H. Ibragimov. Transformation groups applied to mathematical physics. Mathe-
+matics and its Applications (Soviet Series). Translated from the Russian. D. Reidel
+Publishing Co., Dordrecht, 1985, pp. xv+394. url: https://doi.org/10.1007/978-94-009-5243-0.
+[IVV02]
+S. Igonin, A. Verbovetsky, and R. Vitolo. On the formalism of local variational differ-
+ential operators. Memorandum 1641. University of Twente, Department of Applied
+Mathematics, 2002, pp. 1–34. url: https://research.utwente.nl/en/publications/on-the-formalism-of-local-variational-differential-operators.
+[Lor02]
+P. Lorenzoni. “Deformations of bi-Hamiltonian structures of hydrodynamic type”.
+In: J. Geom. Phys. 44.2-3 (2002), pp. 331–375. url: https://doi.org/10.1016/S0393-0440(02)00080-3.
+[LV20]
+P. Lorenzoni and R. Vitolo. “Weakly nonlocal Poisson brackets, Schouten brackets
+and supermanifolds”. In: J. Geom. Phys. 149 (2020), pp. 103573, 8. url: https://doi.org/10.1016/j.geomphys.2019.103573.
+
+26
+REFERENCES
+[LZ05]
+S.-Q. Liu and Y. Zhang. “Deformations of semisimple bihamiltonian structures of
+hydrodynamic type”. In: J. Geom. Phys. 54.4 (2005), pp. 427–453. url: https://doi.org/10.1016/j.geomphys.2004.11.003.
+[LZ11]
+S.-Q. Liu and Y. Zhang. “Jacobi structures of evolutionary partial differential equa-
+tions”. In: Adv. Math. 227.1 (2011), pp. 73–130. url: https://doi.org/10.1016/j.aim.2011.01.015.
+[LZ13a]
+S.-Q. Liu and Y. Zhang. “Bihamiltonian cohomologies and integrable hierarchies I: A
+special case”. In: Comm. Math. Phys. 324.3 (2013), pp. 897–935. url: https://doi.org/10.1007/s00220-013-1822-y.
+[LZ13b]
+S.-Q. Liu and Y. Zhang. “Bihamiltonian cohomologies and integrable hierarchies I: A
+special case”. In: Comm. Math. Phys. 324.3 (2013), pp. 897–935. url: https://doi.org/10.1007/s00220-013-1822-y.
+[Miu68]
+R. M. Miura. “Korteweg-de Vries equation and generalizations. I. A remarkable
+explicit nonlinear transformation”. In: J. Mathematical Phys. 9 (1968), pp. 1202–
+1204. url: https://doi.org/10.1063/1.1664700.
+[MN01]
+A. Y. Maltsev and S. P. Novikov. “On the local systems Hamiltonian in the weakly
+non-local Poisson brackets”. In: Phys. D 156.1-2 (2001), pp. 53–80. url: https://doi.org/10.1016/S0167-2789(01)00280-9.
+[Mok87]
+O. I. Mokhov. “Hamiltonian differential operators and contact geometry”. In: Funk-
+tsional. Anal. i Prilozhen. 21.3 (1987), pp. 53–60, 96.
+[Mok98]
+O. I. Mokhov. “Symplectic and Poisson structures on loop spaces of smooth mani-
+folds, and integrable systems”. In: Uspekhi Mat. Nauk 53.3(321) (1998), pp. 85–192.
+url: https://doi.org/10.1070/rm1998v053n03ABEH000019.
+[Olv88]
+P. J. Olver. “Darboux’s theorem for Hamiltonian differential operators”. In: J. Dif-
+ferential Equations 71.1 (1988), pp. 10–33. url: https://doi.org/10.1016/0022-0396(88)90036-8.
+[Olv93]
+P. J. Olver. Applications of Lie groups to differential equations. Second. Vol. 107.
+Graduate Texts in Mathematics. Springer-Verlag, New York, 1993, pp. xxviii+513.
+url: https://doi.org/10.1007/978-1-4612-4350-2.
+[Pot91]
+G. V. Pot¨emin. “Some questions of differential geometry and algebraic geometry in
+the theory of solitons”. PhD thesis. Moscow State University, 1991.
+[Pot97]
+G. V. Pot¨emin. “On third-order differential-geometric Poisson brackets”. In: Uspekhi
+Mat. Nauk 52.3(315) (1997), pp. 173–174. url: https://doi.org/10.1070/RM1997v052n03ABEH001817.
+[Rog68]
+C. Rogers. “Reciprocal relations in non-steady one-dimensional gasdynamics”. In:
+Z. Angew. Math. Phys. 19 (1968), pp. 58–63.
+[Rog69]
+C. Rogers. “Invariant transformations in non-steady gasdynamics and magneto-
+gasdynamics”. In: Z. Angew. Math. Phys. 20 (1969), pp. 370–382.
+[VV]
+P. Vergallo and R. Vitolo. “Projective geometry of homogeneous second order Hamil-
+tonian operators”. preprint. url: https://arxiv.org/abs/2203.04237.
+[XZ06]
+T. Xue and Y. Zhang. “Bihamiltonian systems of hydrodynamic type and reciprocal
+transformations”. In: Lett. Math. Phys. 75.1 (2006), pp. 79–92. url: https://doi.org/10.1007/s11005-005-0031-9.
+(P. Lorenzoni) Department of Mathematics and Applications, University of Milano Bicocca,
+Via Roberto Cozzi 55, 20125 Milano, Italy and INFN sezione di Milano-Bicocca
+Email address: paolo.lorenzoni@unimib.it
+(S. Shadrin) Korteweg–de Vries Institut for Mathematics, University of Amsterdam, Postbus
+94248, 1090 GE Amsterdam, The Netherlands
+Email address: s.shadrin@uva.nl
+(R. Vitolo) Department of Mathematics and Physics “E. De Giorgi”, University of Salento,
+via per Arnesano, 73100 Lecce, Italy and INFN sezione di Lecce
+Email address: raffaele.vitolo@unisalento.it
+

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+Occupant-Oriented Demand Response
+with Room-Individual Building Control
+Moritz Frahma, Thomas Dengizb, Philipp Zwickela, Heiko Maaßa, J¨org Matthesa, Veit Hagenmeyera
+aKarlsruhe Institute of Technology, Institute for Automation and Applied Informatics, Eggenstein-Leopoldshafen, Germany
+bKarlsruhe Institute of Technology, Institute for Industrial Production, Karlsruhe, Germany
+Abstract
+In future energy systems with high shares of renewable energy sources, the electricity demand of buildings has to react to the
+fluctuating electricity generation in view of stability. As buildings consume one-third of global energy and almost half of this
+energy accounts for Heating, Ventilation, and Air Conditioning (HVAC) systems, HVAC are suitable for shifting their electricity
+consumption in time. To this end, intelligent control strategies are necessary as the conventional control of HVAC is not optimized
+for the actual demand of occupants and the current situation in the electricity grid. In this paper, we present the novel multi-zone
+controller Price Storage Control (PSC) that not only considers room-individual Occupants’ Thermal Satisfaction (OTS), but also the
+available energy storage, and energy prices. The main feature of PSC is that it does not need a building model or forecasts of future
+demands to derive the control actions for multiple rooms in a building. For comparison, we use an ideal, error-free Model Predictive
+Control (MPC) and a conventional hysteresis-based two-point control as upper and lower benchmarks, respectively. We evaluate
+the three controllers in a multi-zone environment for cooling a building in summer and consider two different scenarios that differ
+in how much the permitted temperatures vary. The results show that PSC strongly outperforms the conventional control approach in
+both scenarios with regard to the electricity costs and OTS. It leads to 50 % costs reduction and 15 % comfort improvements while
+the ideal MPC achieves costs reductions of 58 % and comfort improvements of 29 %. Considering that PSC does not need any
+building model or forecast, as opposed to MPC, the results support the suitability of our developed control strategy for controlling
+HVAC systems in future energy systems.
+Keywords: multi-zone, thermal building model, RC model, model predictive control, price storage control, rule-based control,
+occupant behavior, demand response, smart grid
+1. Introduction
+Buildings consume one-third of global final energy [1]. Al-
+most half of this energy is used by Heating, Ventilation, and
+Air Conditioning (HVAC) systems to heat or cool buildings [2].
+Especially the cooling demand is expected to increase signifi-
+cantly in many parts of the world, as the climate warms on av-
+erage [3]. In buildings, the energy consumption results from
+Occupant Behavior (OB) and Occupants’ Thermal Satisfaction
+(OTS) as they interact with the building’s energy systems and
+require comfortable thermal conditions [4]. The energy demand
+of buildings can be covered with renewable energies in order to
+reduce greenhouse gas emissions [5].
+Flexible electrical loads are pivotal for future energy systems
+in view of stability to cope with the increasing share of intermit-
+tent renewable energy sources like solar and wind energy. For
+exploiting flexible electric loads in buildings, the HVAC oper-
+ation can be integrated into Demand Response (DR) programs.
+DR refers to the change of electricity demand in response to in-
+ternal or external factors like the price of electricity [6]. In the
+building sector, electrical HVAC systems, like heat pumps or
+air conditioners, are suitable for DR. They can exploit existing
+infrastructure like the building mass or hot water tanks to shift
+their electricity demand in time [7]. Thus, they can significantly
+contribute to better utilization of renewable energy sources and
+simultaneously help to stabilize the electricity grid. In order
+to use HVAC systems for DR, optimized control strategies are
+necessary.
+In addition to DR, designing the HVAC operation tailored to
+the actual occupants’ needs could significantly reduce energy
+use. For example in office spaces, often, not all rooms are oc-
+cupied. The average occupancy rates of offices are rarely over
+60 % [8]. However, the HVAC control in offices usually does
+not consider the actual occupancy of offices. This leads to un-
+necessary energy use in unoccupied periods. 56 % of the en-
+ergy consumed by buildings is used during unoccupied hours
+and 44 % in occupied hours [9].
+For the optimization of HVAC to consider DR and individual
+OTS, advanced control strategies are required instead of stan-
+dard thermostats [10], for example Model Predictive Control
+(MPC) [11] or heuristic control strategies [12]. MPC finds the
+optimal input trajectory for the HVAC system’s control outputs
+over a future time horizon by solving an optimization problem
+under consideration of future system dynamics, forecasts, and
+constraints. Therefore, it requires a dynamic thermal building
+model and forecasts of OB and weather [13]. The development
+of models and forecasts can make MPC less practicable and
+more expensive for real-world applications [11].
+Preprint submitted to Applied Energy
+January 10, 2023
+arXiv:2301.03376v1  [eess.SY]  9 Jan 2023
+
+Nomenclature
+Acronyms
+BEV
+Battery Electric Vehicle
+DR
+Demand Response
+FMU
+Functional Mock-up Unit
+HVAC Heating, Ventilation, and Air Conditioning
+KPIs
+Key Performance Indicators
+MPC
+Model Predictive Control
+OB
+Occupant Behavior
+OTS
+Occupants’ Thermal Satisfaction
+PI
+Proportional Integral
+PMV
+Predicted Mean Vote
+PPD
+Predicted Percentage of Dissatisfied
+PSC
+Price Storage Control
+PV
+Photovoltaic
+RBC
+Rule-based Control
+RC
+Resistor Capacitor
+Parameters
+χdis
+discomfort factor
+χmod
+heat pump modulation degree
+χp
+price factor
+χsj
+storage factor
+∆t
+time step in s
+εh
+coefficient of performance of heat pump
+ξ j
+allowed deviation buffer in K
+Ci j
+heat capacity of room air in J K−1
+Cm j
+heat capacity of heat accumulating medium in J K−1
+gsj
+solar heat gain factor in m2
+Pmax
+max. electrical power of heat pump in W
+Ra j
+resistance between Ti j and Ta in K W−1
+Ri j
+resistance between Ti j and Tm j in K W−1
+S j
+state of thermal charge
+ymaxj
+maximal comfort temperature in °C
+yminj
+minimal comfort temperature in °C
+yrj
+reference comfort temperature in °C
+Variables
+˙Qhj
+heat flow of heat pump in W
+˙qs
+solar radiation in W m−2
+�F
+empirical distribution function
+Pel
+electrical power of heat pump in W
+t
+time in s
+Ta
+ambient temperature in °C
+Ti j
+room air temperature in °C
+Tm j
+heat accumulating medium temperature in °C
+In contrast, heuristic control strategies are model- and
+forecast-free heuristic algorithms. They iteratively adjust the
+power consumption of HVAC systems in order to archive cer-
+tain goals. In order to do this, they use rule-based control mech-
+anisms and heuristic algorithms that can adapt the HVAC sys-
+tem’s heat flows to internal and external signals. Their core
+advantage is that they do not require a building model to solve
+an optimization problem [12]. Thus, they are applicable to any
+building without significant adjustments.
+1.1. Related Work
+Different control approaches are available in the literature for
+controlling HVAC systems. Tab. 1 compares the most relevant
+studies for the present paper. The most significant difference
+between control strategies is whether they require a model to
+operate or not.
+Most studies in the literature use a model-based approach as
+they can find the optimal solution of an optimization problem
+[7]. Especially MPC is popular in the field of DR. Most authors
+use MPC for controlling HVAC systems, e.g. Maddalena et al.
+[14], Hu et al. [15], Pedersen et al. [16], Blum et al. [17],
+Mork et al. [20], and Zwickel et al. [22]. While model-based
+approaches generally yield adequate results, they suffer from
+execution times and require modeling the thermal behavior of a
+building which is a complex task.
+Fewer studies use model-free control strategies. Compared
+to model-based strategies, the controller design process is sig-
+nificantly simplified, as no building-specific model is required.
+Model-free control algorithms can be found in the studies of
+Dengiz et al. [12], Rodriguez et al. [18], Nolting et al. [19],
+and Michailidis et al. [21]. These approaches are rule-based
+control mechanisms that are in few cases also combined with a
+heuristic approach for optimizing an objective function.
+In all studies, the objective is to reduce the energy costs while
+satisfying OTS. Blum et al. [17] additionally consider the pro-
+vision of ancillary services. Another essential requirement for
+most of the optimized control approaches is the availability of
+forecasts. However, most of the model-free approaches do not
+rely on any forecast.
+Our literature review emphasizes the use of control algo-
+rithms for multiple zones (see Tab. 1). There are also control
+approaches in the literature that consider only buildings with
+one thermal zone (one uniform temperature in the whole build-
+ing). However, the consideration of multiple zones is closer
+2
+
+Table 1: Comparison of relevant papers studying approaches for demand response of HVAC systems
+Literature
+Model-
+free
+control
+Forecast-
+free
+control
+Multi-zone
+control
+Coupling
+of multiple
+buildings
+possible
+Comparison
+with lower
+benchmark
+Comparison
+with upper
+benchmark
+Use
+of
+measured
+data
+Maddalena et al., 2022 [14]
+
+
+
+
+
+
+
+Hu et al., 2014 [15]
+
+
+
+
+
+()
+
+Pedersen et al., 2018 [16]
+
+
+
+()
+
+
+
+Blum et al., 2016 [17]
+
+
+
+
+
+
+
+Dengiz et al., 2019 [12]
+
+
+
+
+
+
+
+Rodr´ıguez et al., 2018 [18]
+
+
+
+
+
+
+
+Nolting et al., 2019 [19]
+
+
+
+
+
+
+
+Mork et al., 2022 [20]
+
+
+
+
+
+
+
+Michailidis et al., 2018 [21]
+
+
+
+()
+
+
+
+Zwickel et al., 2022 [22]
+
+
+
+
+
+
+
+Present work
+
+
+
+()
+
+
+
+to the real thermal behavior of buildings and it also increases
+the complexity of the optimization problem. Another essential
+feature of control algorithms for DR is their capability of cou-
+pling multiple buildings in a coordinated way. While most of
+the listed studies use a central controller for this, Dengiz et al.
+[12] define a hybrid control architecture. Zwickel et al. [22]
+compare central and decentral control approaches for multiple
+buildings.
+To evaluate the performance of the developed control ap-
+proach, all studies, except for two, use a conventional control
+approach, like simple rule-based control, hysteresis-based two-
+point controller, or a Proportional Integral (PI) controller as a
+lower benchmark. The studies using MPC for controlling the
+heating or cooling device define their results also as an upper
+benchmark for the optimization problem, as usually a MPC ap-
+proach is solved by finding the global optimal solution. Most
+studies use simulated synthetic data for defining the building
+model and setting up the simulation. Only Maddalena et al.
+[14] and Michailidis et al. [21] also use measured data for eval-
+uating the OTS.
+1.2. Contribution of this Paper
+The main contribution of the present paper is the introduc-
+tion of a novel heuristic multi-zone control approach, called
+Price Storage Control (PSC). It combines external factors (e.g.
+electricity price) and internal factors (temperatures of different
+zones in the building) to determine when and how much elec-
+tricity should be consumed for the generation of heat flows. The
+approach is model-free and does not need any forecasts. To the
+best of our knowledge, our study is the only one that introduces
+a novel control approach for buildings with multiple zones that
+does not need any model or forecasts and that allows for a co-
+ordinated coupling of multiple buildings. This is because of its
+capability to use any external factor for deriving the HVAC con-
+trol output. Our study is the first that evaluates an introduced
+model-free and forecast-free control algorithm by using a lower
+and upper benchmark that are derived from the use of measured
+data (see Tab. 1).
+To evaluate the PSC control performance in terms of OTS
+and energy costs, we compare three different control strategies
+in a multi-zone thermal building model. In the evaluation, we
+use two scenarios with different degrees of variable room us-
+age. In the base scenario, the temperature range is scheduled
+between comfort and standby mode. The second scenario also
+allows room-individual temperature ranges, based on the use
+case for each room. For comparison, we use an ideal, error-free
+MPC and a hysteresis-based two-point controller as upper and
+lower benchmarks.
+1.3. Structure of this Paper
+We develop and implement three different control strategies
+and an evaluation environment in the present work. We present
+the models in Sec. 2, the controllers in Sec. 3, and evaluate them
+in Sec. 4. Finally, we conclude the evaluation results in Sec. 5.
+2. Models
+In this section, we present the models that we apply for
+the evaluation (see Sec. 4) of different control strategies (see
+Sec. 3). The model-based control strategy, the MPC, also uses
+the models internally to predict future system dynamics (see
+Sec. 3.2). The modeling section Sec. 2 is separated into three
+parts: the model for thermal dynamics of building in Sec. 2.1,
+for the heat pump in Sec. 2.2, and for OTS in Sec. 2.3.
+2.1. Multi-Zone Thermal Building Model
+In this section, we develop a multi-zone thermal building
+model to evaluate room-individual control strategies in Sec. 4.
+The model applies the Resistor Capacitor (RC) analogy to de-
+scribe the heat flows between temperature nodes by resistors R
+and thermal dynamics by capacitors C, as exemplarily shown
+in Eq. (1).
+3
+
+C dT(t)
+dt
+= ˙Qin(t) − ˙Qout(t)
+˙Qxy(t) = Tx(t) − Ty(t)
+R
+(1)
+We illustrate our thermal building model structure in Fig. 1.
+Applying this structure to each room j (j = 1 . . . n) results in a
+decentral multi-zone model [23]. Mathematically, each room is
+gsj ˙qs
+˙Qhj
+Ta
+Cij
+Ti j
+Cm j
+Ri j
+Ra j
+Tm j
+Ta
+Cm j
+Ci j
+Tm j
+Ti j
+Raj
+Ri j
+gsj ˙qs
+˙Qhj
+Figure 1: Thermal building model for each room j ( j = 1 . . . n), obtained from
+[23] (modified from [24])
+thermally defined by the two differential equations Eq. (2) and
+(3).
+Ci j
+dTi j
+dt
+=
+Tm j − Ti j
+Ri j
++
+Ta − Ti j
+Raj
++ gsj ˙qs + ˙Qhj,
+(2)
+Cm j
+dTm j
+dt
+=
+Ti j − Tm j
+Ri j
+.
+(3)
+2.2. Heat Pump Model
+The modeled air-source heat pump has a maximum electrical
+power Pmax and an energy efficiency ratio εh which are both de-
+pendent on the ambient temperature. We use the model AERO
+SLM 3-11 HGL from the Austrian heat pump manufacturer iDM
+Energiesysteme GmbH [25] with a supply temperature of the
+cooling system of 18◦C. To calculate the efficiency and the
+maximum cooling power at every time slot, we use the data
+from the manufacturer’s technical fact sheet and linear interpo-
+lation.
+The heat pump can modulate its power consumption Pel and
+thus the heat flow ˙Qh with χmod between 20 % and 100 %. This
+leads to the following relation between the heat pump’s electri-
+cal power Pel and the thermal building model’s heat pump heat
+flows:
+n
+�
+j=1
+��� ˙Qhj
+��� =
+��� ˙Qh
+��� = εh · Pel = εh · (χmod · Pmax),
+(4)
+χmod ∈ {0, [0.2, 1]},
+(5)
+Pel = χmod · Pmax
+(6)
+2.3. Occupants’ Thermal Satisfaction (OTS) Model
+In this section, we define the temperature ranges [ymin, ymax]
+based on international standards for Occupants’ Thermal Satis-
+faction (OTS) modeling. The three most frequently cited OTS
+standards are ASHRAE Standard 55 [26], ISO 7730:2005 [27],
+and EN 16798-1:2019 [28]. These standards are fundamentally
+based on the Predicted Mean Vote (PMV) standard scale, which
+was first introduced by Fanger’s model [29].
+The PMV is a static model evaluated from a large group
+of people with a given combination of thermal environmental
+and personal parameters. These parameters include metabolic
+activity, clothing, air temperature, radiant temperature, air ve-
+locity, and relative humidity. In a survey, occupants express
+their thermal sensations on a scale from -3 (too cold) to +3 (too
+warm), where 0 is optimum. Fanger also developed an equation
+that relates the PMV to the Predicted Percentage of Dissatisfied
+(PPD). The standard OTS guidelines aim for a PMV from -0.5
+to +0.5 (OTS level II, see Tab. 2). The OTS level can also be
+within closer or wider PMV boundaries, e.g. ±0.2 for level I
+or ±0.7 for level III. Wider temperature limits result in lower
+energy consumption of HVAC systems.
+Based on these OTS levels in Tab. 2, we calculate the corre-
+sponding lower ymin and upper ymax temperature limits that are
+required for Eq. (21). For the calculation of the temperature
+limits, we use the CBE Thermal Comfort Tool [30] with EN-
+16798 standard and summer clothing. In this tool, we set the
+mean radiant temperature equal to the air temperature Ti. This
+implies the assumption that the operative temperature is close to
+the air temperature. For more information about the operative
+temperature, we refer to our previous work [31]. The resulting
+temperature limits for different levels of OTS are presented in
+Tab. 2.
+Table 2: OTS categories, obtained from CBE Thermal Comfort Tool [30] with
+EN-16798 and sommer clothings
+OTS level
+PMV
+PPD
+ymin in °C
+ymax in °C
+I
+±0.2
+< 6 %
+25.6
+26.6
+II
+±0.5
+< 10 %
+24.8
+27.4
+III
+±0.7
+< 15 %
+24.2
+27.9
+Based on the temperature limits, we calculate the reference
+comfort temperature yr j in Eq. (7). This reference temperature
+is required for the controller design of the PSC in Sec. 3.1.
+yr j =
+ymin j + ymax j
+2
+(7)
+3. Control Strategies
+This section describes the development of three different con-
+trol strategies: PSC in Sec. 3.1, MPC in Sec. 3.2, and hysteresis-
+based two point control in Sec. 3.3. The objective is to mini-
+mize the electricity costs given by a time-variable electricity
+price and to maximize the OTS. While we develop the PSC
+as a novel control methodology for occupant-oriented demand-
+response with room-individual building control, the MPC and
+hysteresis-based two-point controller are used as upper and
+lower benchmarks, respectively. The MPC was implemented
+in Python with a prediction horizon of 16 hours and solved us-
+ing Gurobi. Also, PSC and the two-point controller are imple-
+mented in Python.
+4
+
+In general, the three control strategies are applicable to cool-
+ing or heating. For both cases, we use the generic term heat
+flows. A heat flow is the rate of net heat energy transfer be-
+tween hot and cold sides and can be positive or negative for
+heating or cooling, respectively.
+3.1. Price Storage Control (PSC)
+The PSC is a heuristic control algorithm for modulating
+HVAC or heat pump heat flows ˙Qhj in a multi-zone building.
+It essentially consists of 4 steps which it executes in every time
+slot.
+1. Determine the price factor χp(t) based on [12].
+2. Determine the storage factor χs(t).
+3. Calculate the modulation degree χmod using the price fac-
+tor χp(t) and the storage factor χs(t).
+4. Distribute the generated heat flow to the different rooms of
+the multi-zone building.
+Price Factor
+To obtain the price factor χp, the algorithm calculates the em-
+pirical distribution function �F(p) for the future electricity prices
+p(t) of the next 24 hours at the beginning of each day. We as-
+sume that we have an electricity tariff with predetermined prices
+for the next 24 hours (for more information see Section 4.1.1).
+At every time slot of the day, the value of the �F(p) is calculated
+for the current price p(t). The calculation of the empirical dis-
+tribution function �F(p) is illustrated in Fig. 2, exemplarily for
+one day. �F(p) quantifies the share of electricity prices for the
+current day that have a lower or equal value compared to the
+price p of the current time slot. PSC sets the price factor at time
+slot t as in Eq. (8). A low price results in a high price factor
+(due to a high value of �F(p)) and vice versa.
+χp(t) = 1 − �F(p(t))
+(8)
+Storage Factor
+For the calculation of the storage factor χs(t), the state of
+thermal charge S j(t) from Eq. (9) is needed for each room. The
+state of thermal charge S j(t) quantifies the “stored” tempera-
+ture room individually and results in values between 0 and 1.
+Although the PSC method is applicable for heating or cooling
+heat flows, we explain this method exemplarily for the cooling
+case in the following.
+S j(t) =
+yr j + ξ j − Ti j(t − ∆t)
+ξj
+(9)
+If the temperature of the room j from the last time slot
+Ti j(t − ∆t) is lower than the reference temperature yr j the state
+of thermal charge S j(t) is set to 1. This means that the thermal
+storage of this room is full and there is no necessity for applying
+heat flows to the room 1.
+1As we are considering cooling in the present work it has to be noted that
+full thermal storage, in this case, means, that the temperature in the room is low
+and thus the room already has enough ”cooling energy”.
+If the temperature of the room is higher than the reference
+comfort temperature yr j plus an allowed deviation buffer ξ j, for
+sufficiently high OTS, the state of thermal charge S j(t) is set to
+0. In the cooling case, this results in empty thermal storage as
+the temperature in the room is too high.
+For every room temperature that is between the reference
+temperature and the upper OTS limit (yr j + ξj), the algorithm
+uses Eq. (9) to calculate the state of thermal charge S j(t) of
+room j ( j = 1 . . . n). The reference temperature for every room
+yrj is calculated as in Eq. (7). This value depends on the inves-
+tigated scenarios (see Sec. 4) 2.
+After having determined the state of thermal charge S j(t) for
+every room n, the algorithm calculates the storage factor χs(t)
+by using Eq. (10). If the temperatures in the different rooms
+are close to the lower limit, their corresponding state of thermal
+charge will be high resulting in a low storage factor χs(t) and
+vice versa.
+χs(t) = 1 −
+�n
+j=1 S j(t)
+n
+(10)
+Modulation Degree of the HVAC system
+The third step of the algorithm is the calculation of the heat
+pump’s modulation degree and thus the heat flow and the elec-
+trical power using Eq. (11). The modulation degree χmod(t) re-
+sults from the multiplication of the price factor χp and storage
+factor χs. Because both factors can have values between 0 and
+1, the modulation degree χmod(t) likewise varies between 0 and
+1. We choose a multiplication of the two factors instead of a
+weighted sum as this leads to better results in our case studies.
+Based on the modulation degree, Eq. (4)) and Eq. (5) calculates
+the generated heat flows and electrical power.
+χmod(t) = χp(t) · χs(t)
+(11)
+Two factors influence the heat pump power output. A high
+electricity price leads to a low price factor which leads to low
+values of the modulation degree. This results in low electricity
+consumption at that time. On the contrary, a low price leads to a
+high price factor which incentives the heat pump to cool down
+the room. This is desired as we want to generate heat flows
+when the electricity prices are low.
+Next to the price factor, the storage factor impacts the gen-
+erated heat flows and thus consumed electricity. If the temper-
+atures in the rooms are generally low, the storage factor has
+low values due to the high values of the state of thermal charge
+S j(t). A low storage factor leads to low power consumption
+and vice versa. This is also a desired property of the control
+algorithm. If the room temperatures are already low, there is no
+urgent need for cooling whereas high room temperatures tend to
+lead to higher generation of heat flow using the PSC algorithm.
+Distribution of Heat Flows
+In the final step, the algorithm distributes the generated heat
+flows to the different rooms j ( j = 1 . . . n). To do this, the
+2For this internal parameter of the algorithm, a buffer value of ξ j = 2 K
+yields adequate results in the present work.
+5
+
+p(t)
+�F(p)
+Figure 2: Empirical distribution function of the electricity prices
+caused thermal discomfort of each room dc j(t) due to possibly
+too high temperatures is determined. If the temperature of a
+room from the previous time slot Ti j(t − ∆t) is higher than the
+upper temperature limit ymaxj, Eq. (12) and Eq. (14) quantify the
+caused discomfort of the room j and the total caused discomfort
+dc,total(t) from Eq. (13).
+dc j(t) = Ti j(t − ∆t) − ymaxj
+(12)
+dc,total(t) =
+n
+�
+j=1
+dc j(t)
+(13)
+Based on the total caused discomfort dc,total(t) the PSC algo-
+rithm distributes the generated heat flows ˙Qh of time t to each
+room j with ˙Qhj using Eq. (14). This mechanism assures that
+especially rooms that have high temperatures, get more heat
+flow (cooling) than rooms with less need for cooling. If the
+heat pump generates heat flows although no room has violated
+its temperature boundaries in the last time slot, it equally dis-
+tributes the generated heat flows to every room.
+˙Qhj(t) =
+dcj(t)
+�n
+j=1 dc j(t) · ˙Qh(t)
+(14)
+Overall, PSC executes the four mentioned steps for every
+time slot of the day while updating the empirical distribution
+function of the prices at the beginning of each day.
+3.2. Model Predictive Control (MPC)
+In contrast to PSC, MPC requires a model which is obtained
+by restructuring the thermal building model from Sec.2 into
+state-space notation. The resulting equations for n rooms are
+˙x(t) = f (x(t), u(t), z(t)) = Ax(t) + Bu(t) + Ez(t),
+y(t) = g(x(t)) = Cx(t)
+(15)
+where x describes the temperature states of the buildings’ air
+temperatures (Ti j) and thermal masses (Tm j), u the control in-
+puts ( ˙Qhj), z the measurable disturbances (˙qs, Ta), and y the
+outputs (Ti j) with
+x = �Ti1 Tm1 Ti2 Tm2 . . . Tin Tmn
+�T ,
+u =
+� ˙Qh1 ˙Qh2 . . . ˙Qhn
+�T ,
+z = (˙qs Ta)T ,
+y = �Ti1 Ti2 . . . Tin
+�T .
+(16)
+Given the heat pump model (see Eq. (4) and Eq. (5)) and the
+control model (Eq. (15)), we formulate the optimization prob-
+lem with a prediction horizon of N time steps, which must be
+solved at each sampling instant t, based on [22] as
+min
+t+N−1
+�
+k=t
+l
+�
+k, χdis(k|t), Pel(k|t)
+�
+(17a)
+subject to ∀k ∈ [0, N − 1] :
+x(k+1|t) = Adx(k|t) + Bdu(k|t) + Edz(k|t)
+(17b)
+y(k|t) = Cdx(k|t)
+(17c)
+x(0|t) = x(t),
+(17d)
+Pel(k|t) = χmod(k|t) · Pmax(k),
+(17e)
+χmod(k|t) ∈ {0, [0.2, 1]},
+(17f)
+u(k|t) ∈ U, y(k|t) ∈ Y
+(17g)
+where l (k, ·, ·) is the stage-cost, (17b) and (17c) are the
+discrete-time control model, (17d) is the initial condition, and
+(17e) and (17f) are the heat pump model. x(t) is typically mea-
+sured at the time t, and U as well as Y are input and output
+constraint sets (see (17g)).
+The k-step ahead prediction for
+the states, inputs, disturbances, outputs, discomfort factor, heat
+pump modulation degree, and electric power, based on the cur-
+rent initial condition are denoted by x(k|t), u(k|t), z(k|t), y(k|t),
+χdis(k|t), χmod(k|t), and Pel(k|t), respectively.
+We consider the following stage cost
+l
+�
+k, χdis(k), Pel(k)
+�
+= λ
+�
+n
+�
+j=1
+χdisj(k)
+�
++ (1-λ)
+�
+P′
+el(k)p′(k)
+�
+(18)
+where λ ∈ [0, 1] is a user-defined weighting coefficient and
+p(k), k ∈ t : t + N − 1 is a time-dependent price signal (future
+6
+
+electricity prices). P′
+el(k|t) and p′(k) are the min-max normal-
+ization of Pel(k|t) and p(k) 3, respectively, calculated as
+P′
+el(k) =
+Pel(k) − min{Pel}
+max{Pel} − min{Pel} ,
+p′(k) =
+p(k) − min{p}
+max{p} − min{p} .
+(19)
+Furthermore, the control inputs u are limited to cooling with
+the maximum total power constraint by the heat pump model,
+as formulated in Eq. (4), which leads to
+U =
+�
+u ∈ Rn | (∀ j ∈ [1, n] : u j ≤ 0) ∧
+n
+�
+j=1
+|u| = εh · Pel
+�
+. (20)
+In addition, the control outputs should meet predefined time-
+variant temperature ranges [ymin j, ymaxj] for each room j. This
+leads to the following soft constraints:
+Y =
+�
+y ∈ Rn | (∀ j ∈ [1, n] : ymin j-χdis j ≤ yj ≤ ymaxj+χdis j)
+∧ χdis j ≥ 0
+�
+.
+(21)
+3.3. Hysteresis-based Two-point Controller
+The hysteresis-based two-point control serves as the lower
+benchmark for the evaluation. This is a conventional control
+strategy for cooling (or heating) devices that cools down a room
+until a lower temperature limit. Afterward, the device switches
+off and waits until the temperature in the room has reached an
+upper limit. This triggers the control system to start cooling
+down again. We use an adaptive hysteresis that uses the up-
+per and lower temperature limits [yminj(t), ymaxj(t)] depending
+on the scenarios. These predefined temperature limits for OTS
+are described in the evaluation scenarios in Sec. 4.1.2.
+4. Evaluation
+In this section, we compare the control algorithms and de-
+scribe the used evaluation environment. First, we introduce the
+used data, scenarios, and metrics in Sec. 4.1. Then, we present
+the results in Sec. 4.2, discuss them in Sec. 4.3, and show limi-
+tations in Sec. 4.4
+4.1. Data, Scenarios, and Metrics
+4.1.1. Data
+We evaluate the control strategies by using weather data dur-
+ing the summer of 2022, obtained from a weather station on
+an experimental building [32]. This building is located in the
+KIT EnergyLab 2.0 (Karlsruhe, Germany), and is presented in
+Fig. 3a. It has a design similar to a single-family home and is
+used as an office space. For the evaluation, we use the measure-
+ments of the weather station (the solar radiation ˙qs and the am-
+bient temperature Ta) over a period of 13 weeks (05/30/2022 –
+3In the present study, we apply min{Pel} = 0 W, max{Pel} = 3500 W,
+min{p} = 6.9 Cent/kW h, and max{p} = 58.1 Cent/kW h.
+08/22/2022). Due to measurement gaps, we had to sort out three
+weeks (8th, 10th, and 13th week) and could only use the remain-
+ing ten weeks. We use the ten weeks of data in time steps of
+∆t = 15 min. For the variable electricity tariff, we use the data
+of the day-ahead market in Germany from the ENTSO-E Trans-
+parency Platform [33]. The price is different for every hour of
+the day and we assume that these prices are directly forwarded
+to the customers.
+In Sec. 4.2, we evaluate the control algorithms from Sec. 3
+on an evaluation environment that is illustrated in Fig. 3. While
+the evaluation environment is inspired by the building design in
+Fig. 3a, the floor plan in Fig. 3b is generated artificially. The
+floor plan illustrates the different temperature demands of the
+rooms in Tab. 3 and 4. In the present work, we explicitly focus
+on the evaluation of control strategies instead of model parame-
+ter estimation from measurements. For more information about
+parameter identification, we refer to our previous work [23].
+For the results in Sec 4.2, the model parameters of the model
+in Sec. 2 are based on parameters from the literature [24] and
+scaled to consider different room sizes. The used model and pa-
+rameters can be obtained from GitHub (as a Modelica file .mo
+and as a Functional Mock-up Unit (FMU) .fmu) 4.
+4.1.2. Scenarios
+For the evaluation of the control algorithms, we consider two
+scenarios, namely (a) base scenario and (b) multi-zone adap-
+tive scenario. The scenarios differ by their variability of tem-
+perature ranges [ymin j(t), ymaxj(t)]. The base scenario applies the
+same ranges for all rooms, while the second allows individual
+occupancy profiles. The temperature ranges are presented in
+Tab. 3 and 4 and applied for each day. Despite the office sce-
+nario, we do not differentiate between the different days, e.g.
+between weekdays and weekends, to simplify the evaluation.
+(a) Base scenario: We use two different control modes in the
+base scenario (see Tab. 3): a comfort mode and a standby mode
+(inspired by Peng et al. [34]). We apply the comfort mode
+during working hours from 8AM to 5PM and the standby mode
+else.
+Table 3: (a) Base scenario, room air temperatures in °C
+Period
+Room
+1
+2
+3
+4
+5
+8AM to 5PM
+ymin j
+24.8
+24.8
+24.8
+24.8
+24.8
+ymaxj
+27.4
+27.4
+27.4
+27.4
+27.4
+else
+ymin j
+16.0
+16.0
+16.0
+16.0
+16.0
+ymaxj
+30.0
+30.0
+30.0
+30.0
+30.0
+All rooms j ( j = 1 . . . 5) apply the same modes during the
+entire evaluation in the base scenario. As a result, the temper-
+ature ranges [yminj(t), ymaxj(t)] in Tab. 3 are all equal over the
+different rooms. In contrast, the second scenario uses different
+modes in different rooms, depending on the use case of each
+room.
+4https://github.com/Occupant-Oriented-Demand-Response/Conrol-Results
+7
+
+(a) Experimental building with a design similar to a single-family home
+Bath-
+room
+Hallway
+Technical
+room
+Kitchen
+Printer and storage
+Second office
+Main office
+Hallway
+3
+4
+5
+Ground floor
+Upper floor
+1
+2
+(b) Floor plan
+Figure 3: Evaluation environment
+(b) Multi-zone adaptive scenario: The temperature ranges
+[yminj(t), ymaxj(t)] in all rooms j ( j = 1 . . . 5) can be different
+(see Tab. 4). In addition to the comfort and standby mode,
+we also use an eco mode that schedules the reference temper-
+ature by 2 K (+2 K/-2 K for cooling/heating) difference com-
+pared to the comfort mode [34]. This eco mode saves energy
+compared to the comfort mode and also enables fast re-cooling
+/ re-heating compared to the standby mode. In addition, the eco
+mode can save energy in rooms that are less frequently used
+than office rooms, e.g. bathrooms or kitchens.
+Table 4: (b) Multi-zone adaptive scenario, room air temperatures in °C
+Period
+Room
+1
+2
+3
+4
+5
+8AM to 12AM
+ymin j
+24.8
+24.8
+22.8
+22.8
+22.8
+ymaxj
+27.4
+27.4
+29.4
+29.4
+29.4
+12AM to 1PM
+ymin j
+22.8
+22.8
+22.8
+24.8
+22.8
+ymaxj
+29.4
+29.4
+29.4
+27.4
+29.4
+1PM to 5PM
+ymin j
+24.8
+22.8
+22.8
+22.8
+22.8
+ymaxj
+27.4
+29.4
+29.4
+29.4
+29.4
+else
+ymin j
+16.0
+16.0
+16.0
+16.0
+16.0
+ymaxj
+30.0
+30.0
+30.0
+30.0
+30.0
+In the multi-zone adaptive scenario (see Tab. 4), we let the
+control operate with a high focus on OTS in occupied rooms
+and energy saving in unoccupied. Therefore, we use the com-
+fort mode in the offices (rooms 1 and 2) during working hours
+and the kitchen (room 4) during lunch breaks from 12AM to
+1PM. In this scenario, the first office (room 1) is used over the
+entire working day, except lunch break, and the second office
+(room 2) only from 8AM to 12AM (part-time job). The bath-
+room (room 5) and storage (room 3) should be operated in eco
+mode during working hours (8AM to 5PM).
+4.1.3. Metrics
+We use two Key Performance Indicators (KPIs) to evalu-
+ate (i) how accurately a controller meets the desire OTS and
+(ii) how much energy the control strategy therefore consumes.
+Mathematically we define the KPIs as the weekly costs cm,week
+in Eq. (22) and mean weekly discomfort dm, week in Eq. (23),
+cm,week =
+M
+�
+k=1
+�
+p(k)
+�
+k
+Pel(k) dtk
+�
+(22)
+dm, week = 1
+M
+��������
+M
+�
+k=1
+n
+�
+j=1
+dcj(k)
+�������� .
+(23)
+The KPIs consider energy costs and OTS during each time-
+step k for all time steps M = 672 of each week. The energy
+costs cm,week depend on a dynamic energy tariff p(k) and the
+consumed electric power Pel(k). The discomfort dm, week evalu-
+ates the discomfort dc j(k) of the actual room temperature from
+the allowed OTS range. This permitted temperature range is
+time-variant, depending on room-individual usage/attendance
+profiles, as introduced in the scenarios in Sec. 4.1.2.
+Both KPIs are competing, which means when one is im-
+proved, the other is usually deteriorating. It is the objective
+to minimize both KPIs simultaneously, to have low costs and
+low discomfort.
+4.2. Results
+We present the results of the three different control algo-
+rithms, MPC (ideal and error-free) , PSC, and hysteresis-based
+two-point controller (see Sec. 3), in Fig. 4 and A.5. The overall
+results for both scenarios over the entire evaluation period of
+ten weeks can be obtained from Fig. 4. Fig. A.5 illustrates the
+dynamic response of the thermal building model to the three ap-
+plied control strategies, exemplarily for the base scenario dur-
+ing one week.
+Control Results for One Week
+Fig. A.5 illustrates the dynamic behavior of the three con-
+trollers, MPC, PSC, and hysteresis-based two-point controller,
+on the multi-zone thermal building model. The x-axis uses the
+time in days for one week in June/July 2022. On the y-axis, we
+present the air temperatures Ti j in °C for five rooms (j = 1 . . . 5)
+and three control strategies. The blue area shows the permitted
+8
+
+temperature ranges for the air temperatures [yminj(t), ymaxj(t)].
+The bottom y-axes present the controlled variable Pel, the dis-
+turbance variables Ta and ˙qs, and the dynamic electric price
+function p.
+The control characteristics of the three controllers are distin-
+guishable, although, for all controllers, the controlled variable
+Pel of the heat pump operates only in three of seven days no-
+ticeably (06-29, 06-30, and 07-03). In general, the temperature
+trajectories controlled by MPC and PSC are more similar than
+those controlled by the hysteresis-based two-point controller.
+The MPC cools most frequently but at a lower power Pel. In a
+few cases, the MPC exceeds the upper temperature limits, e.g.
+on the 06-30. Therefore, the MPC meets temperature ranges
+more adequately on the next day (07-01), where the tempera-
+tures of the PSC and hysteresis-based two-point controller are
+too low.
+Overall Results for Ten Weeks
+We perform evaluations for the three controllers in two sce-
+narios over ten different weeks and summarize the results in
+Fig. 4. On the y-axis in Fig. 4, we visualize the two KPIs, the
+mean weekly costs (”costs”) and the mean discomfort (”discom-
+fort”) from Eq. (22) and (23). The results are shown for the two
+scenarios, the (a) base scenario and the (b) multi-zone adaptive
+scenario, where (a) and (b) are based on the temperature ranges
+in Tab. 3 and 4, respectively.
+When evaluating the three control strategies in Fig. 4, the
+MPC and PSC show superior results in terms of costs and dis-
+comfort, compared to the hysteresis-based two-point controller.
+In both scenarios, the MPC and PSC have lower discomfort and
+approximately half the costs of the hysteresis-based two-point
+controller (e.g. in (a) from 2.18 to 1.08 or 1.18).
+The performance of the MPC depends more on the evaluated
+scenario (a) vs. (b) than for the other two controllers. In the
+base scenario, The MPC and PSC have a similar overall perfor-
+mance (1.18 vs. 1.08 costs and 0.52 vs. 0.59 discomfort). In
+contrast, in the multi-zone adaptive scenario, the MPC outper-
+forms the PSC with 38.5 % lower costs (from 1.09 to 0.67) and
+also lower discomfort (from 0.19 to 0.13).
+In summary, we obtain the highest overall performance re-
+garding costs and discomfort with the MPC and PSC, while the
+hysteresis-based two-point controller shows the lowest perfor-
+mance. The performance difference between MPC and PSC
+varies depending on the evaluation scenario. In the (a) base
+scenario, the MPC and PSC have similar control results, while
+in the (b) multi-zone adaptive scenario, the MPC outperforms
+the PSC with 38.5 % lower costs and also lower discomfort.
+4.3. Discussion
+The results in Sec. 4.2 evaluate the performance of the
+PSC by comparison with the upper and lower benchmarks us-
+ing ideal, error-free MPC and hysteresis-based two-point con-
+troller, respectively. Overall, the control performance of the
+PSC is significantly superior to the hysteresis-based two-point
+controller and close to the ideal MPC. In the following, we dis-
+cuss the differences in control performance.
+The three controllers differ in their complexity and how much
+knowledge about future system behavior they require.
+The
+hysteresis-based two-point controller uses only minimal and
+maximal temperatures [ymin j(t), ymax j(t)] without any forecasts
+or models. When a maximal temperature is reached, it cools
+over a defined period. The PSC, on the other hand, tries to
+meet a reference temperature that is in the middle of the min-
+imal and maximal ranges. The PSC requires knowledge about
+the temperature ranges, but also about the energy tariff and the
+heat pump modulation. Exploiting this knowledge reduces the
+energy costs of the PSC compared to the hysteresis-based two-
+point controller because the PSC can apply cooling during pe-
+riods of low energy prices.
+The MPC uses the largest amount of available information,
+which increases its performance accordingly. It does not only
+use temperature ranges, energy tariffs, and heat pump modu-
+lation. In addition, the MPC needs a thermal building model
+and weather forecasts. With that internal control model and
+the forecasts, the MPC can predict future system behavior in
+advance and schedule the cooling load optimally. As a result,
+MPC outperforms the PSC when high variations in the temper-
+ature ranges [yminj(t), ymaxj(t)] occur, as in the (b) multi-zone
+adaptive scenario.
+The MPC exploits the knowledge about
+thermal storage inside the building, which enables finding an
+optimal cooling trajectory.
+The PSC is suitable for tracking a reference temperature
+when fewer variations in the temperature ranges occur. In the
+(a) base scenario, the PSC and MPC show similarly high per-
+formance. It should be noted that in the base scenario, a major
+part of the discomfort results from a too-cold temperature, in-
+stead of a too-warm one. None of the three controllers was
+allowed to heat the building. Especially in the morning periods,
+the low ambient temperatures make it challenging for the con-
+trollers to meet warm enough thermal conditions in the build-
+ing. The MPC can only mitigate temperatures that are too low
+by allowing temperatures that are too warm at other times (see
+Fig. A.5, 06-30 and 07-01). Overall, the MPC cannot show its
+advantage as an optimum controller to its best advantage in the
+base scenario.
+In summary, the controllers perform as expected where a
+higher complexity and use of more information improve the
+control quality.
+While the PSC outperforms the hysteresis-
+based two-point controller, the differences between PSC and
+ideal MPC are much smaller. On the one hand, the MPC has
+a superior performance in one of the two scenarios. On the
+other hand, the MPC is significantly more complex to design,
+requiring a thermal model for each room and a forecast, which
+we both assumed to be error-free for our case study. Compared
+to the conventional hysteresis-based two-point controller, PSC
+leads to a reduction of the two combined criteria costs and dis-
+comfort of 44 % (50 % costs reduction and 15 % comfort im-
+provements) while the MPC achieves combined improvements
+of 53 % (58 % cost reduction and 29 % comfort improvements).
+4.4. Limitations
+The evaluation of control strategies in this work is based
+on simulation results, which can neglect several effects from
+9
+
+MPC
+PSC
+Hysteresis
+0
+1
+2
+3
+1.18
+1.08
+2.18
+0.52
+0.59
+0.7
+Mean weekly costs [EUR]
+Mean discomfort [K]
+(a) Base scenario evaluation
+MPC
+PSC
+Hysteresis
+0
+1
+2
+3
+0.67
+1.09
+2.19
+0.13
+0.19
+0.22
+Mean weekly costs [EUR]
+Mean discomfort [K]
+(b) Multi-zone adaptive scenario evaluation
+Figure 4: Control results of three controllers, evaluated in two different scenarios (a) and (b)
+the real application. The control strategies are performed on a
+multi-zone thermal building model instead of a real building.
+The model parameters are based on literature values instead
+of identification from parameter identification. The model and
+weather forecasts of the MPC are assumed as error-free.
+The evaluation is limited to a cooling scenario of a single
+building. Weather data is used for ten weeks during summer in
+Karlsruhe, Germany. The cooling demand in Germany is lower
+than in other regions of the world. A heating scenario is not
+investigated. The evaluated scenarios consider no Photovoltaic
+(PV), battery, Battery Electric Vehicle (BEV), or thermal water
+storage in the optimization.
+5. Conclusion
+In this study, we investigate how a novel multi-zone Price
+Storage Control (PSC) can provide Demand Response (DR)
+while considering room-individual Occupants’ Thermal Sat-
+isfaction (OTS) without using a thermal building model and
+weather forecasts. Therefore, we develop three different con-
+trol strategies, a multi-zone evaluation environment, and two
+different scenarios to compare the controllers. We compare the
+PSC with an ideal, error-free Model Predictive Control (MPC)
+and hysteresis-based two-point controller as upper and lower
+benchmarks, respectively.
+The ideal MPC and PSC achieve higher control performance
+than the hysteresis-based two-point controller in terms of en-
+ergy costs and mean discomfort. The PSC leads to a reduction
+of both criteria combined of 44 % while the MPC achieves im-
+provements of 53 %. Under consideration that the PSC requires
+no models and no forecasts, this control strategy seems espe-
+cially beneficial for real-world control applications. Our devel-
+oped control approach is easy to implement and can be used for
+every building without large-scale adjustments. Further, it can
+include other external signals in its decision-making like the
+load of the electricity grid or a generation signal of renewable
+energy sources. Thus, it can contribute to balancing electricity
+demand and supply and lead to better utilization of renewable
+energy sources in future energy systems.
+In future work, we want to apply the developed control strat-
+egy to a real-world application. For the MPC real-world ap-
+plication, we need to perform parameter identification and de-
+sign a state estimator. For a more realistic scenario, we plan
+to include more relevant components into the optimization, e.g.
+thermal water storage, Photovoltaic (PV) self-production and
+-consumption, and batteries. Finally, we plan to evaluate the
+controller in a heating scenario.
+Data Availability
+We added the following supplementary materials to an open-
+source online repository on GitHub:
+• results of the three control strategies for all individual
+weeks in both scenarios,
+• used input data for the electricity price and weather data,
+• commented Python code of the three control strategies.
+https://github.com/Occupant-Oriented-Demand-
+Response/Conrol-Results.
+Acknowledgment
+This work was conducted within the project FlexK¨alte,
+funded by the German Federal Ministry for Economic Affairs
+and Climate Action (BMWK). The authors would like to thank
+their colleagues from the Energy Lab 2.0 and the Institute for
+Automation and Applied Informatics (IAI) for all the fruitful
+discussions and collaborations.
+Appendix A. Control Results for One Week
+The control results for one week, as discussed in Sec. 4.2, are
+presented in Fig. A.5.
+10
+
+References
+[1] IEA, Buildings - a source of enormous untapped efficiency potential, (ac-
+cessed: 15.12.2022) (2022).
+URL https://www.iea.org/topics/buildings
+[2] L. Yang, H. Yan, J. C. Lam, Thermal comfort and building energy con-
+sumption implications – a review, Applied Energy 115 (2014) 164–173.
+doi:https://doi.org/10.1016/j.apenergy.2013.10.062.
+[3] R. Mutschler, M. R¨udis¨uli, P. Heer, S. Eggimann, Benchmarking cool-
+ing and heating energy demands considering climate change, population
+growth and cooling device uptake, Applied Energy 288 (2021) 116636.
+doi:https://doi.org/10.1016/j.apenergy.2021.116636.
+[4] T. Hong, D. Yan, S. D’Oca, C. fei Chen, Ten questions concerning
+occupant behavior in buildings: The big picture, Building and Envi-
+ronment 114 (2017) 518–530.
+doi:https://doi.org/10.1016/j.
+buildenv.2016.12.006.
+[5] F. M. Vieira, P. S. Moura, A. T. de Almeida, Energy storage system for
+self-consumption of photovoltaic energy in residential zero energy build-
+ings, Renewable Energy 103 (2017) 308–320. doi:https://doi.org/
+10.1016/j.renene.2016.11.048.
+[6] M. H. Albadi, E. F. El-Saadany, Demand response in electricity markets:
+An overview, in: 2007 IEEE Power Engineering Society General Meet-
+ing, 2007, pp. 1–5. doi:10.1109/PES.2007.385728.
+[7] T. Dengiz, Optimization approaches for exploiting the load flexibility of
+electric heating devices in smart grids, Ph.D. thesis, Karlsruher Institut
+f¨ur Technologie (KIT) (2021). doi:10.5445/IR/1000131495.
+[8] A. Mahdavi, A. Mohammadi, E. Kabir, L. Lambeva, Occupants’ op-
+eration of lighting and shading systems in office buildings, Journal of
+Building Performance Simulation 1 (1) (2008) 57–65. doi:10.1080/
+19401490801906502.
+[9] O. Masoso, L. Grobler, The dark side of occupants’ behaviour on building
+energy use, Energy and Buildings 42 (2) (2010) 173–177. doi:https:
+//doi.org/10.1016/j.enbuild.2009.08.009.
+[10] P. H. Shaikh, N. B. M. Nor, P. Nallagownden, I. Elamvazuthi, T. Ibrahim,
+A review on optimized control systems for building energy and com-
+fort management of smart sustainable buildings, Renewable and Sustain-
+able Energy Reviews 34 (2014) 409–429. doi:https://doi.org/10.
+1016/j.rser.2014.03.027.
+[11] J. Drgoˇna, J. Arroyo, I. Cupeiro Figueroa, D. Blum, K. Arendt, D. Kim,
+E. P. Oll´e, J. Oravec, M. Wetter, D. L. Vrabie, L. Helsen, All you need
+to know about model predictive control for buildings, Annual Reviews in
+Control 50 (2020) 190–232. doi:10.1016/j.arcontrol.2020.09.
+001.
+[12] T. Dengiz, P. Jochem, W. Fichtner, Demand response with heuristic con-
+trol strategies for modulating heat pumps, Applied Energy 238 (2019)
+1346–1360.
+doi:https://doi.org/10.1016/j.apenergy.2018.
+12.008.
+[13] M. Frahm, P. Zwickel, J. Wachter, F. Langner, P. Strauch, J. Matthes,
+V. Hagenmeyer, Occupant-Oriented Economic Model Predictive Control
+for Demand Response in Buildings, in: 13th ACM International Confer-
+ence on Future Energy Systems, e-Energy’22, Association for Computing
+Machinery, 2022. doi:10.1145/3538637.3538864.
+[14] E. T. Maddalena, S. A. M¨uller, R. M. dos Santos, C. Salzmann, C. N.
+Jones, Experimental data-driven model predictive control of a hospital
+hvac system during regular use, Energy and Buildings 271 (2022) 112316.
+doi:https://doi.org/10.1016/j.enbuild.2022.112316.
+[15] J. Hu, P. Karava, A state-space modeling approach and multi-level op-
+timization algorithm for predictive control of multi-zone buildings with
+mixed-mode cooling, Building and Environment 80 (2014) 259–273.
+doi:https://doi.org/10.1016/j.buildenv.2014.05.003.
+[16] T. H. Pedersen, S. Petersen, Investigating the performance of scenario-
+based model predictive control of space heating in residential buildings,
+Journal of Building Performance Simulation 11 (4) (2018) 485–498.
+doi:10.1080/19401493.2017.1397196.
+[17] D. H. Blum, N. Xu, L. K. Norford, A novel multi-market optimization
+problem for commercial heating, ventilation, and air-conditioning sys-
+tems providing ancillary services using multi-zone inverse comprehen-
+sive room transfer functions, Science and Technology for the Built En-
+vironment 22 (6) (2016) 783–797.
+doi:10.1080/23744731.2016.
+1197718.
+[18] L. Romero Rodr´ıguez, J. S´anchez Ramos, S. ´Alvarez Dom´ınguez,
+U. Eicker, Contributions of heat pumps to demand response: A case
+study of a plus-energy dwelling, Applied Energy 214 (2018) 191–204.
+doi:https://doi.org/10.1016/j.apenergy.2018.01.086.
+[19] L. Nolting, A. Praktiknjo, Techno-economic analysis of flexible heat
+pump controls, Applied Energy 238 (2019) 1417–1433. doi:https:
+//doi.org/10.1016/j.apenergy.2019.01.177.
+[20] M. Mork, A. Xhonneux, D. M¨uller, Nonlinear distributed model predic-
+tive control for multi-zone building energy systems, Energy and Buildings
+264 (2022) 112066.
+doi:https://doi.org/10.1016/j.enbuild.
+2022.112066.
+[21] I. T. Michailidis, T. Schild, R. Sangi, P. Michailidis, C. Korkas, J. F¨utterer,
+D. M¨uller, E. B. Kosmatopoulos, Energy-efficient hvac management us-
+ing cooperative, self-trained, control agents: A real-life german build-
+ing case study, Applied Energy 211 (2018) 113–125.
+doi:https:
+//doi.org/10.1016/j.apenergy.2017.11.046.
+[22] P. Zwickel, M. Frahm, J. Galenzowski, K.-H. H¨afele, H. Maaß, S. Wac-
+zowicz, V. Hagenmeyer, Demand response in smart districts: Model pre-
+dictive control of building cooling, in: 2022 IEEE PES Innovative Smart
+Grid Technologies Europe (ISGT-Europe), 2022.
+[23] M. Frahm, S. Meisenbacher, E. Klumpp, R. Mikut, J. Matthes, V. Ha-
+genmeyer, Multi-Zone Grey-Box Thermal Building Identification with
+Real Occupants, in: 9th ACM International Conference on Systems for
+Energy-Efficient Buildings, Cities, and Transportation, BuildSys’22, As-
+sociation for Computing Machinery, 2022.
+doi:10.1145/3563357.
+3567403.
+[24] H. Madsen, J. Holst, Estimation of continuous-time models for the heat
+dynamics of a building, Energy and Buildings 22 (1) (1995) 67–79. doi:
+https://doi.org/10.1016/0378-7788(94)00904-X.
+[25] iDM Energiesysteme GmbH, Aero slm luftw¨armepumpe - idm en-
+ergiesysteme gmbh (2020).
+URL
+https://www.idm-energie.at/
+aero-slm-luftwaermepumpe/
+[26] ANSI/ASHRAE, Standard 55, Thermal Environmental Conditions for
+Human Occupancy (2017).
+[27] ISO, 7730 - Ergonomics of the thermal environment — Analytical deter-
+mination and interpretation of thermal comfort using calculation of the
+PMV and PPD indices and local thermal comfort criteria (2005).
+[28] CEN, EN 16798-1 - Energy performance of buildings - Ventilation for
+buildings. Part 1: Indoor environmental input parameters for design and
+assessment of energy performance of buildings addressing indoor air
+quality, thermal environment, lighting and acoustics (2019).
+[29] P. O. Fanger, Thermal Comfort Analysis and Applications in Environ-
+mental Engineering, Mcgraw-Hill, New York, 1970.
+[30] F. Tartarini, S. Schiavon, T. Cheung, T. Hoyt, Cbe thermal comfort
+tool: Online tool for thermal comfort calculations and visualizations,
+SoftwareX 12 (2020) 100563.
+doi:https://doi.org/10.1016/j.
+softx.2020.100563.
+[31] M. Frahm, F. Langner, P. Zwickel, J. Matthes, R. Mikut, V. Hagenmeyer,
+How to Derive and Implement a Minimalistic RC Model from Thermo-
+dynamics for the Control of Thermal Parameters for Assuring Thermal
+Comfort in Buildings, in: Open Source Modelling and Simulation of En-
+ergy Systems (OSMSES), IEEE, 2022. doi:10.1109/OSMSES54027.
+2022.9769134.
+[32] V. Hagenmeyer et al., Information and communication technology in en-
+ergy lab 2.0, Energy Technology 4 (1) (2016). doi:https://doi.org/
+10.1002/ente.201500304.
+[33] Entso-e transparency platform (21.07.2022).
+URL https://transparency.entsoe.eu/
+[34] Y. Peng, A. Rysanek, Z. Nagy, A. Schl¨uter, Using machine learning tech-
+niques for occupancy-prediction-based cooling control in office buildings,
+Applied Energy 211 (2018) 1343–1358. doi:https://doi.org/10.
+1016/j.apenergy.2017.12.002.
+11
+
+20
+30
+Ti1
+[◦C]
+20
+30
+Ti2
+[◦C]
+20
+30
+Ti3
+[◦C]
+20
+30
+Ti4
+[◦C]
+20
+30
+Ti5
+[◦C]
+0
+3.5
+Pel
+[kW]
+10
+30
+Ta
+[◦C]
+0
+1.2
+˙qs
+[kW/m2]
+2022−06−27
+2022−06−28
+2022−06−29
+2022−06−30
+2022−07−01
+2022−07−02
+2022−07−03
+0
+15
+p
+[cent/kwh]
+Hysteresis
+PSC
+MPC
+Figure A.5: Control results for three different controllers in the (a) base scenario over a period of one week
+12
+

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+Preprint version of: Ghorbani, Fatemeh, Mahsa Farshi Taghavi, and Mehdi Delrobaei. "Towards an intelligent assistive system based on augmented reality and 
+serious games." Entertainment Computing 40 (2022): 100458. https://doi.org/10.1016/j.entcom.2021.100458.  
+ 
+ 
+ 
+ 
+ 
+Towards an Intelligent Assistive System Based on 
+Augmented Reality and Serious Games 
+Fatemeh Ghorbani1, Mahsa Farshi Taghavi1, Mehdi Delrobaei1, 2 
+1 Faculty of Electrical Engineering, K. N. Toosi University of Technology, Tehran 1631714191, Iran. 
+2 Department of Electrical and Computer Engineering, Western University, London, ON N6A 5B9, Canada. 
+ 
+E-mail addresses:  
+fatemeghorbani@email.kntu.ac.ir, m.farshitaghavi@email.kntu.ac.ir, delrobaei@kntu.ac.ir, https://orcid.org/my-orcid?orcid=0000-0002-4188-6958 
+ 
+ 
+ 
+ 
+ 
+Abstract - Age-related cognitive impairment is generally characterized by gradual memory loss and decision-making difficulties. 
+The aim of this study is to investigate multi-level support and suggest relevant helping means for the elderly with mild cognitive 
+impairment as well as their caregivers as the primary end-users. This work reports preliminary results on an intelligent assistive 
+system, achieved through the integration of Internet of Things, augmented reality, and adaptive fuzzy decision-making methods. 
+The proposed system operates in different modes, including automated and semi-automated modes. The former helps the user 
+complete their daily life activities by showing augmented reality messages or making automatic changes; while the latter allows 
+manual changes after the real-time assessment of the user’s cognitive state based on the augmented reality serious game score. We 
+have also evaluated the accuracy of the serious game score with 37 elderly participants and compared it with users’ paper-based 
+cognitive test results. We further noted that there is an acceptable correlation between the paper-based test and users’ serious game 
+scores. Moreover, we observed that the system response in the semi-automated mode causes less data loss compared with the 
+automated mode, as the number of active devices decreases. 
+ 
+Keywords: Augmented reality, Fuzzy decision-making, intelligent assistive technology, Internet of things, serious game. 
+ 
+ 
+ 
+1. Introduction 
+Recent projections of the world population indicate obvious trends 
+tending towards more elderly people with the proportion of the global 
+population aged 65 years or over is expected to increase from 9.3 percent in 
+2020 to 16.0 percent by 2050 [1]. Moreover, age-related cognitive 
+impairment is a degenerative neurological disorder charac- terized by 
+progressive loss of memory affecting the elderly [2]. Loss of memory and 
+lacking capacity to make decisions are two main difficulties 
+experienced by the elderly and patients with cognitive impairments. Nearly 
+5.8 million patients are currently suffering from Alzheimer’s disease (AD), 
+as the common cause of cognitive impairments, in the 
+United States alone [3]. They generally have problems in remembering recent 
+information and completing everyday tasks. Therefore, they should always be 
+reminded of the required tasks that improve their confidence and quality of 
+life [4]. 
+In response to the increasing number of elder people with mild 
+cognitive impairment (MCI) and a lack of treatment that slows or stops its 
+progression, the pervasive deployment of intelligent assistive (IA) systems 
+could have a disruptive effect on the elderly and their care- 
+givers’ daily life [5]. IA systems can (i) reduce the burden on public 
+finances throughout the postponement or removal of institutional care, 
+(ii) lessen the psychological burden on formal caregivers and families, 
+(iii) compensate for the lack of human caregivers while improving and 
+optimizing the quality of care, and (iv) empower older adults with MCI and 
+thereby enhancing their confidence [6]. For example, in [7], the authors 
+proposed MED-AR system based on the intelligent augmented reality (AR) 
+for helping older adults in the medication task. The system was designed to 
+present a research methodology for tracking and distributing prescribed 
+medicines for older adults in a home health care scenario. 
+On the other hand, serious games have had positive impacts on 
+reconstructing a functional environment where the user could poten- 
+tially encounter real-world scenarios [8–10]. Moreover, researchers 
+ 
+ 
+ 
+
+2 
+ 
+ 
+have shown that it is a valid evaluation tool for activities of daily living 
+[11] as well as cognitive screening [12]. For instance, researchers used to 
+evaluate activities of daily living through a serious game platform by 
+integrating game playing with a serious purpose. The aim of the study was to 
+assess functional independence in older people through five tasks using the 
+Smart Aging game [13]. In another study, the authors devel- oped a serious 
+game based on virtual reality to estimate cognitive dis- abilities during 
+navigational tasks. Users had to follow arrows to get to special places and 
+find the way back home on their own [14]. Even though the number of 
+researches in this area has been significantly expanding, few researchers have 
+addressed the development of IA sys- tems with possible application into 
+age-related MCI care based on the 
+fact that users’ daily cognitive abilities could be changed. 
+The aim of our study is to suggest an IA system for elderly people 
+with MCI, and improve their ability to complete everyday tasks on their own. 
+In our previous study [15], we implemented a task prompting system based 
+on AR messages without any adaptive decision-making 
+engine to evaluate the user’s general cognitive condition. To increase the 
+system’s intelligence for making a correct reminder, we take advantage of an 
+AR-based serious game assessment tool. 
+The suggested AR-based serious game in this work is potentially capable 
+of providing entertainment and mental state examination simultaneously. Our 
+target population and primary end-users are in- dividuals experiencing MCI, 
+who can understand the technology they are using and its expected 
+assistance and risks. This preliminary study is 
+focused on assessing the proposed system’s usability by the older adults 
+(with and without MCI) and considering their feedback. 
+In more detail, based upon the above suggestions, this paper pro- poses a 
+system the following features: 
+To implement the Internet of Things (IoT) system, the system bene- fits 
+from the Message Queuing Telemetry Transport (MQTT) con- nectivity 
+protocol for communication. All data values, such as indoor positioning 
+data, embedded sensors, and cognitive serious game 
+scores, can be published to the server in the system’s automatic 
+mode. Family or caregivers can also subscribe to each particular 
+topic and override the message for the user to send reminders if needed. 
+In the automatic mode, once an event happens in a specific indoor 
+location, all the relevant fuzzy rules are checked on the server. Thus, 
+following the decision-making algorithm, data values are updated. Both 
+the user and the caregiver could receive appropriate notifications. 
+To monitor the user’s real-time location, the user wears an indoor 
+localization tag. The localization tag sends the positioning informa- 
+tion to the monitor module, and the monitor module receives all the 
+published data. 
+According to the AR serious game score, the performing mode of the 
+IA system changes to the manual mode by real-time assessment of the 
+user’s cognitive state. This approach leads to saving the devices’ energy, 
+enhancing accuracy and performance. 
+ 
+2. Materials and methods 
+2.1. System architecture 
+As the daily routine of an elderly with MCI is contingent on their 
+surrounding objects in the environment, environment conditions such as 
+temperature, humidity, and CO2 level is essential for designing a smart home 
+environment. In our work, such data is published on the cloud layer by 
+placing different sensors and actuators in several locations such as kitchen, 
+bedroom and washing room. 
+Furthermore, Wireless Sensor Networks (WSNs) have been deployed in 
+the local fog layer to collect data about the user’s location. Three anchors are 
+placed in the home environment to estimate the user’s real- 
+time location. 
+All the communication to our IA system is done over MQTT with data 
+serialized in JavaScript Object Notation format (JSON). To avoid un- 
+necessary transmission of the data, the performing mode of the IA sys- tem 
+can be changed to the semi-automated mode based on the decision- making 
+algorithm results. 
+We propose our model based on the localization tag worn by the user for 
+positioning system. The application for making interaction with the end-user 
+and assessing their cognitive functions is developed on an Android operating 
+system, implemented on a smartphone. 
+In an attempt to send all the data to the cloud, we consider MQTT 
+protocol as the messaging protocol. MQTT is a commonly used appli- cation 
+layer protocol to transmit data between the devices in IoT ar- chitecture due 
+to its simplicity and scalability [16]. We have utilized HiveMQ server as a 
+MQTT message broker and Wemos D1 Mini modules for each sensor, 
+actuator, and data coordinator in the positioning system 
+to collect user’s indoor data. Fig. 1 shows the general architecture of the 
+proposed IA system in which the IA system works in its automated mode. 
+We have also designed a service-based application (based on the C# 
+cloud service) for caregivers to present the data related to the events detected 
+by the sensors and to control the actuators, embedded in the smart home 
+environment [17]. The user cognitive functions state and game results can 
+also be monitored by the caregivers. 
+2.2. Adaptive decision-making process 
+In this section, we consider an adaptive fuzzy method to convert our 
+system into an intelligent and context-aware system that promotes monitoring 
+of users and improving their ability to perform their daily tasks. To interact 
+with the user correctly, the decision-making process should be adaptive and 
+precise to avoid false notifications or false- positive AR messages. In some 
+situations, such as completing daily tasks, people occasionally require 
+assistance for memory recall. 
+In order to update each variable of the IA system on the cloud and 
+making the right decision, we employed an adaptive fuzzy logic model that 
+selects the most proper values. The model of the fuzzy adaptive decision-
+making is shown in Fig. 2. 
+A fuzzy controller has four main components: rule base, inference 
+system, fuzzifier interface, and defuzzifier interface. In our system, we have 
+designed two knowledge bases elicited from the user’s cognitive state and the 
+AR game score. These rule-bases are necessary for deciding when to provide 
+aids and predict the user’s target accurately. Now we go through each part of 
+them in our model. 
+2.2.1. Variables and membership functions 
+In our proposed model, the input variables are several types of data from 
+embedded devices, the user’s real-time location, and their cogni- tive state. 
+The output variables are different types of AR messages which 
+are received by the user or their caregivers. Table 1 shows input and output 
+parameters, data types, and their membership functions in our work. 
+We can add other home objects and calculate the distance as a fuzzy input 
+variable. By considering each input and output variable, mem- bership 
+functions are defined based on the data types and fuzzy rules. 
+Fig. 3 shows the membership function for the user’s movement time 
+as well as the AR game score. Output variables are voice, image, and text 
+messages, defined as fuzzy singleton membership functions. Each mes- sage 
+has an identification number (ID) and can be activated by particular inputs. A 
+triangular membership function describes the states of the relay actuators. 
+2.2.2. Fuzzy rule-bases 
+After defining each variable’s membership function, we must build the 
+rule-base, including all of the expert IF-THEN rules. In our IA system, 
+two main rule-bases are essential to creating an adaptive decision- making 
+process and multi-level support. The first one helps the user complete the 
+activities independently in their daily life by showing AR 
+• 
+• 
+• 
+• 
+
+3 
+ 
+ 
+ 
+Device 
+Layer 
+ 
+ 
+User 
+Interface 
+ 
+Patient 
+ 
+ 
+ 
+ 
+Expert 
+ 
+ 
+Device layer 
+(sensors and 
+actuators) 
+ 
+Fig. 1. Intelligent Assistive system architecture. All IoT data values, such as indoor positioning data, embedded sensors, and serious game scores, are published to the server. 
+Following the fuzzy decision-making algorithm and serious game score, data values are updated in the automatic mode. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. 2. Fuzzy adaptive decision-making model. Two knowledge bases are obtained from the user’s cognitive state and the serious game score. User and caregiver receive 
+appropriate notifications based on the fuzzy rules’ outputs. 
+ 
+Table 1 
+Description of Inputs and outputs of the fuzzy system. 
+ 
+Parameter 
+Fuzzy membership function 
+Data type 
+ 
+ 
+Game score 
+Low, high 
+Linguistic 
+Game 
+Start, Stop 
+Boolean 
+ 
+Once an event occurs (for example, medication time alarm), input data 
+values such as location, AR game score, and sensors data value are published 
+on the cloud via MQTT protocol. Moreover, fuzzy rules are checked, and 
+corresponding output variables are updated and published on their topics. 
+Then, according to which rule-bases are activated, the 
+user receives different types of messages, or the user’s manual changes 
+Time 
+Early morning, morning, early afternoon, late 
+Linguistic 
+can be enabled. 
+Distance 
+Humidity 
+afternoon, evening, night, midnight 
+Near, far, very far 
+Linguistic 
+Very dry, dry, humid, very humid 
+Linguistic 
+The adaptive fuzzy decision-making algorithm is capable of scaling 
+up to more rules to improve the user’s independence. The location of 
+Temperature 
+Very cold, cold, cool, mild, warm, hot, very hot 
+Linguistic 
+Reminder 
+Yes, No 
+Boolean 
+Movement 
+Low, medium, high 
+Linguistic 
+different objects and the most used ones can also be added to the 
+database. 
+Therefore, new rules or new objects can be added, or the other rules 
+Flame 
+detection 
+Gas detection 
+Yes, No 
+Boolean 
+ 
+Yes, No 
+Boolean 
+can be removed and updated by the caregivers remotely based on the 
+knowledge and dependent of each user’s lifestyle, and they can manu- 
+Relay status 
+Yes, No 
+Boolean 
+Voice message 
+1, 2, … 
+Integer 
+ally start a scenario or send simple reminders to the user. The fuzzy rules 
+must be endowed with a base of knowledge of each user provided by the 
+Image 
+1, 2, … 
+Integer 
+caregiver, so all the activities are adapted to their preferences [18]. 
+ 
+message 
+ 
+ 
+messages or generating automatic changes such as actuators activation. The 
+second one allows manual changes after real-time assessment of the 
+user’s cognitive state according to the AR-based serious game score. In 
+this situation, some of the smart home sensors and reminders can be 
+turned off or disabled. 
+Some parts of such scenarios and their fuzzy rules are presented in 
+Table 2. These scenarios are precisely defined to choose the most proper ones 
+for the first experiment with individuals with cognitive impair- ments [4,18]. 
+According to [4], elderly people with MCI often lose their ability to 
+sequence in activities such as putting on clothes or cooking. When the user 
+walks in different rooms and tries to interact with other objects, the 
+Fuzzy decision-making 
+engine based on 
+MQTT protocol 
+Indoor 
+localization 
+data 
+Cloud 
+layer 
+Game 
+score/AR 
+messages 
+Service- 
+based 
+application 
+Wireless sensor 
+network 
+(fog layer) 
+Input 
+Output 
+Inference System 
+Data 
+Base 
+Rule 
+Base 
+ 
+ 
+ 
+Application 
+Layer 
+ 
+Fuzzifier 
+ 
+Defuzzifier 
+
+4 
+ 
+ 
+ 
+ 
+Fig. 3. Gaussian membership function of the user’s (a) movement and (b) game score. 
+IA system tries to predict the user’s target based on the distance between the 
+user and predefined objects and other inputs. 
+Lack of movement is also an important issue that should be consid- ered 
+in the users’ daily life. The system measures the user’s movement to check 
+this factor by localization tag worn by the user. If he moves be- 
+tween 3.5 and 4.5 h per day, the IA system sends a reminder to the user and 
+their caregiver [19]. In another scenario, once the user is near the drawer of a 
+wardrobe or closet in the bedroom, the IA system can suggest to them what to 
+wear. This approach can help the user make everyday decisions before they 
+get nervous. 
+ 
+2.3. Real-time position estimation 
+To provide users’ interaction with the objects and monitoring plat- form 
+for caregivers, we need the real-time location of the user. There are 
+many challenges introduced by GPS into the system such as high energy 
+consumption and vulnerabilities that may cause poor accuracy rates in 
+specific environments, for example, indoor locations. Based on the literature, 
+the most accurate beacon-based local positioning system (LPS) solutions are 
+those using ultrasound [20], or ultrawideband (UWB) radio signals [21]. 
+However, ultrasound has a weak capability of a limited maximum 
+range (about 10 m) and cannot pass through walls; thus, its coverage is 
+limited by the number of beacons to install [22]. UWB can penetrate walls in 
+buildings and is able to resolve individual multipath mecha- nisms due to 
+its large bandwidth. Therefore, we have designed our 
+positioning system based on the UWB, because user’s location can be 
+changed over time through different rooms. To achieve this goal, a new 
+prototypical indoor monitoring method developed on a low-cost UWB 
+WSNs, based on IEEE 802.15.4 and IEEE 802.15.4e standards, has been 
+created as a part of our IA system [23]. 
+It is mentioned that the location of the frequently used objects is 
+predetermined; therefore, the distance from the objects can be simply 
+estimated. Locations of the objects are supposed to be fixed and stored 
+on the server. Thus, to calculate the user’s distance from a predefined 
+object, the Euclidean distance is easily assessed between the user and 
+locations of the objects. A true-range multilateration technique is employed 
+as a positioning algorithm in the implemented indoor posi- tioning system 
+[24]. 
+Position calculation consists of two main parts; finding distances from 
+the tag to the anchors, and solving the location inverse geometry problem. 
+The WSNs based on UWB is implemented using a single-chip wireless 
+transceiver, Decawave (Dublin, Ireland) Sensor DW1000 Module [22]. The 
+impulse-based UWB facilitates a precise ranging and 
+
+7
+Low
+Medium
+High
+0.8
+0.6
+区
+0.4
+0.2
+0
+3
+4
+5
+6
+7
+8
+9
+Movement(hour)
+(a)
+1
+LoW
+High
+0.8
+0.6
+0.4
+0.2
+0
+0
+20
+40
+60
+80
+100
+120
+GameScore(%)
+(b)5 
+ 
+ 
+ 
+Table 2 
+Examples of fuzzy rules. 
+ 
+ 
+ 
+ 
+assessment 
+localization tag wearing by the user sends position information to the monitor 
+module, and the monitor module receives all the positioning data. 
+All the data related to the localization system and AR messages can be 
+stored on the cloud for further studies. To improve the user’s safety, 
+caregivers can add the location of their home’s danger zone. Hence, the 
+IA system can provide warnings if the user is near to these locations. The 
+second user interface is built on Android Studio and SDK tools to interact 
+Medication 
+schedule 
+ 
+Leaving the stove 
+Not 
+specific 
+ 
+Not 
+Time is 
+“morning” 
+ 
+Gas detection 
+Voice message 
+is “1′′ and 
+Image message 
+is “1” 
+Reminder 
+ 
+ 
+Alert 
+with the individual with MCI as a personal assistant device [31]. 
+Android Studio is the official integrated development environment (IDE) for 
+the Android operating system created particularly for Android 
+on 
+specific 
+Relay status is 
+is “Yes” 
+��Yes” and 
+Voice message 
+is “2” 
+development. In this work, two different AR messages are defined to send 
+reminders or alerts to the user, including visual and audio 
+messages. 
+Trouble with 
+cooking 
+Kitchen 
+Distance* is 
+“near” 
+Voice message 
+is “3′′ and 
+Image message 
+is “2” 
+Reminder 
+The AR messages show information using virtual data based on the 
+predefined fuzzy rules. These multimedia contents are for taking medication, 
+playing AR cognitive training games, reminding meal times, 
+Putting on clothes 
+Bedroom 
+Distance is 
+“near” 
+Voice message 
+is “4′′ and 
+Image message 
+Reminder 
+and alerting danger zones. For example, to recall medication schedule, a voice 
+message, “Let’s take our medicine,” and an image message indi- 
+cating the drug’s picture are defined. These messages must not remind 
+ 
+Meal time 
+Not 
+specific 
+ 
+Lack of movement 
+Not 
+specific 
+ 
+High game score 
+Not 
+ 
+Time is 
+“early 
+afternoon” 
+Time is 
+“early 
+afternoon” 
+Game score 
+is “3” 
+Voice message 
+is “5′′ and 
+Image message 
+is “4” 
+Voice message 
+is “6′′ 
+Reminder is 
+ 
+Reminder 
+ 
+ 
+Alert 
+ 
+Disable 
+users’ disability. 
+Based on the similar studies and their recommendations on designing 
+serious games for people with MCI [32–34], we have also considered other 
+designing factors in the AR interaction, including: 
+A summary screen showing scores, errors or are included in the game time 
+to foster competition and make an appropriate interaction. 
+ 
+specific 
+is “high” 
+“No” 
+reminders   
+*Distance between the user and particular object (refrigerator, drawer, etc.) 
+ 
+high-accurate localization of the network nodes [25,26]. 
+Our range finding algorithm is a double-sided and two way ranging based 
+on time difference of arrival (TDoA) value [27]. In the designed local fog, 
+calculated ranges are always transmitted between the tag (processor: 
+STM32F4 Cortex-M4) and the anchors (processor: STM32F1 Cortex-M3), 
+sniffed by the data coordinator. 
+Furthermore, the data coordinator concentrates and encapsulates all the 
+transferred distances on the local fog and publishes them in JSON format via 
+MQTT protocol to the cloud server. For activity recognition, 
+pedometer for estimating user’s total movement and fall detection, all 
+the motion processing libraries available in X-CUBE-MEMS1 and can be 
+simply implemented. 
+ 
+2.4. Visualizations 
+The IA system uses two different user interfaces (UI) to interact with the 
+users and their family. Both user interfaces are developed by uti- lizing 
+ARCore SDK (Software Development Kit) for Unity and Android features 
+[28]. In this section, we consider the applications and their features for 
+making interaction with the user, monitoring service for the caregivers, and 
+the development process of designing a serious game environment. 
+2.4.1. AR interaction 
+Unity is a cross-platform game engine that can be applied to design 
+games and simulations for computers, consoles, and smartphones [29]. 
+To display the user’s localization information clearly, a 3D indoor 
+positioning system based on the Unity 3D platform is created. This 
+configurable user interface can represent complex indoor area geographic 
+information and enhance the user experience [30]. 
+Furthermore, to monitor and track the user’s real-time location and 
+provide their interaction with the objects, the user wears an indoor 
+localization tag. For presenting and transmitting positioning data in JSON 
+format via MQTT protocol to the cloud server, MQTTnet and Json. NET 
+libraries were also employed in the Unity application. The 
+Instructions before the game about how to solve game tasks are kept brief 
+and concise. 
+• Only one active screen area to interact with is designed. 
+• Large and highly contrasted icons are used. 
+The game is played from the user’s perspective to prevent confusion with 
+avatars. 
+The difficulty level of the game increases when the user completes the 
+first task correctly, so this can result in early success to foster motivation 
+and prevent frustration. 
+2.4.2. AR-based serious game and scenarios 
+Initiating treatment at the early stage of cognitive problems would slow 
+the progression of the situation and expand the individual’s quality of life 
+[35]. In this study, an AR-based serious game provides enter- 
+tainment and mental state examination simultaneously. The IA system 
+suggests playing the game several times per day to get the user involved and 
+keep them active. Some parts of the game’s features are as follows: 
+It is designed based on “World Tracking” that allows users to put an AR 
+object anywhere they would like in the camera view. 
+Before starting each task, the system provides some details within the 
+questions to trigger memory through voice interactions. 
+After performing each task, users’ score is calculated based on the 
+number of right and wrong decisions automatically. 
+The time duration of each task and total time response are measured 
+automatically. 
+The AR serious game consists of a simulation of daily living situations 
+with five tasks: finding a particular object, remembering colors, and 
+arranging numbers. These tasks are defined to evaluate different cognitive 
+functions such as pattern separation and completion, visuo- spatial and 
+episodic memory, decision-making ability, concentration, and overall 
+processing speed by measuring response time [33]. 
+Performing such tasks may also result in improving the user’s 
+cognitive abilities ultimately. Furthermore, based on the fuzzy rules, the 
+tasks can be suggested to the user to perform several times per day for 
+assessing their cognitive status and rehabilitation program. To prevent 
+reappearance, we have designed three series of the game scenarios in which 
+each task contains different 3D objects. In every session, the IA 
+• 
+• 
+• 
+• 
+• 
+• 
+• 
+• 
+Event 
+User’s 
+Fuzzy rules 
+ 
+Command 
+ 
+location 
+If 
+Then 
+type 
+Entertainment/ 
+cognitive 
+Not 
+specific 
+Time is 
+“evening” 
+Game is “start” 
+Launching 
+AR game 
+ 
+
+6 
+ 
+ 
+system randomly selects one of the game series for recommending to the 
+user. Table 3 describes the tasks and outputs measures. 
+Before beginning the assessment process and game scenarios, the user 
+should look at the whole objects for 10 s (second) to memorize their location 
+and color, as shown in Fig. 4 (a). Audio messages are provided 
+before starting each task to help the user realize the task requirements, for 
+example, which object’s color should be recalled [33]. Fig. 4 illus- trates the 
+game steps based on the tasks presented in Table 3. 
+In more detail, Fig. 5 (a) shows the task 4 in which the user should 
+identify the unnatural placement of objects, for instance, headphones in a 
+sink [33]. In the next step, the user observes a series of numbers on the screen 
+for 5 s, and after that, he must choose the correct sequence of numbers in the 
+picture frames. This step is proposed based on the fact that people with mild 
+cognitive impairments usually lose their ability to 
+sequence [4]. Fig. 5 (b) demonstrates this step of the game. Finally, the user’s 
+final score according to the total time response and the number of true (T) 
+and false (F) answers are calculated. 
+The total response time must not be more than 10 min, which is the 
+normal cognitive assessment test duration [34]. Fig. 5 (c) indicates an 
+example of the assessment result that is sent to the fuzzy decision- making 
+system for evaluating the level of support, this result is not included in the 
+statistical analysis. 
+The IA system is designed based on error-less learning to encourage the 
+user to recall different events. If the user gives more correct answer, the 
+system vanishes reminders and turns off some smart home sensors. 
+Thus, it can improve the user’s self-management and self-care, and it 
+would also slow the progression of the disease [3]. 
+To make a user-friendly interaction, the game is presented on a touch 
+screen for performing different tasks, because most of the older adults can 
+interact with the touch screen without any training [36]. We have used 
+different 3D objects for creating an AR environment, and an event- based 
+creation supporting unit for defining interaction between the virtual space 
+object in a format of event script [37]. 
+To build the AR serious game, different virtual objects are incorpo- rated 
+into the web-based AR platform, ZapWorks Studio and ZapWorks Designer 
+(Zappar Ltd., United Kingdom) [38]. The platform allows us to upload 
+multimedia contents to create an AR experience that lets users view the 
+virtual objects through the mobile device application, Zappar. 
+This 
+application could be efficiently run on both Android and iOS operating 
+systems. In the current study, we have used a Samsung Galaxy S9 Plus 
+smartphone, with an Octa-core processor, 6 GB RAM, and dual 12MP rear 
+camera to view the AR experience, with each virtual object being displayed 
+on the screen. 
+ 
+Table 3 
+Descriptions of tasks and outcomes measures. 
+Tasks 
+Outcome measures 
+ 
+Task 1: retrieving objects location 
+Time on the task 1 
+Task completion: selecting the correct 
+location 
+Score: 1 point for the correct answer 
+Task 2: memorizing objects color 
+Time on the task 2 
+Task completion: selecting the correct color 
+Score: 1 point for the correct answer 
+Task 3: recognizing an extra object 
+Time on the task 3 
+Task completion: selecting the correct object 
+Score: 1 point for the correct or wrong answer 
+Task 4: identifying unnatural 
+placement of objects 
+ 
+ 
+Task 5: recalling sequence of numbers 
+Time on the task 4 
+Task completion: selecting the correct 
+picture 
+Score: 1 point for the correct or wrong 
+answer 
+Time on the task 5 
+Task completion: selecting the correct 
+numbers sequentially 
+Score: 1 point for the correct answer 
+Fig. 4. Screenshots of the five augmented tasks. (a) Overview of the main 
+environment, (b) Task 1: retrieving the object location, (c) Task 2: memorizing the 
+object color, (d) Task 3: recognizing the extra object, (e) Task 4: identifying 
+unnatural placement of the objects in the picture frames, and (f) Task 5: recalling 
+sequence of the numbers on the screen. 
+ 
+ 
+
+(a)
+(b)
+(d)
+(e)
+3-1-2-4-5
+(f) 
+ 
+7 
+ 
+ 
+× 
+× 
+×
+ 
+× 
+ 
+ 
+Fig. 5. Illustration of response of a user (a) identify the unnatural placement of 
+objects, (b) choose the correct sequence of numbers in the picture frames, and 
+(c) final score is sent to the decision-making system. 
+ 
+2.5. Study participants 
+We set out to validate the IA system and the serious game in a group of 
+individuals aged above 50 years to be age-matched with the end- users. They 
+comprised 37 older adults, including 16 females and 21 males. All 
+participants were volunteers recruited from universities and social media. 
+They all lived independently and had active social and cognitive lives. 
+Persian was the mother tongue for all participants in the study. Before 
+playing the AR serious game, participants underwent the MoCA [39]. The 
+MoCA is a 30-point test, including a set of cognitive subtests, concerning six 
+different cognitive domains. 
+Based on the literature, a cutoff of 26 (scores of 25 or below indicate 
+impairment) yields the best balance between sensitivity and specificity for 
+the MCI and other groups [40]. Thus, in our study, we have divided 
+participants into two groups based on their MoCA results: (A) control group 
+(MoCA ≥ 26) and (B) MCI group (MoCA < 26). Table 4 shows the 
+ 
+Table 4 
+Group characteristic in the study (A: control group, B: MCI group). 
+ 
+Participants (n = 37) 
+Total number 
+Female 
+MoCA score 
+Group A 
+Group B 
+26 
+11 
+12 
+4 
+≥26 
+<26 
+ 
+number of people from each group and their gender. 
+All participants first filled a questionnaire including demographic 
+information, personal and medical history. They then completed a training in 
+the game in order to get familiarize with the interactions of 
+virtual objects for final evaluation. Table 5 describes all the participants’ 
+demographics in details. 
+ 
+2.6. Statistical analysis 
+We have analyzed all the data and game scores to consider serious game 
+accuracy compared to MoCA scores. We chose the MoCA as our reference 
+screening tool as it verified to be sensitive to recognize MCI and it is an 
+accurate screening measurement of cognitive ability. The 
+significance level was set α < 0.05 for all analyses. The SPSS 26.0 sta- 
+tistical software package was used to perform all the statistical analyses. 
+The data was normal distribution and homogeneous, so Pearson corre- lation 
+coefficient was used for categorical variables. 
+3. Results and discussions 
+ 
+In this section, we present the technical feasibility and usability of the 
+system via simulations and user experiences. First, the IA system’s accuracy 
+in detecting the danger zones and making appropriate rec- 
+ommendations have been assessed. Then, the serious game results with 
+37 participants compared with their MoCA test scores have been pro- vided. 
+Finally, we evaluated the system’s performance in working in different 
+modes. 
+ 
+3.1. Forbidden zones 
+In this experience, we evaluated the system’s accuracy in detecting 
+danger zones area. Four different danger areas have been defined with equal 
+sizes of 0.52  0.7  0.23 m3 in an 8.5  4.6  2 m3 room space, as 
+shown in Fig. 6. Fig. 7 illustrated these zones, which were determined in the 
+Unity game engine to display the user’s localization information clearly. 
+In a typical scenario, once the user entered a dangerous zone (for 
+example, the fireplace), they received an image message alarm. This alarm 
+reminded them to keep away from the area. The near membership function 
+was defined from 1 to 5 dm (the danger zone). Fig. 8 demon- strates the AR 
+message showing the fire alert sent to the user by acti- vating the fuzzy rule 
+number 16. Other reminders could be for pharmacological management and 
+completing daily tasks. 
+The positioning data and the movement time of the user was recor- ded 
+during the experiment. According to the system’s operating mode and the 
+users’ cognitive state, if the users’ movement duration was 
+lower than 4.5 h, they received a voice message. This audio message (ID: 
+13), which was sent by the IA system, reminded the users their lack of 
+movement [19]. If the movement duration was higher than 4.5 or the 
+ 
+Table 5 
+Participants’ demographics. 
+ 
+ 
+Participants 
+Average 
+Minimum 
+Maximum 
+Standard 
+deviation 
+Age of group A 
+60.07 
+51 
+81 
+7.98 
+Age of group B 
+65.73 
+56 
+80 
+7.91 
+MoCA score of group 
+27.71 
+26 
+29 
+1.069 
+A 
+MoCA score of group B 
+23.36 
+21 
+25 
+1.36 
+
+(a)
+3
+2
+4
+(b)
+Level
+T
+F
+Time
+1
+1
+14
+2
+0
+14
+3
+0
+13
+1
+4
+1
+0
+8
+5
+1
+18
+4
+0
+Total
+162
+(c) 
+ 
+8 
+ 
+ 
+- 
+±
+ 
+ 
+± 
+= - 
+±
+ 
+ 
+ 
+ 
+Fig. 6. Four danger zone areas are defined for evaluation of the localization 
+system in the experimental condition. 
+ 
+ 
+Fig. 7. Model of the 3D-indoor positioning system and four danger zone areas 
+simulated in the Unity game engine. 
+user’s serious game result was high, they did not receive any messages or 
+alarms. Table 6 summarizes the results of these experiments in different 
+conditions based on the data values. 
+ 
+3.2. Serious game 
+We have evaluated the serious game’s accuracy with 37 participants and 
+compared the game score with the users’ MoCA test scores. Table 7 
+compares both scores in group A and group B. 
+We used the Pearson correlation coefficient for control group par- 
+ticipants with respect to the parametricity of the data and applied the 
+Spearman correlation coefficient due to the nonparametric data for the 
+MCI group. These correlations have been calculated between the score of the 
+MoCA and the score of the game, and the total time and participants’ age. 
+The results are given in Table 8 and Table 9. 
+According to the results, the total score of MoCA had a significant 
+correlation (0.911) with the total score of the game in the control group, 
+while the correlation coefficient of Spearman in the MCI group has a lower 
+correlation (0.722). Furthermore, there was no correlation be- tween the total 
+score of MoCA and the total time of the game in the control group. While in 
+the MCI Group, there was a relative correlation ( 0.773) between the total 
+score of MoCA and the total game time duration. This may indicate that the 
+response time in the MCI group is a more effective factor than the control 
+group. 
+Also, in both groups, there was no correlation between the score of the 
+game and the age of the participants, which means that the age of the 
+participants was not a determining factor for the game score. 
+Table 10 compares the scores of MoCA and serious game results. These 
+grades showed that the participants with the MoCA score higher than 27, 
+could get at least 4 scores in the serious game. 
+Spearman correlation coefficient between the total score of MoCA with 
+total time was 0.692 for the participants who scored 3 points in the game. 
+This number increased to 0.841 for the participants who scored 2 points and 1 
+point in the game. However, a comprehensive evaluation cannot be 
+concluded from these correlations due to the very small amount of data for 
+each group of participants who received the same score of 1 to 3 from the 
+game. 
+We also performed a T-test for the total score of the game to deter- mine a 
+meaningful performance difference between the control and MCI group 
+members. The results of this evaluation showed that the MCI group had a 
+lower overall game score (2.09  0.893) compared to the 
+control group participants (4.21  0.70); t (37)  6.939, p < 0.001. 
+Also, the result of the T-test for the total game duration time between the 
+control and MCI groups indicated that the MCI group had a higher total 
+game time duration (5.13  2.03) than the control group (3.35 
+1.9); t (37)   2.275, p < 0.05. 
+This means that there is a considerable difference between the con- trol 
+group and MCI group participants’ cognitive performance. Our evaluation 
+findings were in line with previous results indicating that the 
+touchscreen was well accepted by participants without computer expe- rience 
+based on the participants’ feedback [41,42]. 
+ 
+3.3. System operation modes 
+We considered a wide range of devices that must cover the maximum 
+ 
+Table 6 
+Result of the experimental test. 
+Movement 
+(hour) 
+Serious game 
+result (out of 
+100) 
+Voice 
+message ID 
+Danger zone 
+number 
+Image 
+message ID 
+ 
+Fig. 8. Sample of an image message received by the user when a specific fuzzy rule 
+was activated, picture message number 16 (fire alert). 
+ 
+1.15 
+55 
+13 
+1 
+16 
+5.30 
+48 
+– 
+2 
+17 
+5.25 
+97 
+– 
+3 
+18 
+3.40 
+76 
+13 
+4 
+19 
+
+TERRACE
+ONNIO
+STOF
+HLO 
+ 
+9 
+ 
+ 
+55 
+5 8 7 
+15 
+3 2 4 10 
+ 
+Table 7 
+Comparison between group A and group B game results. 
+ 
+ 
+ 
+5) group A 
+ 
+100 
+97 
+80 
+60 
+40 
+Serious game score (out of 
+5) group B 
+Serious game time duration 
+(mins) group A 
+Serious game time duration 
+(mins) group B 
+2.09 
+1 
+3 
+0.83 
+ 
+3.35 
+1.38 
+7.47 
+1.9 
+ 
+5.13 
+2.42 
+8.83 
+2.03 
+20 
+0 
+automated 
+mode 
+ 
+ 
+ 
+semi- 
+automated 
+mode 
+ 
+Table 8 
+Pearson correlation coefficient between MoCA scores and total serious game 
+score, total response time and age in the control group. 
+ 
+serious game score 
+number of alarms 
+number of reminders number of actuators 
+number of sensors 
+Pearson Correlation in 
+group A 
+Total Serious 
+Game Score 
+Total Response 
+Time 
+Participant 
+Age 
+ 
+Fig. 9. Workload scenarios description. The number of alarms and reminder 
+MoCA Total Score 
+0.911 
+-0.343 
+0.003 
+messages decreased as the user’s game score increased. 
+Total Serious Game 
+Score 
+Table 9 
+1 
+-0.115 
+0.031 
+3.4. Limitations 
+The target population and primary end-users of our proposed system are 
+older adults experiencing cognitive impairments who can under- 
+Spearman Rank Correlation (Spearman’s Rho) between MoCA scores and total 
+serious game score, total response time and age in MCI group. 
+ 
+ 
+stand the technology they are using and its expected assistance and risk. 
+This study was focused on assessing the proposed system’s usability by 
+Pearson Correlation in 
+group A 
+Total Serious 
+Game Score 
+Total Response 
+Time 
+Participant 
+Age 
+the users without cognitive impairments and considering their feedback. 
+However, this study has some limitations. 
+ 
+MoCA Total Score 
+0.722 
+-0.773 
+-0.278 
+The space validity of this work was limited, for part of the study was 
+Total Serious Game 
+Score 
+ 
+ 
+Table 10 
+1 
+-0.477 
+0.017 
+presented in a laboratory environment. The IA system prototype can be 
+evaluated with more users, and it includes more scenarios and func- 
+tionalities. Therefore, although the scenarios technical requirements are taken 
+into account in our work, the particular content of the messages 
+must be customized for each user before the system’s implementation. 
+Comparison between the scores of MoCA and serious game results. 
+ 
+ 
+Group 
+Total MoCA score 
+Total serious game score 
+Control Group 
+29≤ 
+5 
+28 
+4–5 
+27 
+4 
+To achieve this goal, the IA system is capable of adding more fuzzy rules and 
+being customized for each person. 
+4. Conclusions and future work 
+MCI Group 
+26 
+3 
+25 
+3 
+24 
+2–3 
+23 
+2 
+22 
+1–2 
+21 
+1–2 
+Memory loss and decision-making difficulties are the main symptoms of 
+MCI experienced by elderly people. The person forgets recent in- cidents and 
+have problems with remembering the information. Such situations have a 
+considerable effect on the confidence and quality of life of both the elderly 
+people and their caregivers. The aim of this study was to introduce an IA 
+system for individuals with age-related MCI, and 
+area of the user’s environment. The number of devices ranged from 10 to 15 
+to realize different workloads, including various sensors and actua- 
+tors as we mentioned in Section 2.1. Specifically, for each number of 
+devices, we simulated the two different cases based on the system’s 
+operation mode. The complexity of the decision-making process 
+decreased from automated mode to semi-automated mode, as the number of 
+active devices declined from 22 to 14. 
+In the functionality test, we consider the number of audio messages 
+sent to the user, and the number of IoT devices enabled during the 
+experiment. Fig. 9 illustrates the numbers relevant to different types of AR 
+messages included in each case and received by the user. As shown in Fig. 9, 
+the number of alarms and reminder messages decreased as the 
+user’s game score increased. 
+The results indicated that the user required lower reminders while 
+their cognitive test score was higher than previous session. These also led to 
+less data loss in the decision-making algorithm and during the process of 
+publishing or subscribing to the messages. Moreover, the operation in semi-
+automated mode can cause less need for battery recharging or replacement 
+[43]. 
+improve their ability to complete everyday tasks on their own without 
+compromising their privacy. We designed, implemented, and evaluated an IA 
+system based on the AR serious game and IoT to help users make everyday 
+decisions more independently through interaction and recall memory of 
+events. 
+Several ready-to-use libraries were used to facilitate system imple- 
+mentation on smartphones and computers. We evaluated the system’s 
+technical feasibility and usability via simulations and users’ experiences 
+with 37 elderly participants. The result of our evaluation implied that 
+the total score of the serious game had a high correlation with the total score 
+of MoCA in the control group, and there was an acceptable cor- relation in 
+the MCI group. Also, our game was capable of showing a considerable 
+difference between the two groups of control and MCI 
+participants’ cognitive performance. This indicates the high accuracy of the 
+system in estimating users’ cognitive states compared to the tradi- tional 
+assessment tools. 
+Moreover, the touchscreen interaction and verbal reminders created a 
+natural way of interaction and were well accepted by the participants without 
+computer experiences. The system response in the semi- automated mode 
+also caused less data loss than the automated mode, 
+Participants 
+Average 
+Minimum 
+Maximum 
+Standard 
+deviation 
+Serious game score (out of 
+4.21 
+3 
+5 
+0.70 
+ 
+
+ 
+10 
+ 
+ 
+as the number of active devices decreased. 
+The findings of this study support the idea that an IA system can 
+potentially help older adults in the home environment and can be evaluated 
+by users with cognitive impairments according to partici- 
+pants’ feedback. Serious games also have a positive impact on present- 
+ing the possibility of reconstructing a functional environment for the 
+users. Some future plans could be the study of historical data, including daily 
+serious game scores, to evaluate the user’s mental and physical health 
+condition. 
+Declaration of Competing Interest 
+The authors declare that they have no known competing financial 
+interests or personal relationships that could have appeared to influence the 
+work reported in this paper. 
+Acknowledgments 
+The authors gratefully acknowledge the families and the individuals that 
+participated in this study. 
+References 
+[1] United Nations Department of Economic and Social Affairs, 2020. United Nations 
+Department of Economic and Social Affairs, World Population Ageing 2020 
+Highlights: Living Arrangements Of Older Persons. United Nations Publications 
+(2020) Retrieved April 8, 2021, from United Nations in https://www.un.org/de 
+velopment/desa/pd/sites/www.un.org.development.desa.pd/files/undesa_pd-20 
+20_world_population_ageing_highlights.pdf. 
+[2] S. Rattan, M. Kyriazi, The science of hormesis in health and longevity, (2018). 
+[3] A.s. Association, 2019 Alzheimer’s disease facts and figures, Alzheimer’s & 
+Dementia, 15 (2019) 321–387. 
+[4] V. Bell, D. Troxel, A Dignified Life: The Best FriendsTM Approach to Alzheimer’s 
+Care: A Guide for Care Partners, Health Communications, Inc., 2012. 
+[5] M.E. Pollack, Intelligent assistive technology: the present and the future, in: 
+International Conference on User Modeling, Springer, 2007, pp. 5–6. 
+[6] M. Ienca, J. Fabrice, B. Elger, M. Caon, A. Scoccia Pappagallo, R.W. Kressig, 
+T. Wangmo, Intelligent assistive technology for Alzheimer’s disease and other 
+dementias: a systematic review, J. Alzheimer’s Dis. 56 (2017) 1301–1340. 
+[7] E. Guerrero, M.-H. Lu, H.-P. Yueh, H. Lindgren, Designing and evaluating an 
+intelligent augmented reality system for assisting older adults’ medication 
+management, Cognit. Syst. Res. 58 (2019) 278–291. 
+[8] M. Pasch, N. Bianchi-Berthouze, B. van Dijk, A. Nijholt, Movement-based sports 
+video games: investigating motivation and gaming experience, Entertain. Comput. 1 
+(2009) 49–61. 
+[9] I.T. Paraskevopoulos, E. Tsekleves, C. Craig, C. Whyatt, J. Cosmas, Design 
+guidelines for developing customised serious games for Parkinson’s Disease 
+rehabilitation using bespoke game sensors, Entertain. Comput. 5 (2014) 413–424. 
+[10] K. Seaborn, D.I. Fels, Gamification in theory and action: a survey, Int. J. Hum 
+Comput Stud. 74 (2015) 14–31. 
+[11] Ioannis Tarnanas, Winfried Schlee, Magda Tsolaki, Ren´e Müri, Urs Mosimann, 
+Tobias Nef, Ecological validity of virtual reality daily living activities screening for 
+early dementia: longitudinal study, JMIR Serious Games 1 (1) (2013) e1, https:// 
+doi.org/10.2196/games.2778. 
+[12] H. Tannous, D. Istrate, A. Perrochon, J.-C. Daviet, A. Benlarbi-Delai, J. Sarrazin, 
+M.-C.H.B. Tho, T.T. Dao, GAMEREHAB@ HOME: a new engineering system 
+using serious game and multi-sensor fusion for functional rehabilitation at home, IEEE 
+Trans. Games (2019). 
+[13] S. Bottiroli, C. Tassorelli, M. Lamonica, C. Zucchella, E. Cavallini, S. Bernini, 
+E. Sinforiani, S. Pazzi, P. Cristiani, T. Vecchi, Smart aging platform for evaluating 
+cognitive functions in aging: a comparison with the MoCA in a normal population, 
+Front. Aging Neurosci. 9 (2017) 379. 
+[14] V. Vallejo, P. Wyss, L. Rampa, A.V. Mitache, R.M. Müri, U.P. Mosimann, T. Nef, 
+Evaluation of a novel Serious Game based assessment tool for patients with 
+Alzheimer’s disease, PLoS One 12 (2017), e0175999. 
+[15] F. Ghorbani, M. Kia, M. Delrobaei, Q. Rahman, Evaluating the Possibility of 
+Integrating Augmented Reality and Internet of Things Technologies to Help Patients 
+with Alzheimer’s Disease, in: National and 4th International Iranian Conference on 
+Biomedical Engineering (ICBME), IEEE, 2019, pp. 139–144. 
+[16] A. Al-Fuqaha, M. Guizani, M. Mohammadi, M. Aledhari, M. Ayyash, Internet of 
+things: a survey on enabling technologies, protocols, and applications, IEEE Commun. 
+Surv. Tutorials 17 (2015) 2347–2376. 
+[17] A. Jindal, A. Dua, N. Kumar, A.K. Das, A.V. Vasilakos, J.J. Rodrigues, Providing 
+healthcare-as-a-service using fuzzy rule based big data analytics in cloud computing, 
+IEEE J. Biomed. Health. Inf. 22 (2018) 1605–1618. 
+ 
+[18] M.A. Salichs, I.P. Encinar, E. Salichs, A´. Castro-Gonza´lez, M. Malfaz, Study of 
+scenarios and technical requirements of a social assistive robot for Alzheimer’s 
+disease patients and their caregivers, Int. J. Soc. Robot. 8 (2016) 85–102. 
+[19] A. Al-Adhab, H. Altmimi, M. Alhawashi, H. Alabduljabbar, F. Harrathi, H. 
+ALmubarek, IoT for remote elderly patient care based on Fuzzy logic, in: 2016 
+International Symposium on Networks, Computers and Communications (ISNCC), 
+IEEE, 2016, pp. 1–5. 
+[20] F. Seco, A.R. Jim´enez, F. Zampella, Fine-grained acoustic positioning with 
+compensation of CDMA interference, in: 2015 IEEE International Conference on 
+Industrial Technology (ICIT), IEEE, 2015, pp. 3418–3423. 
+[21] L. Zwirello, T. Schipper, M. Jalilvand, T. Zwick, Realization limits of impulse-based 
+localization system for large-scale indoor applications, IEEE Trans. Instrum. Meas. 64 
+(2014) 39–51. 
+[22] Antonio Ramon Jimenez Ruiz, Fernando Seco Granja, Comparing ubisense, 
+bespoon, and decawave uwb location systems: Indoor performance analysis, IEEE 
+Trans. Instrum. Meas. 66 (8) (2017) 2106–2117. 
+[23] I. Group, Part 15.4: Low-Rate Wireless Personal Area Networks (LR-WPANs). 
+Amendment 1: MAC sublayer, IEEE, IEEE Standard for Local and metropolitan area 
+networks IEEE Std, 802 (2012) 4e-2012. 
+[24] C.L. Sang, M. Adams, T. Korthals, T. Ho¨rmann, M. Hesse, U. Rückert, 
+A bidirectional object tracking and navigation system using a true-range multilateration 
+method, in: 2019 International Conference on Indoor Positioning 
+and Indoor Navigation (IPIN), IEEE, 2019, pp. 1–8. 
+[25] M.Z. Win, D. Dardari, A.F. Molisch, W. Wiesbeck, W. Jinyun Zhang, History and 
+Applications of UWB, Institute of Electrical and Electronics Engineers, 2009. 
+[26] D. Neuhold, C. Bettstetter, A.F. Molisch, HiPR: High-Precision UWB Ranging for 
+Sensor Networks, in: Proceedings of the 22nd International ACM Conference on 
+Modeling, Analysis and Simulation of Wireless and Mobile Systems, 2019, pp. 103–
+107. 
+[27] G. Oguntala, R. Abd-Alhameed, S. Jones, J. Noras, M. Patwary, J. Rodriguez, 
+Indoor location identification technologies for real-time IoT-based applications: an 
+inclusive survey, Comput. Sci. Rev. 30 (2018) 55–79. 
+[28] J. Linowes, K. Babilinski, Augmented Reality for Developers: Build practical 
+augmented reality applications with Unity, ARCore, ARKit, and Vuforia, ARKit, 
+and Vuforia, Packt Publishing Ltd, 2017. 
+[29] J. Hocking, Unity in action: Multiplatform game development in C# with Unity 5, 
+Manning Publ., 2015. 
+[30] G. Chen, X. Meng, Y. Wang, Y. Zhang, P. Tian, H. Yang, Integrated 
+WiFi/PDR/ Smartphone using an unscented kalman filter algorithm for 3D indoor 
+localization, 
+Sensors 15 (2015) 24595–24614. 
+[31] K.M. Kanno, E.A. Lamounier Jr, A. Cardoso, E.J. Lopes, S.A. Fakhouri 
+Filho, Assisting individuals with Alzheimer’s disease using mobile augmented 
+reality with voice interaction: an acceptance experiment with individuals in the 
+early stages, Res. Biomed. Eng. 35 (2019) 223–234. 
+[32] C. Dietlein, B. Bock, Recommendations on the design of serious games for people 
+with Dementia, EAI Endorsed Trans. Serious Games 5 (2019). 
+[33] A. Vovk, A. Patel, D. Chan, Augmented reality for early Alzheimer’s disease 
+diagnosis, in: Extended Abstracts of the 2019 CHI Conference on Human Factors in 
+Computing Systems, 2019, pp. 1–6. 
+[34] C. Dietlein, S. Eichberg, T. Fleiner, W. Zijlstra, Feasibility and effects of serious 
+games for people with dementia: a systematic review and recommendations for future 
+research, Gerontechnology 17 (2018) 1–17. 
+[35] T. Pradeep, M.J. Bray, S. Arun, L.N. Richey, S. Jahed, B.R. Bryant, C. LoBue, C. 
+G. Lyketsos, P. Kim, M.E. Peters, History of traumatic brain injury interferes with 
+accurate diagnosis of Alzheimer’s dementia: a nation-wide case-control study, Int. 
+Rev. Psychiatry 32 (2020) 61–70. 
+[36] J. Asad, S. Kousar, N.Q. Mehmood, Dementia-related serious games: a comparative 
+study, Univ. Sindh J. Inf. Commun. Technol. 3 (2019) 171–177. 
+[37] L. Jun-Sup, S.-W. Lee, J.-Y. Youn, L. Suk-Hyun, G.-H. Lee, Apparatus and method 
+for creating spatial augmented reality content, in, Google Patents, 2014. 
+[38] G. Terzopoulos, I. Kazanidis, M. Satratzemi, A. Tsinakos, A comparative study of 
+augmented reality platforms for building educational mobile applications, in: 
+Interactive Mobile Communication, Technologies and Learning, Springer, 2019, 
+pp. 307–316. 
+[39] M. Abd Razak, N. Ahmad, Y. Chan, N.M. Kasim, M. Yusof, M.A. Ghani, M. 
+Omar, 
+F. Abd Aziz, R. Jamaluddin, Validity of screening tools for dementia and mild 
+cognitive impairment among the elderly in primary health care: a systematic review, 
+Public Health 169 (2019) 84–92. 
+[40] Z.S. Nasreddine, N.A. Phillips, V. B´edirian, S. Charbonneau, V. Whitehead, 
+I. Collin, J.L. Cummings, H. Chertkow, The Montreal Cognitive Assessment, MoCA: 
+a brief screening tool for mild cognitive impairment, J. Am. Geriatr. Soc. 53 (2005) 
+695–699. 
+[41] V. Vallejo, A.V. Mitache, I. Tarnanas, R. Müri, U.P. Mosimann, T. Nef, Combining 
+qualitative and quantitative methods to analyze serious games outcomes: A pilot study 
+for a new cognitive screening tool, in: 2015 37th Annual International Conference of 
+the IEEE Engineering in Medicine and Biology Society (EMBC), IEEE, 
+2015, pp. 1327–1330. 
+[42] V. Vallejo, I. Tarnanas, T. Yamaguchi, T. Tsukagoshi, R. Yasuda, R. Müri, U. 
+P. Mosimann, T. Nef, Usability assessment of natural user interfaces during serious 
+games: adjustments for dementia intervention, J Pain Manage. 9 (2016) 333–339. 
+[43] G. Maselli, M. Piva, J.A. Stankovic, Adaptive communication for battery-free 
+devices in smart homes, IEEE Internet Things J. 6 (2019) 6977–6988. 
+

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+Understanding the Complexity and Its Impact on
+Testing in ML-Enabled Systems
+A Case Sutdy on Rasa
+Junming Cao∗, Bihuan Chen∗, Longjie Hu∗, Jie Gao† Kaifeng Huang∗ and Xin Peng∗
+∗Fudan University, Shanghai, China
+†Singapore University of Technology and Design, Singapore
+Abstract—Machine learning (ML) enabled systems are emerg-
+ing with recent breakthroughs in ML. A model-centric view
+is widely taken by the literature to focus only on the analy-
+sis of ML models. However, only a small body of work takes
+a system view that looks at how ML components work with
+the system and how they affect software engineering for ML-
+enabled systems. In this paper, we adopt this system view, and
+conduct a case study on Rasa 3.0, an industrial dialogue system
+that has been widely adopted by various companies around the
+world. Our goal is to characterize the complexity of such a large-
+scale ML-enabled system and to understand the impact of the
+complexity on testing. Our study reveals practical implications
+for software engineering for ML-enabled systems.
+I. INTRODUCTION
+The recent advances in machine learning (ML) have attracted
+an increasing interest in applying ML across a breadth of busi-
+ness domains, e.g., self-driving cars, virtual assistants, robotics,
+and health care. According to the Global AI Adoption Index by
+IBM [34], 35% of companies around the world have deployed
+AI in their business, while 42% of companies are exploring AI.
+Such a trend has caused the emergence of ML-enabled systems
+which are composed of ML and non-ML components. ML com-
+ponents are often important, but usually only a part of many
+components in ML-enabled systems [38].
+The previous research on software engineering for machine
+learning often takes a model-centric view that focuses only on
+the analysis of ML models [38, 55]. For example, many ad-
+vances have been made for DL model testing (e.g., [4, 19, 25,
+40, 59, 68, 72, 82, 84]), verification (e.g., [9, 57, 58, 66, 73])
+and debugging (e.g., [43, 49, 54, 71]). Only a small body of
+work takes a holistic system view, e.g., architectural design
+[64, 80], technical debt [63, 70], ML component entanglement
+[53, 78, 83], feature interaction [1, 2, 8], and model interactions
+in Apollo [60]. However, the lack of system-level understanding
+of ML-enabled systems may hide problems in engineering ML-
+enabled systems and hinder practical solutions.
+In this paper, we adopt this system view, and conduct a case
+study on Rasa 3.0 [11] to characterize the complexity of such a
+large-scale ML-enabled system as well as to understand the im-
+pact of the complexity on testing. Rasa is a task-oriented in-
+dustrial dialogue system that has been widely used by various
+companies around the world. Therefore, we believe Rasa is a
+good representative of real-world ML-enabled systems.
+We first investigate the complexity of Rasa at three levels. At
+the system level, we explore how ML components are adopted
+across the modules in Rasa. We find that there are 23 ML mod-
+els in 15 ML components across 6 modules. At the interaction
+level, we analyze how ML components interact with other com-
+ponents in Rasa. We find that there are 43 interaction patterns
+and 230 interaction instances across 4 major categories and 8
+inner categories. At the component level, we investigate how
+the code of ML components is composed by what kinds of code.
+We find that 57.1% of the code inside components are data
+processing code, and there are 8 composition patterns between
+data processing code and model usage code.
+We then explore the impact of the complexity on test-
+ing from two perspectives. From the testing practice perspec-
+tive, we analyze how is the characteristic of test cases, and
+how well they cope with the complexity. We find that the
+test coverage of component interactions is low because of the
+complexity from huge configuration space and from hidden
+component interactions. From the mutation testing perspective,
+we study how is the bug-finding capability of test cases and test
+data (i.e., the data for testing models), and how well they cope
+with the complexity. We find that there may be many potential
+bugs in data processing code that can only be detected by test
+cases, due to the complexity from data processing code. The
+capability of test data to kill mutants is limited because of the
+complexity from huge configuration space.
+Based on our case study, we highlight practical implications
+to improve software engineering for ML-enabled systems. For
+example, the configuration space of ML-enabled systems should
+be tested adequately, and configuration suggestions should be
+provided to developers. A general taxonomy of data processing
+code should be constructed, and then the maintaining and test-
+ing tools for it can be developed. More integration-level test
+cases should be created to cover component interactions. Test
+cases and test data should be used in combination to detect
+both non-ML specific and ML-specific bugs.
+In summary, this paper makes the following contributions.
+• We conduct an in-depth case study on Rasa to characterize
+its complexity and the impact of its complexity on testing.
+• We highlight practical implications to improve software
+engineering for ML-enabled systems.
+arXiv:2301.03837v1  [cs.SE]  10 Jan 2023
+
+II. BACKGROUND AND STUDY DESIGN
+We present the architecture of a typical task-oriented dialogue
+systems, an overview of Rasa, and our study design.
+A. Architecture of a Typical Task-Oriented Dialogue System
+A task-oriented dialogue system (TDS) aims to assist users in
+performing specific tasks, such as restaurant booking and flight
+booking [15]. A pipeline-based TDS consists of four parts, i.e.,
+natural language understanding (NLU), dialogue state tracking
+(DST), dialogue policy (Policy) and natural language generation
+(NLG) [86]. NLU parses a user utterance into a structured se-
+mantic representation, including intent and slot-values. The in-
+tent is a high-level classification of the user utterance, such as
+Inform and Request. Slot-values are task-specific entities
+that are mentioned in the user utterance, such as restaurant price
+range and location preference. After tokenization and featuriza-
+tion of the user utterance, NLU applies classification models to
+recognize intent, and named entity extraction models to extract
+slot-values. DST takes the entire history of the conversation,
+including both user utterances with predicted intents and slot-
+values and system responses, to estimate the current dialogue
+state, which is usually formulated as a sequential prediction task
+[77]. Dialogue state is typically the probability distribution of
+user intent and slot-values till the current timestamp. Given the
+estimated dialogue state, Policy generates the next system ac-
+tion, such as Query Database and Utter Question.
+As Policy determines a series of actions sequentially, sequential
+models such as Recurrent Neural Network (RNN) are applied.
+For actions that require a response, NLG converts the action
+into a natural language utterance, which is often considered as
+a sequence generation task [76].
+B. An Overview of Rasa 3.0
+Rasa is a popular open-source ML-enabled TDS, which is
+fully implemented with Python and used by many well-known
+companies in customer service for real users, including Adobe,
+Airbus, and N26 [11]. An architecture overview of Rasa 3.0 is
+shown in Fig. 1. Each module consists of one single component
+or multiple semantically similar components. Apart from the
+modules in a typical TDS, Rasa proposes the Selector module to
+select candidate intents and responses for FAQ questions [16].
+We present some concepts in Rasa 3.0 to ease our presentation.
+Components in Rasa. There are two types of components in
+Rasa. We define ML components as components that are im-
+plemented with ML models, and rule-based components as
+components that are implemented with rule-based code logic.
+General utils code in Rasa is not considered in this paper, such
+as command line and database access code.
+Configuration File and Component Graph in Rasa. As
+there are multiple available components in each module, de-
+velopers need to choose components that are actually used in
+the Rasa pipeline with a configuration file to build a chatbot.
+Parameters of each component are specified in the configuration
+file (e.g., ML model used by a component and hyperparamers of
+a ML model). Rasa applies Dask [6] to compile a configuration
+file into a component graph. Each node in the component graph
+denotes a component, and the edges connected with it denote up-
+stream and downstream components with input and output data
+dependency. Execution of components obeys the topological
+order specified by edges. These components interact with each
+other through fields in shared Message class instances. An
+upstream component stores outputs to Message instances, and
+a downstream component retrieves them for further processing.
+ML Stages in Rasa. Different from Apollo, which uses
+trained model files from external systems, and therefore only
+contains the inference stage of ML models [60], the training,
+evaluation and inference stages of ML models are all present in
+Rasa. Given a configuration file, Rasa separately compiles it to
+a training component graph and an inference component graph.
+In training stage, the trainable upstream components are first
+trained, and then process the training data used by downstream
+components. In evaluation stage, only the performance metrics
+of IntentClassifier, EntityExtractor and Policy are reported, as
+there is no ground truth for evaluation data in other modules.
+C. Study Design
+Our goal is to understand the complexity and its impact on
+testing in Rasa. To achieve this goal, we propose five RQs.
+• RQ1 System Complexity Analysis: how ML compo-
+nents are adopted across the modules in Rasa?
+• RQ2 Interaction Complexity Analysis: how ML compo-
+nents interact with other components in Rasa?
+• RQ3 Component Complexity Analysis: how the code of
+ML components is composed by what kinds of code?
+• RQ4 Testing Practice Analysis: how is the characteristic
+of test cases, and how well they cope with the complexity?
+• RQ5 Mutation Testing Analysis: how is the bug-finding
+capability of test cases and test data (i.e., the data for testing
+models), and how well they cope with the complexity?
+RQ1 aims to identify ML components in Rasa and broadly
+view them from the perspective of dependent libraries and ML
+models. RQ2 aims to summarize a comprehensive taxonomy
+of component interaction patterns. RQ3 aims to inspect the
+source code inside every component to characterize the statistics
+and composition patterns of different code types, including
+data processing code, model usage code, etc. Our findings
+from RQ1, RQ2 and RQ3 could reveal how the complexity
+originates and manifests in real world large-scale ML-enabled
+systems, which provide both practitioners and researchers with
+insights to overcome the complexities involved in implementing,
+maintaining, debugging and testing such complex systems.
+RQ4 aims to quantitatively assess Rasa’s test cases from code
+coverage, test case statistics (i.e., granularity levels, oracle types,
+and ML stages), and component interaction coverage perspec-
+tives. RQ5 aims to generate mutants (i.e., artificial bugs) and
+check whether these mutants can be killed (i.e., detected) by test
+cases. Further, for the survived mutants, we train Rasa with 3
+default configuration files on MutiWoz [15], a widely used multi-
+domain TDS dataset. We calculate the statistical significance
+between the performance metrics from mutated Rasa code with
+metrics from pipelines trained with clean code. Our findings
+from RQ4 and RQ5 evaluate the testing practice in Rasa, and
+
+DST
+“I'd like a moderate 
+Thai restaurant.”
+NLG
+[[0, 0, …, 1, 0]
+…,
+[1, 1, …, 0, 0 ]]
+Past user and
+bot events
+ask_location,0.957
+ask_numpeople, 0.013
+ask_moreupdates,0.002
+…
+"where?"
+Postprocess
+Tokenizer
+Preprocess
+Tokenizer 
+Module
+Token
+Postprocess
+Featurizer
+Preprocess
+Featurizer
+Module
+Entity
+Extractor
+Preprocess
+EntityExtractor
+Module
+Postprocess
+Entity
+Synonym 
+Mapper
+Intent
+Classifier
+Preprocess
+IntentClassifier
+Module
+Postprocess
+Fallback
+Classifier
+Postprocess
+Selector
+Preprocess
+Selector 
+Module
+[[entiy: price, value: mid,
+confidence_score: 0.9997],
+[entiy: cuisine, value:thai,
+confidence_score: 0.9993]]
+inform, 1.0
+request_info, 4.122e-18
+thankyou, 2.408e-18
+greet, 2.364e-18
+Policy
+Preprocess
+Policy 
+Module
+Postprocess
+Policy
+Ensemble
+NLU
+[I,d,like,a,moderate,
+Thai,restaurant]
+Selected 
+Response
+Features
+Intent 
+List
+Entity 
+list
+Action 
+List
+Bot
+Utterance
+Fig. 1: The Modules and Workflow of Rasa
+shed light on automated test generation, bug localization and
+bug repairing techniques for complex ML-enabled systems.
+III. RQ1: SYSTEM COMPLEXITY ANALYSIS
+A. Methodology
+To answer RQ1, we identified ML and rule-based com-
+ponents in Rasa and characterized them through a detailed
+examination of Rasa’s source code and documentation. We
+excluded DST and NLG as they are fully implemented with
+rule-based code logic in Rasa without ML components.
+All the modules we identified are listed in Table I, except for
+a special module, Shared, as it contains general data processing
+code and ML model definition code (e.g., Transformer), while
+does not contain any independent components. We will include
+it in the last three RQs. Specifically, for each component, we
+recursively tracked methods within it to manually extract the
+model or rule definition code. We examined implementation
+details of APIs in ML libraries by reading the documentations
+and source code of external libraries, including ML model type
+and number of candidate models.
+In particular, we analyzed whether ML components are
+implemented by using external direct libraries or indirect
+libraries, whether the components can be trained (notice that
+not only some of ML components can be trained, but also
+some rule-based components can be trained as long as they
+update internal parameters when processing training data),
+whether Rasa implements components with its own code and
+provides built-in model and rule definition code, and the lines
+of code (LoC) of each component excluding blank lines, code
+comments and import statements.
+B. Results
+The results are summarized in Table I. Components shown
+in gray color are ML components, and others are rule-based
+components. There are 6 modules in total, including 15 ML and
+14 rule-based components. These components contain 23 ML
+models and are implemented with 7 directly dependent external
+ML libraries and 3 indirectly dependent external ML libraries.
+In particular, all ML components in Tokenizer and Featurizer
+are not trainable because pre-trained language models are
+applied. All components in Policy are implemented in Rasa’s
+own code, because there are no ready-to-use Policy models
+provided by existing libraries. There are a total of 5348 LoC
+in ML components and 2980 LoC in rule-based components.
+In addition, the general module Shared contains 5375 LoC,
+which is not listed in the table.
+Notably, we find that classical machine learning models
+(e.g., Support Vector Machine and Conditional Random Field)
+together with deep learning models (e.g., Convolutional Neural
+Networks and Transformer) play an important role in Rasa.
+This is different from the previous study [60] on Apollo, which
+is focused on deep learning models.
+Next, we introduce components used in each module.
+Tokenizer. Tokenizer splits the user utterance into tokens
+with component specific split symbols (e.g., whitespace and
+punctuation). (1) SpacyTokenizer provides the richest token
+information, including splitting tokens with rules, lemmatizing
+tokens with a look-up table, and performing part-of-speech tag-
+ging with a multi-layer perceptron (MLP). (2) JiebaTokenizer
+is the only component that tokenizes non-English sentences
+using Hidden Markov Model (HMM) [20]. (3) MitieTokenizer
+and WhitespaceTokenizer toeknize text with predefined rules.
+Featurizer. As shown in Fig. 1, Featurizer converts tokens
+into features for downstream module inference. (1) ConveRT-
+Featurizer loads TFHub’s [27] pre-trained ConveRT (Conversa-
+tional Representations from Transformers) TensorFlow model
+to featurize tokens [29]. (2) LanguageModelFeaturizer loads
+
+TABLE I: System Complexity Analysis of Rasa
+Module
+Component
+Direct Lib.
+Indirect Lib.
+Model Type
+No. Model
+Trainable
+Rasa Imp.
+LoC
+Tokenizer
+JiebaTokenizer
+Jieba
+N/A
+HMM
+1
+False
+False
+85
+SpacyTokenizer
+Spacy
+Thinc
+MLP
+1
+False
+False
+39
+MitieTokenizer
+Mitie
+N/A
+N/A
+N/A
+False
+False
+43
+WhitespaceTokenizer
+N/A
+N/A
+N/A
+N/A
+False
+True
+52
+Featurizer
+ConveRTFeaturizer
+TensorFlow
+N/A
+Transformer
+1
+False
+False
+269
+LanguageModelFeaturizer
+Transformers
+TensorFlow
+Transformer
+6
+False
+False
+378
+MitieFeaturizer
+Mitie
+Dlib
+CCA
+1
+False
+False
+98
+SpacyFeaturizer
+Spacy
+Thinc
+CNN
+2
+False
+False
+66
+CountVectorsFeaturizer
+Scikit-learn
+N/A
+N/A
+N/A
+True
+False
+520
+LexicalSyntacticFeaturizer
+N/A
+N/A
+N/A
+N/A
+True
+True
+319
+RegexFeaturizer
+N/A
+N/A
+N/A
+N/A
+True
+True
+151
+IntentClassifier
+DIETClassifier
+TensorFlow
+N/A
+Transformer
+1
+True
+True
+1217
+MitieIntentClassifier
+Mitie
+Dlib
+SVM
+1
+True
+False
+89
+SklearnIntentClassifier
+Scikit-learn
+N/A
+SVM
+1
+True
+False
+173
+FallbackClassifier
+N/A
+N/A
+N/A
+N/A
+False
+True
+91
+KeywordIntentClassifier
+N/A
+N/A
+N/A
+N/A
+True
+True
+132
+EntityExtractor
+DIETClassifier
+TensorFlow
+N/A
+Transformer
+1
+True
+True
+1217
+CRFEntityExtractor
+Scikit-learn
+N/A
+CRF
+1
+True
+False
+438
+MitieEntityExtractor
+Mitie
+Dlib
+SVM
+1
+True
+False
+164
+SpacyEntityExtractor
+Spacy
+Thinc
+MLP
+1
+False
+False
+52
+DucklingEntityExtractor
+N/A
+N/A
+N/A
+N/A
+False
+False
+134
+RegexEntityExtractor
+N/A
+N/A
+N/A
+N/A
+True
+True
+124
+EntitySynonymMapper
+N/A
+N/A
+N/A
+N/A
+True
+True
+102
+Selector
+ResponseSelector
+TensorFlow
+N/A
+Transformer
+2
+True
+True
+560
+Policy
+TEDPolicy
+TensorFlow
+N/A
+Transformer+CRF
+1
+True
+True
+1262
+UnexpecTEDIntentPolicy
+TensorFlow
+N/A
+Transformer+CRF
+1
+True
+True
+458
+MemoizationPolicy
+N/A
+N/A
+N/A
+N/A
+True
+True
+207
+AugmentedMemoizationPolicy
+N/A
+N/A
+N/A
+N/A
+True
+True
+65
+RulePolicy
+N/A
+N/A
+N/A
+N/A
+True
+True
+818
+PolicyEnsemble
+N/A
+N/A
+N/A
+N/A
+False
+True
+150
+pre-trained language models from Hugging Face Transformers
+[22], including BERT [17], GPT [31], XLNet [79], Roberta
+[46], XLM [41] and GPT2 [56]. (3) MitieFeaturizer combines
+Canonical Correlation Analysis (CCA) feature and word mor-
+phology features together. (4) SpacyFeaturizer applies HashEm-
+bedCNN or Roberta to convert tokens to features, depending
+on the pre-trained Spacy pipeline specified in the configuration
+file. (5) CountVectorsFeaturizer, LexicalSyntacticFeaturizer and
+RegexFeaturizer create sparse features with n-grams, sliding
+window and regex patterns, respectively.
+IntentClassifier. IntentClassifier generates a predicted intent
+list ordered by confidence scores based on tokens and features
+from upstream modules. (1) DIETClassifier implements Dual
+Intent and Entity Transformer (DIET) to perform intent clas-
+sification and entity recognition simultaneously, and is there-
+fore included in both IntentClassifier and EntityExtractor
+modules. (2) MitieIntentClassifier and SklearnIntentClassifier
+apply a multi-class Support Vector Machine (SVM) [65] with
+a sparse linear kernel using Scikit-learn and Mitie, respectively.
+(3) KeywordIntentClassifier classifies user intent with keywords
+extracted from training data. (4) FallbackClassifier is a post-
+processing component to check the results of other components
+in DIETClassifier. It identifies a user utterance with the intent
+nlu_fallback if the confidence scores are not greater than
+threshold, or the score difference of the two highest ranked
+intents is less than the ambiguity_threshold.
+EntityExtractor. EntityExtractor extract entities such as the
+restaurant’s location and price. (1) DIETClassifier also serves
+as an EntityExtractor. (2) CRFEntityExtractor, MitieEntityEx-
+tractor and SpacyEntityExtractor utilize a conditional random
+fields (CRF) model, a multi-class linear SVM, and a MLP to
+predict entities, respectively. (3) DucklingEntityExtractor and
+RegexEntityExtractor extract entities using a duckling server
+[23] and regex patterns. (4) EntitySynonymMapper is a post-
+processing component to convert synonymous entity values into
+a same value. As Fig. 1 shows, the value of “price” entity,
+“moderate”, is coverted to “mid” by EntitySynonymMapper.
+Selector. ResponseSelector aims to directly select the re-
+sponse from a set of candidate responses, which is also known
+as response selection task in the literature [16]. It embeds user
+inputs and candidate responses in the same vector space, using
+the same neural network architecture as DIETClassifier.
+Policy. Policy decides the action the system takes on each
+conversation based on dialogue states. (1) TEDPolicy proposes
+a Transformer Embedding Dialogue (TED) model to embed
+dialogue states and system actions into a single semantic vector
+space, and select the action with the max similarity score with
+the current dialogue states [75]. (2) MemoizationPolicy, Aug-
+mentedMemoizationPolicy and RulePolicy match the current
+conversation history with examples in the training data and
+predefined rules to predict system actions. (3) UnexpecTEDIn-
+tentPolicy decides on the possibility of the intent predicted by
+IntentClassifier given current dialogue states, which follows
+the same model architecture as TEDPolicy. (4) PolicyEnsemble
+is a post-processing component to select the proper system
+action from output actions of different policies.
+
+C. Implications
+The system complexity of Rasa poses challenges for devel-
+opers using Rasa (i.e., application developers) and developers
+creating Rasa (i.e., system developers).
+Complexity from ML supply chain. Rasa depends on 10
+external
+ML
+libraries
+directly
+or
+indirectly.
+Less
+than 100 (0.03%) projects out of 355392 projects using
+TensorFlow on GitHub depend on 10 more DL libraries
+[69]. It could be inferred that relying on 10 more ML
+libraries is also less common. For application developers,
+it is difficult to understand the implementation details of
+components that rely on external ML libraries, not to mention
+selecting proper components and parameters. For example,
+due to the lack of documentations of MitieFeaturizer in
+Rasa, application developers need to inspect Mitie’s source
+code to learn that it implements CCA using Dlib APIs. For
+system developers, vulnerabilities [81] and dependency bugs
+[32] may arise because of outdated or incompatible library
+versions. Therefore, future work should provide supports for
+the management of components and corresponding dependent
+ML libraries for ML-enabled systems, similar to traditional
+software component analysis [26].
+Complexity from configurations. It could be extremely
+complex to configure Rasa with 29 components and hundreds
+of parameters, making it easy to misconfigure and thus affect
+functionality and performance. This kind of misconfiguration
+is similar to what happens in traditional configurable software
+systems [74]. Additionally, finding optimal configurations for
+application developers’ specific TDS scenarios is difficult, also
+known as configuration debt [63]. Although AutoML has been
+extensively studied to select appropriate ML models and param-
+eters for specific tasks, they all focus on selecting a single ML
+model without considering the combination of multiple ML
+models and rules [28]. Another challenge is to detect potential
+bugs by testing a huge set of configuration settings. Existing
+studies on ML model testing mainly focus on testing a single
+ML model with predefined hyperparameters [82].
+IV. RQ2: INTERACTION COMPLEXITY ANALYSIS
+A. Methodology
+The workflow in Fig. 1 only shows the general flow among
+different modules. The details of interactions among compo-
+nents are still uncovered. We consider the interaction among
+two or more components, with at least one ML component.
+An interaction pattern contains a module placeholder, which
+could be instantiated with components in the module to gener-
+ate interaction instances. For example, pattern (PolicyEnsemble,
+[Policy]) could be instantiated as (PolicyEnsemble, TEDPolicy)
+or (PolicyEnsemble, (TEDPolicy, RulePolicy)). To answer RQ2,
+we conducted a qualitative and quantitative analysis of the
+component interaction patterns and instances of components.
+Step 1: Extract interaction patterns. The interaction can
+be divided into two categories: inter-module interaction and
+intra-module interaction. (1) Inter-module interaction: the inter-
+action between two adjacent modules (e.g., Featurizer with To-
+kenizer) was considered. We analyzed the usages of Message
+Component Interaction
+(43/230)
+Inter-Module
+(33/172)
+Intra-Module 
+(8/56)
+Component
+Instantiation (N/A)
+Component
+Inheritance (2/2)
+Data 
+Dependency 
+(25/95)
+Confidence 
+Checking (2/50)
+Output 
+Selection (1/18)
+Output 
+Refinement 
+(1/4)
+Usage 
+Constraints (4/5)
+Functionality 
+Equivalence 
+(2/18)
+Priority Order 
+(1/7)
+Usage 
+Constraints 
+(5/31)
+Both ML-ML and
+ML-Rule
+Only ML-Rule
+Only ML-ML
+Fig. 2: Taxonomy of Component Interactions
+in the component code because components use the Message
+class to transfer data. Specifically, we extracted all interaction
+patterns of its upstream and downstream component. We also
+considered the interaction between post-processing compo-
+nent (i.e., FallbackClassifier, EntitySynonymMapper and Poli-
+cyEnsemble) and other components in their residing modules
+as inter-module interaction. (2) Intra-module interaction: we
+identify the interaction pattern for components in each module.
+Step 2: Generate interaction instances. For each inter-
+module interaction pattern, we instantiated the module place-
+holder with every component in the module. For every intra-
+module interaction, we extracted the Cartesian product of all
+components in each module as interaction instances. We then
+filtered out component instances that do not contain ML com-
+ponents, or do not meet the constraints specified in Rasa doc-
+umentation. For example, CRFEntityExtractor could not use
+features of SparseFeaturizer other than RegexFeaturizer.
+Step 3: Summarize the interaction taxonomy. For gen-
+erated component patterns and instances, we analyzed their
+semantics and summarized a component interaction taxonomy.
+B. Results
+The component interaction taxonomy is shown in Fig. 2. It
+is divided into 4 high-level categories (i.e. Inter-Module, Intra-
+Module, Component Instantiation and Component Inheritance)
+and 8 inner categories. Note that only Inter-Module interactions
+contain components with direct data dependency, while other
+categories contain components with indirect interactions (e.g.,
+two featurizers are used together). The number of interaction
+patterns and interaction instances in each category is listed as
+pattern count/instance count in Fig. 2. There are a total of
+43 interaction patterns and 230 interaction instances. Almost
+all categories include both ML to ML components and ML to
+rule-based components interactions. On the contrary, previous
+work on Apollo [60] also presented 4 of the 8 inner categories,
+but did not provide a taxonomy and quantitative analysis.
+Inter-Module. Components from multiple modules interact
+through data transfers. In particular, Output Selection means
+that the downstream component selects the proper output from
+multiple upstream outputs based on configurable criteria, e.g.,
+PolicyEnsemble with policies. Output Refinement denotes that
+
+the downstream component complements the imperfect outputs
+of upstream components with rules, e.g., EntitySynonymMapper
+with entity extractors. Confidence Checking means that the
+downstream component checks reliability of the output from up-
+stream components using ML models (e.g., UnexpecTEDIntent-
+Policy with intent classifiers) or rules (e.g., FallbackClassifier
+with IntentClassifiers). If the outputs are marked as not reliable,
+fallback behaviors such as the fall_back system action are
+triggered. Usage Constraints defines components that should
+or should not be used together under certain circumstances. For
+example, SpacyTokenizer is required by CountVectorsFeaturizer
+when applying use_lemma option and LexicalSyntacticFea-
+turizer when applying pos_tag option. Data Dependency
+includes the rest of inter-module interaction patterns that do
+not fall into any of the above categories, which are relatively
+“trivial” interactions with no specific semantics.
+Intra-Module. The interaction mode of components within a
+module differs from Inter-Module. These components interact
+indirectly when used together. Priority Order means that the
+outputs of components within a module are selected according
+to priority order, e.g., the priority order of policies. Usage
+Constraints is similar to Usage Constraints in the inter-module
+category. For example, only one component in any of Tokenizer,
+IntentClassifier and EntityExtractor should be used in each
+configuration file, otherwise outputs of additional components
+will be overwritten. Functionality Equivalence includes all
+intra-module interaction patterns that do not belong to any of
+the above categories, which are relatively “trivial” interactions
+involving components used together with no specific semantics.
+Component Instantiation. Rasa supports creating multiple
+instances of a component within a configuration setting. For
+example, multiple CountVectorFeaturizers instances with dif-
+ferent ngram settings, and multiple LanguageModelFeaturizer
+instances with different language models could be used together.
+We did not count this category of interaction patterns and
+instances, since developers could specify infinite instances of
+a component within a configuration setting.
+Component Inheritance. The class inheritance mechanism
+allows ML models to be shared among components. For ex-
+ample, ML model definition class in UnexpecTEDIntentPolicy
+is a subclass of the ML model definition class in TEDPolicy.
+C. Implications
+Lack of specifications for interactions. The outputs of ML
+components for specific inputs are not guaranteed due to the
+stochastic nature of ML models [8]. Thus, it is more difficult
+to formulate interaction semantics in ML-enabled systems than
+in traditional systems. When testing samples are predicated
+wrongly, it is challenging to localize the exact faulty component.
+Moreover, even if the faulty component has been fixed and
+performance of it has been improved, the overall performance
+of the entire system may degrade [78]. Therefore, training
+and evaluation should be extended from component-level to
+system-level to consider interactions among components. In
+summary, we need to pay more attention to addressing the
+TABLE II: LoC of Different Code Categories
+Module
+Data
+Model
+Rule
+Pre.
+Post. Usage Definition Usage Definition
+Tokenizer
+8
+80
+27
+0
+25
+25
+Featurizer
+390
+323
+92
+0
+162
+119
+IntentClassifier
+441
+131
+113
+298
+3
+69
+EntityExtractor
+746
+311
+120
+298
+24
+30
+Selector
+48
+55
+9
+16
+0
+0
+Policy
+1332
+540
+64
+543
+167
+283
+Shared
+996
+314
+112
+1673
+0
+43
+Total
+3961 1754
+537
+2828
+381
+569
+challenges caused by lack of specifications in bug localization
+and repairing for ML-enabled systems.
+Hidden interactions. It is non-trivial to identify all inter-
+actions even for system developers of Rasa. For example, the
+Data Dependency interaction between RegexFeaturizer and CR-
+FEntityExtractor is not marked in documentations and can only
+be identified from source code. Application developers may
+misuse components and get confused with the poor performance
+of the system without understanding the hidden interactions,
+especially for interaction categories like Usage Constraints,
+Output Selection and Priority Order. Techniques like data flow
+analysis can be explored to automatically reveal component
+interactions in ML-enabled systems [62].
+Furthermore, our results on component interaction complex-
+ity could be helpful to guide developers to build a better ML-
+enabled system. For example, developers can follow interaction
+patterns Output Selection and Output Refinement to improve
+the outputs of components at system level, as well as utilizing
+Confidence Checking to detect cases that ML models can not
+handle, and then triggering fallback rules, which is very impor-
+tant in safety-critical systems like self-driving systems [60].
+V. RQ3: COMPONENT COMPLEXITY ANALYSIS
+A. Methodology
+To answer RQ3, we classified categories of code snippets
+in each component and explored their composition patterns.
+Step 1: Label code snippets. We segmented each source
+code file into code snippets according to semantic meaning, and
+then classified them into 6 categories: (1) model definition, the
+definition code of ML models; (2) rule definition, the definition
+code of rules in rule-based components; (3) model usage, the
+usage code of ML models; (4) rule usage, the usage code
+of rules; (5) data pre-processing, the input data processing
+code before model or rule usages; (6) data post-processing, the
+output data processing code after model or rule usages. Two of
+the authors labeled code snippets independently, and the third
+author was involved to resolve disagreements. The Cohen’s
+Kappa coefficient of the two authors reached 0.830.
+Step 2: Summarize composition patterns of code snip-
+pets. Based on labeled code snippets, we summarized the
+composition patterns of data processing code, and model or
+rule usage code in each component.
+B. Results
+The statistics of different code categories are shown in Table
+II. We only considered the LoC of labeled code snippets, while
+
+Select
+Select
+One
+Select
+Select
+One
+Select
+Fig. 3: Non-Sequential Code Composition Patterns within Components
+ignoring general utils code such as class initialization. Data
+processing code contributes a total of 5715 (57.1%) LoC, while
+model usage&definition code and rule usage&definition code
+contribute 3365 (33.5%) and 950 (9.4%) LoC, respectively.
+1673 (59.2%) of the 2828 LoC of model definition code is in
+Shared module, which shows that the reuse of model definition
+code between different components is quite common. There is
+no model definition code in Tokenizer and Featurizer, because
+ML components are all built on top of external ML libraries.
+We classified data pre-processing and data post-processing
+categories into more specific types, due to the dominant propor-
+tion of data processing code in Rasa. Specifically, Validation
+code intends to validate the input or output data of components.
+Format Transformation code transforms data format, such as
+constructing vectors from Python arrays and reshaping vectors.
+Component Input/Output Filter code filters data that does not
+meet the specified criteria, such as the absence of certain at-
+tributes. Data Scale/Padding/Encoding/Decoding code changes
+the value of data, while Data Split/Shuffle/Balance/Batch/Rank
+code changes the organization of data for better training and
+inference of components. We provide the complete codebook
+and statistics of data processing types at our website [7].
+Moreover, we find that composition patterns of code snippets
+include sequential code composition pattern and various non-
+sequential composition patterns. In a typical sequential compo-
+sition pattern, data is first pre-processed, and then processed
+by model or rule usage code, and finally post-processed. The
+non-sequential code composition patterns are summarized in
+Fig. 3. The black box is a data processing code snippet, the
+red box is a model or rule usage code snippet, and the green
+diamond means to select one or multiple downstream code
+snippets. The first 5 patterns consist of multiple model or rule
+usages in one component. The last 3 patterns consist of a single
+model or rule usage with multiple possible data processing
+snippets, decided by configurations or input data.
+C. Implications
+Data processing. Data processing code is scattered at dif-
+ferent granularity levels, unlike the well-documented and struc-
+tured code of ML models and rules. In detail, data processing
+code includes data processing components (e.g., PolicyEnsem-
+ble), general data processing classes and functions in Shared
+module, and specific data processing snippets in components
+entangled with model or rule usages. On the one hand, it could
+become troublesome to identify and understand the semantics of
+all data processing code for application developers. A specific
+example is that data pre-processing code also exists in model
+definition class of TransformerRasaModel, including
+Formant Conversion and Data Batch code. It could be explicitly
+helpful to automatically extract and analyze the semantics of
+data processing code with techniques like program analysis [61].
+On the other hand, it would be challenging to maintain and test
+data processing code for system developers, possibly resulting
+in severe consequences with ML development paradigm shift
+from model-centric to data-centric [45]. In general, building
+a taxonomy of data processing code would be helpful for the
+maintaining and testing of data processing code.
+Code composition patterns. These non-sequential compo-
+sition patterns could introduce additional dynamic complexity
+for ML-enabled systems, e.g., it is too expensive to capture
+all possible run-time compositions of code snippets with static
+analysis. Although dynamic testing is widely adopted to com-
+plement the limitations of static analysis in traditional software
+[24], most existing testing techniques tailored for ML only
+target at ML model level [82]. It would be beneficial to extend
+them to include data processing code and composition patterns.
+VI. RQ4: TESTING PRACTICE ANALYSIS
+A. Methodology
+To answer RQ4, test cases were inspected in three steps.
+Step 1: Label test cases. We manually labeled the granular-
+ity level, oracle type and ML stage of each test case. There are
+three different granularity levels of test cases: (1) Method-level:
+testing single or multiple methods; (2) Component-level: testing
+the complete process of a component in training or inference
+stage; (3) Integration-level: testing the current component with
+upstream components. There are four test oracle types: (1)
+Given input-output pairs: the input and expected output data are
+given; (2) Component-specific constraints: the constraints must
+be satisfied according to the implementation of a component,
+i.e., the sum of confidence scores of the intent list generated by
+classifiers should equal to 1; (3) Differential executions: outputs
+of executions under different settings should change or remain
+the same, i.e., given the same input, the outputs of an original
+ML model and its loaded version from disk should remain
+the same; (4) Exception: whether or not the test case throws
+exceptions for certain inputs and configurations. Finally, the
+ML stages covered by test cases include training, inference and
+evaluation stages. To test the training stage of a component,
+test cases must first train it and check whether any test oracle
+is violated. Note that there may exist several oracle types and
+ML stages but only one granularity level for each test case in
+
+TABLE III: Code Coverage and Labeled Statistics of Test Cases
+Module
+Total
+Test Case Type
+Test Case Stage
+Oracle Type
+Code Coverage
+Meth.
+Comp.
+Integ.
+Infer.
+Train
+Eval.
+I-O
+C-S
+Diff.
+Exception
+Stat. Cov.
+Bran. Cov.
+Tokenizer
+27
+7
+20
+0
+25
+14
+0
+24
+1
+0
+3
+97.4%
+96.8%
+Featurizer
+62
+13
+14
+35
+56
+40
+0
+46
+5
+3
+8
+95.7%
+94.9%
+IntentClassifier
+36
+7
+15
+14
+29
+30
+0
+18
+11
+7
+1
+92.5%
+89.5%
+EntityExtractor
+41
+6
+19
+16
+34
+31
+0
+18
+14
+8
+1
+92.3%
+90.1%
+Selector
+13
+4
+6
+3
+9
+12
+0
+6
+5
+1
+1
+68.3%
+66.4%
+Policy
+165
+77
+88
+0
+105
+127
+0
+117
+51
+0
+20
+95.7%
+94.5%
+Shared
+138
+129
+2
+7
+90
+84
+0
+89
+42
+2
+16
+92.3%
+91.4%
+Total
+461
+240
+156
+65
+331
+317
+47
+312
+123
+15
+49
+93.2%
+92.0%
+TABLE IV: Test Coverage of Component Interactions
+Category
+Sub-Category
+Cov. Patterns Cov. Instances
+Inter-module
+Data Dependency
+9/25
+17/95
+Confidence Checking
+0/2
+0/50
+Output Selection
+0/1
+0/18
+Output Refinement
+1/1
+1/4
+Usage Constraints
+3/3
+3/3
+Intra-module Functionality Equivalence 2/2
+3/18
+Prioriy Order
+1/1
+4/7
+Usage Constraints
+2/2
+2/4
+Total
+18/37
+30/199
+Rasa. Two of the authors labeled test cases independently, and
+the third author was involved to resolve disagreements. The
+Cohen’s Kappa coefficient of granularity level, test oracle and
+ML stage is 0.907, 0.908, and 0.854, respectively.
+Step 2: Collect code coverage of test cases. We collected
+the statement coverage and branch coverage of code via pytest-
+cov, because Rasa uses pytest to run test cases.
+Step 3: Collect interaction pattern coverage of test cases.
+We injected logging statements into methods of every compo-
+nent, and then executed test cases to collect the co-executed
+component sets of each test case. All component interaction
+instances, except Model Inheritance instances and some Usage
+Constraints instances that cannot be used together, were tried to
+be matched with these component sets. The matched interaction
+patterns and instances were considered as covered by test cases.
+B. Results
+The code coverage and labeled statistics of test cases are
+shown in Table III. (1) The total statement coverage and branch
+coverage of code reach 93.2% and 92.0%, which is much higher
+than 21.5% and 13.3% in Apollo [60]. (2) The coverage of
+Selector is only 68.3% and 66.4%, because Selector has two
+candidate ML models while only one of them was tested. (3)
+There are 240 (52.0%) method-level, 156 (33.8%) component-
+level and 65 (14.1%) integration-level test cases. (4) There is no
+integration-level test cases in Policy, because Policy was tested
+with given intents and entities input from developers without
+the dependency of NLU modules. (5) Inference and training
+stages have similar test case quantities. (6) Only test cases in
+Shared module cover evaluation stage, because Shared module
+provides the evaluation code for all components. (7) There are
+312 (67.7%), 123 (26.3%), 15 (3.3%), and 49 (10.6%) test cases
+with given input-output pairs, component-specific constraints,
+differential executions and exception test oracles.
+As Table IV shows, the test coverage of interactions is rela-
+tively low, i.e., 18 (48.6%) of 37 patterns and 30 (15.1%) of 199
+instances are covered. This is because only integration-level
+tests cover components interactions. In particular, Confidence
+Checking and Output Selection are not covered.
+C. Implications
+Low test coverage of component interactions. It is difficult
+to achieve a high test coverage of component interactions, due
+to the complexity caused by huge configuration space and hid-
+den interactions. The only test cases that cover component inter-
+actions (i.e., integration-level test cases) contribute no more than
+15% test cases. Yet, integration-level test cases can cover and
+kill more mutants than component-level and method-level test
+cases, as many mutants do not manifest in non-integration-level
+test cases [42]. Therefore, it is crucial to generate integration-
+level test cases for ML-enabled systems.
+Limited test oracle types. It is challenging and time-
+consuming to write test cases with given input-output pairs and
+component-specific constraints oracles, due to the complexity
+brought by lacking of specification for interactions. As a result,
+those test cases without the need of specification of interactions,
+that is, differential executions and exception test oracles, have
+been widely utilized to tackle the oracle problem in test case
+generation techniques for traditional software, such as differ-
+ential testing [21], fuzzing [44] and search-based testing [50].
+Besides, we find that test cases with these two oracles have a
+similar capability to kill mutants similar to component-specific
+constraints oracle (see RQ5). In spite of this, only 13.9%
+test cases in Rasa are written with differential executions and
+exception test oracles, implying that there could be a big room
+to apply these two test oracle types in test case generation
+techniques for ML-enabled systems.
+VII. RQ5: MUTATION TESTING ANALYSIS
+A. Methodology
+To answer RQ5, we performed an analysis of mutation test-
+ing [37]. It applies mutators to generate versions of faulty code,
+i.e., mutants. For every mutant, test cases were executed to col-
+lect the testing results to decide whether the mutant was killed.
+As Rasa contains both ML components and rule-based com-
+ponents, we considered both mutators for traditional software
+(i.e., syntactic mutators) and ML-specific mutators. As Table V
+shows, we used 9 syntactic mutators from Jia et al. [36] and
+11 ML specific mutators from DeepCrime [33].
+We list steps of mutation analysis in the following.
+Step 1: Generate mutants. We generated syntactic mutants
+using mutmut [51]. We used two groups of syntactic mutators,
+i.e., Logic and Value, which mutate the logic flow and variable
+
+value. Besides, we generated ML specific mutants with Deep-
+Crime [33]. We used 4 of 8 mutation groups in DeepCrime
+(Activation, Regularization, Weights and Optimization). For
+others, mutators in Training Data and Validation groups are not
+the focus of this paper; Hyperparameters group is not included,
+as hyperparameters in Rasa are specified with configuration files
+by developers; and Loss Function group is not applicable, as
+the loss functions in Rasa are all implemented from scratch,
+while the mutators provided by DeepCrime are only to replace
+the Keras loss function API with another one. Besides, we
+only generated mutants for 6 labeled code cateogries in RQ3,
+excluding general utils code. We generated no more than 30
+mutants for every Python class to reduce potential bias. We
+also only modified one AST node for every mutant.
+Step 2: Perform mutation testing analysis with test cases.
+For every mutant, only test cases that cover the mutated line
+were collected (from test coverage data in RQ4) and executed
+to save running time. If any test case fails on a mutant, the
+mutant is considered as killed by the test case. Otherwise, the
+mutant is considered as survived. A test case could fail with
+three symptoms: (1) an assertion fails; (2) an execution or
+runtime error manifests; and (3) the test case times out. The
+maximum time for a test case to run is 10 times of its running
+time in the original clean code version. Besides, test cases
+were executed 3 times for every mutant to avoid flaky tests.
+We found that all test case statuses remain same for three runs.
+Step 3: Perform mutation testing analysis with test data.
+For those survived mutants in Step 2, we assessed the impact
+of them with 3 default configuration files and the restaurant
+domain data in Multiwoz [15], which is a widely used multi-
+domain dataset to evaluate the performance of TDS. Given a
+configuration file, only components specified in it are included
+in the Rasa pipeline, thus mutated nodes of some survived mu-
+tants from Step 2 will not be executed as they are not impacted
+by the configuration. Due to the stochastic nature of machine
+learning programs, we trained both the mutated program and
+original program for 5 times with 80/20 data split into train/test
+data randomly, and decided whether the performance metrics of
+two versions are statistically significant with non-negligible and
+non-small effect size. We followed the same formula to decide
+whether a mutant is killed with the test data as [33, 35], with
+the threshold of significance value is 0.05 and of effect size is
+0.5. We adopted F1 scores of IntentClassifier, EntityExtractor
+and Policy as performance metrics, i.e., if the F1 score in any
+of the three modules is statistically different between two code
+versions, the mutant is marked as killed by test data.
+B. Results
+The mutation testing results by each mutator are shown in Ta-
+ble V. There are 1447 mutants generated, 1106 (76.4%) mutants
+killed by test cases, 341 (23.6%) mutants survived, 146 (10.1%)
+mutants impacted, and 22 (1.5%) mutants killed by test data.
+Only 146 mutants from 341 survived mutants impact the default
+3 Rasa pipelines, which shows that the huge configuration space
+is challenging to be tested adequately. 81.3% syntactic mutants
+and 20.0% ML specific mutants are killed by test cases, while
+TABLE V: Mutation Testing Results
+Mutation Group Operator Total
+Test Case
+Test Data
+Killed Survived Impacted Killed
+Logic
+ArOR
+109
+86
+23
+10
+1
+ComOR
+109
+88
+21
+6
+0
+LogOR
+145
+112
+33
+14
+0
+AsOR
+20
+19
+1
+0
+0
+MemOR
+32
+30
+2
+0
+0
+KVR
+12
+7
+5
+1
+0
+Value
+BVR
+64
+32
+32
+9
+0
+NVR
+224
+180
+64
+18
+2
+AsVR
+582
+525
+67
+10
+0
+Activation
+ACH
+22
+3
+19
+18
+6
+ARM
+2
+0
+2
+1
+0
+AAL
+22
+11
+11
+11
+2
+Regularization
+RAW
+6
+0
+6
+3
+3
+RCW
+10
+0
+10
+10
+0
+RRW
+5
+0
+5
+5
+0
+Weights
+WCI
+24
+10
+14
+13
+1
+WAB
+4
+0
+4
+3
+1
+WRB
+3
+1
+2
+2
+0
+Optimization
+OCH
+24
+2
+22
+9
+6
+OCG
+8
+0
+8
+3
+0
+Total
+1447
+1106
+341
+146
+22
+TABLE VI: Mutant Location Results
+Location
+Total
+Test Case Result
+Test Data Result
+Killed
+Survived
+Impacted
+Killed
+Data Prep.
+385
+326
+59
+23
+0
+Data Post.
+271
+222
+49
+5
+0
+Model Usage
+307
+243
+64
+4
+0
+Model Def.
+364
+224
+140
+99
+22
+Rule Usage
+4
+4
+0
+0
+0
+Rule Def.
+115
+101
+14
+4
+0
+TABLE VII: Test Case Mutation Results
+Category
+Type
+Test Num. Strong Test Num. Covered Killed
+Granularity
+Method
+240
+59
+947
+635
+Component
+156
+31
+1121
+709
+Integration
+65
+29
+903
+613
+Stage
+Infer.
+331
+86
+1358
+995
+Training
+317
+75
+1184
+847
+Evaluation
+47
+11
+772
+476
+Oracle Type
+I-O
+312
+98
+1298
+956
+C-S
+123
+19
+1103
+707
+Diff
+15
+3
+625
+338
+Exception
+49
+6
+686
+352
+4.4% syntactic mutants and 24.4% ML specific mutants from
+impacted mutants are killed by test data. It shows that test
+case is much more effective to detect syntactic mutants and
+slightly less effective to detect ML specific mutants than test
+data. The killed syntactic mutants and ML-specific mutants
+by test data cause the F1 score degradation of IntentClassifier,
+EntityExtractor and Policy by 20.8%, 0.8%, 3.6% and 11.1%,
+13.4%, 5.7% on average.
+The mutation testing results w.r.t. the location of mutants
+are shown in Table VI. 224 (61.5%) of 364 mutants in model
+definition code, and 896 (82.8%) of 1082 mutants in other code
+categories are killed by test cases. In particular, few mutants
+in code categories except model definition are impacted and
+killed by test data, which implies that test data is only effective
+to kill mutants in model definition code.
+We investigated the capability to detect mutants w.r.t. differ-
+ent categories of test cases, by calculating the ratio of strong
+test case number to all test case number, and the ratio of killed
+mutants to covered mutants of them. We define strong test
+
+case as the test case that kills equal or more than 75% of its
+covered mutants. As Table VII shows, test cases in integration
+level have the highest ratio of strong test case (44.6%) and
+highest ratio of killed mutants (67.9%) among three granularity
+levels. Test cases with given input-output test oracle have the
+highest ratio of strong test case (31.4%) and highest ratio of
+killed mutants (73.7%) among four oracle types, while test
+cases with other three oracle types have similar ratios.
+C. Implications
+Non-ML specific bugs and test cases in ML-enabled sys-
+tems. Complexity from data processing code causes that non-
+ML specific bugs are prone to be introduced. Compared with
+test data, test case is more effective to detect syntactic mutants,
+i.e., non-ML specific bugs. Moreover, it is notorious for develop-
+ers to analyze, localize and fix bugs in ML programs according
+to test data, thus interpreting [85], debugging [3] and repairing
+[67] techniques have been developed for ML models. It is easier
+for developers to localize and fix bugs with failed test cases by
+analyzing violated test oracles. Thus, we claim that non-ML
+specific bugs and test cases in ML-enbaled systems should be
+paid more attention to. Although there is a rich set of test cases
+in Rasa that achieve high code coverage, the kill ratio of mutants
+remains to be improved (76.4%), especially of ML-specific
+mutants (29.8%). The applicability and limitations of existing
+test case generation, selection and quality assurance techniques
+in ML-enabled systems are worthwhile to be explored [18, 39].
+Challenges of test data to kill mutants. Existing researches
+on mutation testing for ML programs only evaluated mutants
+with test data to decide whether they can be killed [30, 33, 35,
+36, 48]. However, the capability of test data to kill mutants in
+large-scale ML-enabled systems is limited for two reasons. First,
+due to complexity from configurations, only part of mutants will
+impact the components of actual configured systems. Second,
+the amount and distributions of training data and test data affect
+the results a lot. For example, we tried to train the clean code
+version and mutated version with 75% of original training data,
+the number of killed mutants changed from 22 to 83, which
+means some bugs may only manifest under specific training
+data settings. Therefore, system developers should evaluate and
+test ML-enabled systems under more possible configurations
+and data settings that may be used by application developers
+to detect potential bugs.
+VIII. THREATS
+First, our study conducts a case study on Rasa, a widely used
+task-oriented industrial dialogue system. It is not clear whether
+our results can be generalized to other ML-enabled systems.
+However, we believe it is a good start to take a system view for
+ML-enabled systems. Second, our study involves a lot of man-
+ual analyses of Rasa source code and documentations, which
+may incur biases. To reduce them, two of the authors conduct
+manual analysis separately, and a third author is involved to
+resolve disagreements. Third, the mutators that we adopt may
+not simulate real-world bugs. To mitigate it, we decide to use
+mutators from DeepCrime [33], whose mutators are actually
+summarized from real word ML bugs.
+IX. RELATED WORK
+Study of ML-Enabled Systems. While much of the at-
+tention has been on ML models, less attention has been
+paid on system-level analysis [38]. Peng et al. [60] inves-
+tigated the integration of ML models in Apollo by analyzing
+how ML models interact with the system and how is the
+current testing effort. Besides, Nahar et al. [52] explored
+collaboration challenges between data scientists and software
+engineers through interviews. Amershi et al. [5] and Bernardi
+et al. [10] reported challenges and practices of MLOps
+(from model requirement to model monitoring) at Microsoft
+and Booking.com. Although they still take a model-centric
+view, they emphasize that models can be complexly entangled to
+cause non-monotonic errors [5] and model quality improvement
+does not necessarily indicate system value gain [10]. Further,
+Yokoyama [80] developed an architectural pattern to separate
+ML and non-ML components, while Serban and Visser [64]
+surveyed architectural challenges for ML-enabled systems.
+Sculley et al. [63] identified ML-specific technical debt in ML-
+enabled systems, while Tang et al. [70] further derived new ones
+from real-world code refactorings. In addition, some attempts
+were made on the problem of ML component entanglement [5],
+e.g., performing metamorphic testing on a system with two ML
+components [83], troubleshooting failures in a system with three
+ML components by human intellect [53], and decomposing er-
+rors in a system with two or three ML components [78]. These
+studies explore the interaction among models but only on simple
+systems. Moreover, Abdessalem et al. [1, 2] studied the feature
+interaction failures in self-driving systems, and proposed testing
+and repairing approaches to automatically detect and fix them.
+Apel et al. [8] also discussed feature interactions in ML-enabled
+systems, and suggested strategies to cope with them.
+The main difference from the previous work is that we take a
+large-scale complex ML-enabled system, explore its complexity
+at three levels, and analyze the impact of its complexity on test-
+ing. The closest work is Peng et al.’s [60], but we report a deeper
+complexity analysis and also conduct a testing impact analysis.
+Mutation Testing for DL Models. Jia et al. [36] used syn-
+tactic mutators for traditional programs to DL models. DeepMu-
+tation [48] and DeepMutation++ [30] defined DL-specific mu-
+tators. DeepCrime [33] derived DL-specific mutators based on
+real DL bugs. Jahangirova and Tonella [35] evaluated syntactic
+and DL-specific mutators. These studies are focused on model-
+level mutation, while we target at system-level mutation.
+Testing for Dialogue Systems. Bozic and Wotawa [13] pro-
+posed a security testing approach for chatbots to prevent cross-
+site scripting and SQL injection. Bozic et al. [12] tested a hotel
+booking chatbot via planning. Bozic and Wotawa [14] intro-
+duced a metamorphic testing approach for chatbots. Similarly,
+Liu et al. [47] used semantic metamorphic relations to test the
+NLU module in dialogue systems. Despite the effort, less atten-
+tion has been paid on system-level testing of dialogue systems.
+
+X. CONCLUSION
+We present a comprehensive study on Rasa to characterize
+its complexity at three levels and the impact of its complexity
+on testing from two perspectives. Furthermore, we highlight
+practical implications to improve software engieering for ML-
+enabled systems. All study data and source code used in this
+paper are available at https://rasasystemcomplexity.github.io/.
+REFERENCES
+[1] R. B. Abdessalem, A. Panichella, S. Nejati, L. C. Briand, and T. Stifter,
+“Testing autonomous cars for feature interaction failures using many-
+objective search,” in Proceedings of the 33rd ACM/IEEE International
+Conference on Automated Software Engineering, 2018, p. 143–154.
+[2] ——, “Automated repair of feature interaction failures in automated
+driving systems,” in Proceedings of the 29th ACM SIGSOFT International
+Symposium on Software Testing and Analysis, 2020, pp. 88–100.
+[3] A. Abid, M. Yuksekgonul, and J. Zou, “Meaningfully debugging model
+mistakes using conceptual counterfactual explanations,” in Proceedings
+of the International Conference on Machine Learning, 2022, pp. 66–88.
+[4] A. Aggarwal, P. Lohia, S. Nagar, K. Dey, and D. Saha, “Black box
+fairness testing of machine learning models,” in Proceedings of the 2019
+27th ACM Joint Meeting on European Software Engineering Conference
+and Symposium on the Foundations of Software Engineering, 2019, p.
+625–635.
+[5] S. Amershi, A. Begel, C. Bird, R. DeLine, H. Gall, E. Kamar,
+N. Nagappan, B. Nushi, and T. Zimmermann, “Software engineering for
+machine learning: A case study,” in Proceedings of the 41st International
+Conference on Software Engineering: Software Engineering in Practice,
+2019, pp. 291–300.
+[6] Anaconda. (2022) Dask. [Online]. Available: https://docs.dask.org/en/
+stable/
+[7] Anonymous. (2022) Understanding the complexity and its impact
+on
+testing
+in
+ml-enabled
+systems.
+[Online].
+Available:
+https:
+//rasasystemcomplexity.github.io/
+[8] S. Apel, C. K¨astner, and E. Kang, “Feature interactions on steroids:
+On the composition of ml models,” IEEE Software, vol. 39, no. 3, pp.
+120–124, 2022.
+[9] T. Baluta, Z. L. Chua, K. S. Meel, and P. Saxena, “Scalable quantitative
+verification for deep neural networks,” in Proceedings of the 43rd Inter-
+national Conference on Software Engineering: Companion Proceedings,
+2021, p. 248–249.
+[10] L. Bernardi, T. Mavridis, and P. Estevez, “150 successful machine learning
+models: 6 lessons learned at booking.com,” in Proceedings of the 25th
+ACM SIGKDD International Conference on Knowledge Discovery &
+Data Mining, 2019, p. 1743–1751.
+[11] T. Bocklisch, J. Faulkner, N. Pawlowski, and A. Nichol, “Rasa: Open
+source language understanding and dialogue management,” CoRR, vol.
+abs/1712.05181, 2017.
+[12] J. Bozic, O. A. Tazl, and F. Wotawa, “Chatbot testing using ai planning,”
+in Proceedings of the IEEE International Conference On Artificial
+Intelligence Testing, 2019, pp. 37–44.
+[13] J. Bozic and F. Wotawa, “Security testing for chatbots,” in Proceedings
+of the IFIP International Conference on Testing Software and Systems,
+2018, pp. 33–38.
+[14] ——, “Testing chatbots using metamorphic relations,” in Proceedings
+of the IFIP International Conference on Testing Software and Systems,
+2019, pp. 41–55.
+[15] P. Budzianowski, T.-H. Wen, B.-H. Tseng, I. Casanueva, S. Ultes,
+O. Ramadan, and M. Gaˇsi´c, “Multiwoz - a large-scale multi-domain
+wizard-of-oz dataset for task-oriented dialogue modelling,” in Proceedings
+of the Conference on Empirical Methods in Natural Language Processing,
+2018, pp. 5016–5026.
+[16] D. Chaudhuri, A. Kristiadi, J. Lehmann, and A. Fischer, “Improving
+response selection in multi-turn dialogue systems by incorporating domain
+knowledge,” in Proceedings of the 22nd Conference on Computational
+Natural Language Learning, 2018, pp. 497–507.
+[17] J. Devlin, M.-W. Chang, K. Lee, and K. Toutanova, “Bert: Pre-training
+of deep bidirectional transformers for language understanding,” arXiv
+preprint arXiv:1810.04805, 2018.
+[18] D. Di Nardo, N. Alshahwan, L. Briand, and Y. Labiche, “Coverage-based
+test case prioritisation: An industrial case study,” in 2013 IEEE Sixth
+International Conference on Software Testing, Verification and Validation,
+2013, pp. 302–311.
+[19] S. Dola, M. B. Dwyer, and M. L. Soffa, “Distribution-aware testing
+of neural networks using generative models,” in Proceedings of the
+IEEE/ACM 43rd International Conference on Software Engineering,
+2021, pp. 226–237.
+[20] S. R. Eddy, “Hidden markov models,” Current opinion in structural
+biology, vol. 6, no. 3, pp. 361–365, 1996.
+[21] R. B. Evans and A. Savoia, “Differential testing: a new approach to
+change detection,” in The 6th Joint Meeting on European software
+engineering conference and the ACM SIGSOFT Symposium on the
+Foundations of Software Engineering: Companion Papers, 2007, pp.
+549–552.
+[22] H. Face. (2022) Transformers. [Online]. Available: https://huggingface.
+co/docs/transformers/index
+[23] Facebook. (2022) Duckling. [Online]. Available: https://github.com/
+facebook/duckling/
+[24] R. E. Fairley, “Tutorial: Static analysis and dynamic testing of computer
+software,” Computer, vol. 11, no. 4, pp. 14–23, 1978.
+[25] Y. Feng, Q. Shi, X. Gao, J. Wan, C. Fang, and Z. Chen, “Deepgini:
+Prioritizing massive tests to enhance the robustness of deep neural
+networks,” in Proceedings of the 29th ACM SIGSOFT International
+Symposium on Software Testing and Analysis, 2020, p. 177–188.
+[26] D. Foo, J. Yeo, H. Xiao, and A. Sharma, “The dynamics of software
+composition analysis,” CoRR, vol. abs/1909.00973, 2019.
+[27] Google. (2022) Tensorhub. [Online]. Available: https://tensorflow.google.
+cn/hub
+[28] X. He, K. Zhao, and X. Chu, “Automl: A survey of the state-of-the-art,”
+Knowledge Based Systems, vol. 212, 2021.
+[29] M. Henderson, I. Casanueva, N. Mrkˇsi´c, P.-H. Su, T.-H. Wen, and
+I. Vuli´c, “Convert: Efficient and accurate conversational representations
+from transformers,” CoRR, 2019.
+[30] Q. Hu, L. Ma, X. Xie, B. Yu, Y. Liu, and J. Zhao, “Deepmutation++: A
+mutation testing framework for deep learning systems,” in Proceedings
+of the 34th IEEE/ACM International Conference on Automated Software
+Engineering, 2019, pp. 1158–1161.
+[31] Z. Hu, Y. Dong, K. Wang, K.-W. Chang, and Y. Sun, “Gpt-gnn: Generative
+pre-training of graph neural networks,” in Proceedings of the 26th ACM
+SIGKDD International Conference on Knowledge Discovery & Data
+Mining, 2020, pp. 1857–1867.
+[32] K. Huang, B. Chen, S. Wu, J. Cao, L. Ma, and X. Peng, “Demystifying
+dependency bugs in deep learning stack,” CoRR, vol. abs/2207.10347,
+2022.
+[33] N. Humbatova, G. Jahangirova, and P. Tonella, “Deepcrime: Mutation
+testing of deep learning systems based on real faults,” in Proceedings of
+the 30th ACM SIGSOFT International Symposium on Software Testing
+and Analysis, 2021, p. 67–78.
+[34] IBM. (2022) Ibm global ai adoption index 2022. [Online]. Available:
+https://www.ibm.com/watson/resources/ai-adoption
+[35] G. Jahangirova and P. Tonella, “An empirical evaluation of mutation
+operators for deep learning systems,” in Proceedings of the IEEE 13th
+International Conference on Software Testing, Validation and Verification,
+2020, pp. 74–84.
+[36] L. Jia, H. Zhong, X. Wang, L. Huang, and Z. Li, “How do injected
+bugs affect deep learning?” in Proceedings of the IEEE International
+Conference on Software Analysis, Evolution and Reengineering, 2022,
+pp. 793–804.
+[37] Y. Jia and M. Harman, “An analysis and survey of the development of
+mutation testing,” IEEE Transactions on Software Engineering, vol. 37,
+no. 5, pp. 649–678, 2011.
+[38] C.
+K¨astner.
+(2022)
+Machine
+learning
+in
+production:
+From
+models to systems. [Online]. Available: https://ckaestne.medium.com/
+machine-learning-in-production-from-models-to-systems-e1422ec7cd65
+[39] R. Kazmi, D. N. Jawawi, R. Mohamad, and I. Ghani, “Effective regression
+test case selection: A systematic literature review,” ACM Computing
+Surveys (CSUR), vol. 50, no. 2, pp. 1–32, 2017.
+[40] J. Kim, R. Feldt, and S. Yoo, “Guiding deep learning system testing using
+surprise adequacy,” in Proceedings of the 41st International Conference
+on Software Engineering, 2019, p. 1039–1049.
+[41] G. Lample and A. Conneau, “Cross-lingual language model pretraining,”
+2019. [Online]. Available: https://arxiv.org/abs/1901.07291
+[42] H. Leung and L. White, “A study of integration testing and software
+regression at the integration level,” in Proceedings. Conference on
+Software Maintenance 1990, 1990, pp. 290–301.
+
+[43] Z. Li, X. Ma, C. Xu, J. Xu, C. Cao, and J. L¨u, “Operational calibration:
+Debugging confidence errors for dnns in the field,” in Proceedings of the
+28th ACM Joint Meeting on European Software Engineering Conference
+and Symposium on the Foundations of Software Engineering, 2020, p.
+901–913.
+[44] H. Liang, X. Pei, X. Jia, W. Shen, and J. Zhang, “Fuzzing: State of the
+art,” IEEE Transactions on Reliability, vol. 67, no. 3, pp. 1199–1218,
+2018.
+[45] W. Liang, G. A. Tadesse, D. Ho, F.-F. Li, M. Zaharia, C. Zhang, and
+J. Zou, “Advances, challenges and opportunities in creating data for
+trustworthy AI,” Nature Machine Intelligence, 2022.
+[46] Y. Liu, M. Ott, N. Goyal, J. Du, M. Joshi, D. Chen, O. Levy, M. Lewis,
+L. Zettlemoyer, and V. Stoyanov, “Roberta: A robustly optimized bert
+pretraining approach,” arXiv preprint arXiv:1907.11692, 2019.
+[47] Z. Liu, Y. Feng, and Z. Chen, “Dialtest: automated testing for recurrent-
+neural-network-driven dialogue systems,” in Proceedings of the 30th ACM
+SIGSOFT International Symposium on Software Testing and Analysis,
+2021, pp. 115–126.
+[48] L. Ma, F. Zhang, J. Sun, M. Xue, B. Li, F. Juefei-Xu, C. Xie, L. Li, Y. Liu,
+J. Zhao, and Y. Wang, “Deepmutation: Mutation testing of deep learning
+systems,” in Proceedings of the IEEE 29th International Symposium on
+Software Reliability Engineering, 2018, pp. 100–111.
+[49] S. Ma, Y. Liu, W.-C. Lee, X. Zhang, and A. Grama, “Mode: Automated
+neural network model debugging via state differential analysis and input
+selection,” in Proceedings of the 2018 26th ACM Joint Meeting on
+European Software Engineering Conference and Symposium on the
+Foundations of Software Engineering, 2018, p. 175–186.
+[50] P. McMinn, “Search-based software testing: Past, present and future,”
+in 2011 IEEE Fourth International Conference on Software Testing,
+Verification and Validation Workshops, 2011, pp. 153–163.
+[51] Mutmut. (2022) Mutmut. [Online]. Available: https://pypi.org/project/
+mutmut/
+[52] N. Nahar, S. Zhou, G. Lewis, and C. K¨astner, “Collaboration challenges in
+building ml-enabled systems: Communication, documentation, engineer-
+ing, and process,” in Proceedings of the IEEE/ACM 44th International
+Conference on Software Engineering, 2022, pp. 413–425.
+[53] B. Nushi, E. Kamar, E. Horvitz, and D. Kossmann, “On human intellect
+and machine failures: Troubleshooting integrative machine learning
+systems,” in Proceedings of the Thirty-First AAAI Conference on Artificial
+Intelligence, 2017, pp. 1017–1025.
+[54] A. Odena, C. Olsson, D. Andersen, and I. Goodfellow, “TensorFuzz: De-
+bugging neural networks with coverage-guided fuzzing,” in Proceedings
+of the 36th International Conference on Machine Learning, 2019, pp.
+4901–4911.
+[55] K. O’Leary and M. Uchida, “Common problems with creating machine
+learning pipelines from existing code,” in Proceedings of the Third
+Conference on Machine Learning and Systems, 2020.
+[56] OpenAI. (2022) Gpt2. [Online]. Available: https://openai.com/blog/tags/
+gpt-2/
+[57] B. Paulsen, J. Wang, and C. Wang, “Reludiff: Differential verification
+of deep neural networks,” in Proceedings of the ACM/IEEE 42nd
+International Conference on Software Engineering, 2020, p. 714–726.
+[58] B. Paulsen, J. Wang, J. Wang, and C. Wang, “Neurodiff: Scalable differ-
+ential verification of neural networks using fine-grained approximation,”
+in Proceedings of the 35th IEEE/ACM International Conference on
+Automated Software Engineering, 2020, p. 784–796.
+[59] K. Pei, Y. Cao, J. Yang, and S. Jana, “Deepxplore: Automated whitebox
+testing of deep learning systems,” in Proceedings of the 26th Symposium
+on Operating Systems Principles, 2017, p. 1–18.
+[60] Z. Peng, J. Yang, T.-H. P. Chen, and L. Ma, “A first look at the integration
+of machine learning models in complex autonomous driving systems: A
+case study on apollo,” in Proceedings of the 28th ACM Joint Meeting
+on European Software Engineering Conference and Symposium on the
+Foundations of Software Engineering, pp. 1240–1250.
+[61] V. Salis, T. Sotiropoulos, P. Louridas, D. Spinellis, and D. Mitropoulos,
+“Pycg: Practical call graph generation in python,” in Proceedings of
+the 43rd International Conference on Software Engineering, 2021, p.
+1646–1657.
+[62] F. Sattler, A. von Rhein, T. Berger, N. S. Johansson, M. M. Hardø,
+and S. Apel, “Lifting inter-app data-flow analysis to large app sets,”
+Automated Software Engineering, vol. 25, no. 2, pp. 315–346, 2017.
+[63] D. Sculley, G. Holt, D. Golovin, E. Davydov, T. Phillips, D. Ebner,
+V. Chaudhary, M. Young, J.-F. Crespo, and D. Dennison, “Hidden
+technical debt in machine learning systems,” in Proceedings of the 28th
+International Conference on Neural Information Processing Systems,
+2015, p. 2503–2511.
+[64] A. Serban and J. Visser, “Adapting software architectures to machine
+learning challenges,” in Proceedings of the IEEE International Conference
+on Software Analysis, Evolution and Reengineering, 2022, pp. 152–163.
+[65] A. Shmilovici.
+[66] G. Singh, T. Gehr, M. P¨uschel, and M. Vechev, “An abstract domain
+for certifying neural networks,” Proc. ACM Program. Lang., vol. 3, no.
+POPL, pp. 1–30, 2019.
+[67] B. Sun, J. Sun, L. H. Pham, and J. Shi, “Causality-based neural network
+repair,” in Proceedings of the 44th International Conference on Software
+Engineering, 2022, pp. 338–349.
+[68] Y. Sun, M. Wu, W. Ruan, X. Huang, M. Kwiatkowska, and D. Kroening,
+“Concolic testing for deep neural networks,” in Proceedings of the 33rd
+ACM/IEEE International Conference on Automated Software Engineering,
+2018, p. 109–119.
+[69] X. Tan, K. Gao, M. Zhou, and L. Zhang, “An exploratory study of
+deep learning supply chain,” in Proceedings of the IEEE/ACM 44th
+International Conference on Software Engineering, 2022, pp. 86–98.
+[70] Y. Tang, R. Khatchadourian, M. Bagherzadeh, R. Singh, A. Stewart,
+and A. Raja, “An empirical study of refactorings and technical debt
+in machine learning systems,” in Proceedings of the IEEE/ACM 43rd
+International Conference on Software Engineering, 2021, pp. 238–250.
+[71] G. Tao, S. Ma, Y. Liu, Q. Xu, and X. Zhang, “Trader: Trace divergence
+analysis and embedding regulation for debugging recurrent neural net-
+works,” in Proceedings of the ACM/IEEE 42nd International Conference
+on Software Engineering, 2020, p. 986–998.
+[72] Y. Tian, K. Pei, S. Jana, and B. Ray, “Deeptest: Automated testing of
+deep-neural-network-driven autonomous cars,” in Proceedings of the 40th
+International Conference on Software Engineering, 2018, p. 303–314.
+[73] F. Toledo, D. Shriver, S. Elbaum, and M. B. Dwyer, “Distribution models
+for falsification and verification of dnns,” in Proceedings of the 36th
+IEEE/ACM International Conference on Automated Software Engineering,
+2021, p. 317–329.
+[74] M. Velez, P. Jamshidi, N. Siegmund, S. Apel, and C. K¨astner, “On
+debugging the performance of configurable software systems: Developer
+needs and tailored tool support,” in Proceedings of the IEEE/ACM 44th
+International Conference on Software Engineering, 2022, pp. 1571–1583.
+[75] V. Vlasov, J. E. M. Mosig, and A. Nichol, “Dialogue transformers,”
+CoRR, vol. abs/1910.00486, 2019.
+[76] T.-H. Wen, M. Gaˇsi´c, N. Mrkˇsi´c, P.-H. Su, D. Vandyke, and S. Young,
+“Semantically conditioned LSTM-based natural language generation for
+spoken dialogue systems,” in Proceedings of the Conference on Empirical
+Methods in Natural Language Processing, 2015, pp. 1711–1721.
+[77] J. Williams, A. Raux, and M. Henderson, “The dialog state tracking
+challenge series: A review,” Dialogue & Discourse, vol. 7, no. 3, pp.
+4–33, 2016.
+[78] R. Wu, C. Guo, A. Y. Hannun, and L. van der Maaten, “Fixes that
+fail: Self-defeating improvements in machine-learning systems,” in
+Proceedings of the 35th Conference on Neural Information Processing
+Systems, 2021, pp. 11 745–11 756.
+[79] Z. Yang, Z. Dai, Y. Yang, J. Carbonell, R. R. Salakhutdinov, and Q. V. Le,
+“Xlnet: Generalized autoregressive pretraining for language understanding,”
+Advances in neural information processing systems, vol. 32, 2019.
+[80] H. Yokoyama, “Machine learning system architectural pattern for
+improving operational stability,” in Proceedings of the IEEE International
+Conference on Software Architecture Companion, 2019, pp. 267–274.
+[81] A. Zerouali, E. Constantinou, T. Mens, G. Robles, and J. Gonz´alez-
+Barahona, “An empirical analysis of technical lag in npm package
+dependencies,” in Proceedings of the 17th International Conference
+on Software Reuse, 2018, pp. 95–110.
+[82] J. M. Zhang, M. Harman, L. Ma, and Y. Liu, “Machine learning
+testing: Survey, landscapes and horizons,” IEEE Transactions on Software
+Engineering, vol. 48, no. 1, pp. 1–36, 2022.
+[83] J. Zhang, X. Jing, W. Zhang, H. Wang, and Y. Dong, “Improve the
+quality of arc systems based on the metamorphic testing,” in Proceedings
+of the International Symposium on System and Software Reliability, 2016,
+pp. 137–141.
+[84] P. Zhang, J. Wang, J. Sun, G. Dong, X. Wang, X. Wang, J. S. Dong,
+and T. Dai, “White-box fairness testing through adversarial sampling,”
+in Proceedings of the ACM/IEEE 42nd International Conference on
+Software Engineering, 2020, p. 949–960.
+[85] Y. Zhang, P. Tiˇno, A. Leonardis, and K. Tang, “A survey on neural
+network interpretability,” IEEE Transactions on Emerging Topics in
+
+Computational Intelligence, vol. 5, no. 5, pp. 726–742, 2021.
+[86] Z. Zhang, R. Takanobu, Q. Zhu, M. Huang, and X. Zhu, “Recent
+advances and challenges in task-oriented dialog systems,” Science China
+Technological Sciences, vol. 63, no. 10, pp. 2011–2027, 2020.
+

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+Explaining Imitation Learning through Frames
+Boyuan Zheng1 , Jianlong Zhou1 , Chunjie Liu , Yiqiao Li1 and Fang Chen1
+1University of Technology Sydney
+Boyuan.Zheng-1@student.uts.edu.au, Jianlong.Zhou@uts.edu.au
+Abstract
+As one of the prevalent methods to achieve au-
+tomation systems, Imitation Learning (IL) presents
+a promising performance in a wide range of do-
+mains. However, despite the considerable improve-
+ment in policy performance, the corresponding re-
+search on the explainability of IL models is still
+limited. Inspired by the recent approaches in ex-
+plainable artificial intelligence methods, we pro-
+posed a model-agnostic explaining framework for
+IL models called R2RISE. R2RISE aims to explain
+the overall policy performance with respect to the
+frames in demonstrations. It iteratively retrains the
+black-box IL model from the randomized masked
+demonstrations and uses the conventional evalua-
+tion outcome environment returns as the coefficient
+to build an importance map. We also conducted ex-
+periments to investigate three major questions con-
+cerning frames’ importance equality, the effective-
+ness of the importance map, and connections be-
+tween importance maps from different IL models.
+The result shows that R2RISE successfully distin-
+guishes important frames from the demonstrations.
+1
+Introduction
+Recent advances in Imitation Learning (IL), which leverages
+external demonstration to reproduce the desired behaviours,
+demonstrate a promising performance in fields like 3D game-
+play [Scheller et al., 2020], robotics [Yu et al., 2018], and au-
+tomatic driving [Codevilla et al., 2019]. Most of the research
+on IL keep applying more complex Deep Neural Network
+(DNN) models, such as convolutional neural network (CNN)
+and generative adversarial network (GAN), to achieve greater
+performance under various condition while paying less atten-
+tion to explaining what information the trained agents learned
+from the external demonstration. In this case, despite the suc-
+cess, IL methods are becoming increasingly unexplainable,
+and this problem remains an open challenge in both IL and
+Explainable Artificial Intelligence (XAI).
+Currently, the number of existing research that combines
+XAI and IL is still limited, and they could be roughly catego-
+rized into two approaches to achieve better explainability, i.e.,
+leveraging white-box models and analyzing the pixel-wise
+explainability via existing computer vision techniques. As
+for leveraging white-box models, most existing research sub-
+stitutes the prevalent neural network architecture with other
+white-box models with intrinsic interpretability. For example,
+Leech [2019] proposed a learning framework that aggregates
+IL and logical automata to represent problems as compact fi-
+nite state automata with human-interpretable logic states. Be-
+wley et al. [2020] modelled the behaviour policy of a trained
+black-box agent in the form of a decision tree by analyzing
+its input-output statistics. Zhang et al. [2021] leveraged a hi-
+erarchical structure to explain the model’s decision-making.
+On the other hand, the research related to analyzing the pixel-
+wise explainability aims at the CNN structures in IL models
+that are widely used to capture features from image input.
+Referring to existing research in XAI and computer vision,
+the explainability of the model is commonly represented as
+heatmaps and analyzing the model’s decision-making process
+from the heatmaps. For example, Pan et al. [2020] proposed
+a model-specific method called xGAIL that is based on Gen-
+erative Adversarial Imitation Learning (GAIL) [Ho and Er-
+mon, 2016] and obtains local and global explanations for the
+passenger-seeking problem.
+In fact, before xGAIL was proposed, the research commu-
+nity of IL investigated features in image inputs, but they did
+not highlight the significance of explainability. For example,
+Brown et al. [2019] used attention maps of the input image
+frames to validate the effectiveness of the learning process.
+De Haan et al.
+[2019a] pointed out that the IL agent could
+learn wrong causal correlations between expert behaviours
+and irrelevant features in the input. These methods, including
+xGAIL, demonstrate what features in a single frame are sig-
+nificant for models to learn the desired behaviours. However,
+the above-mentioned methods fail to evaluate the importance
+of frames. Do the input image frames have identical impor-
+tance? If not, how to distinguish frames’ importance?
+To tackle these problems, we attempt to explain the in-
+put demonstrations as a whole by proposing a novel explain-
+ing method called R2RISE, which iteratively masks random
+frames in the demonstrations and evaluates the performance
+of the agents trained by the masked inputs. The intuition is
+that the input demonstrations are regarded as a single im-
+age, and frames in the demonstrations are regarded as pix-
+els. In this case, existing XAI and computer vision meth-
+ods could be directly applied to investigate the importance of
+arXiv:2301.01088v1  [cs.LG]  3 Jan 2023
+
+Figure 1: A diagrammatic representation of a single iteration of R2RISE. The input demonstrations are subject to element-wise multiplication
+(denotes as �) with a random mask which creates a masked demonstration, with greyed frames indicating those which are masked. Sub-
+sequently, the masked demonstration is used to train a black box IL model. The trained model interacts with the test environment to obtain
+returns, the mean of which is element-wise multiplied with the initial mask and accumulated to the existing importance map.
+frames instead of features in a single frame. R2RISE com-
+bines the existing methods RISE [Petsiuk et al., 2018] and
+ROAR [Hooker et al., 2019], and achieves model-agnostic
+explanations for IL models with various architectures.
+Our main contribution is summarized as follows: 1) We
+proposed a model-agnostic method to explain IL models; 2)
+We extended a novel perspective to explain IL with respect
+to the whole input dataset instead of a specific frame; 3) We
+investigated the connection between agents’ overall perfor-
+mance and demonstration frames;
+2
+Preliminaries
+To better illustrate our approach, we first introduce the related
+existing literature in the field of XAI: RISE [Petsiuk et al.,
+2018] and ROAR [Hooker et al., 2019]. We then review an
+insightful method xGAIL [Pan et al., 2020] that aggregates
+XAI with specific IL model GAIL [Ho and Ermon, 2016],
+and discuss its limitations.
+2.1
+Randomized Input Sampling for Explanation
+(RISE)
+Randomized Input Sampling for Explanation (RISE) is one
+of the state-of-the-art XAI methods proposed by Petsiuk et
+al.
+[2018] that explains black-box models. The attractive
+characteristics of RISE are its simplicity and generality. Un-
+like other popular XAI approaches, which calculate the gra-
+dient of image classification outputs, RISE probes the target
+model by randomly masking the input image and recording
+the probability result with respect to the target class. This
+process is repeated multiple times, and the recorded proba-
+bilities for each pixel are linearly combined to generate an
+importance map. This allows for the extraction of the most
+influential region in the input image for the target decision.
+RISE is also significant for explaining IL, as IL typically re-
+quires multiple demonstrations to train the model, which can
+be regarded as a single image. In this case, RISE can be used
+to explain which frames are important for policy training.
+2.2
+RemOve And Retrain (ROAR)
+RemOve And Retrain (ROAR) was proposed by Hooker et al.
+[2019], and its name concisely summarizes the working pro-
+cess of ROAR. By substituting some pixels estimated to be
+important with fixed uninformative values and then retrain-
+ing a new model, ROAR achieves to evaluate feature impor-
+tance for a wide range of models. The motivation of ROAR
+is that if the model demonstrates more sharp degradation in
+performance because of the removal, then we could conclude
+the proposed model is more accurate. The authors also ar-
+gued that the retraining process is essential as machine learn-
+ing models commonly assume that the training and test dis-
+tribution is similar, and repeating the training several times
+could ensure a low variance in performance. The intuition of
+ROAR is valuable for explaining IL models, as training a sin-
+gle model to determine the importance of frames is risky, and
+the research community commonly uses ensemble methods to
+deal with the distribution shift. Retraining serval models un-
+der the same removal rate could improve the reliability of the
+importance along the demonstration trajectories. More im-
+portantly, since the conventional evaluation of IL problems
+is different from the classical CNN-involved XAI tasks, the
+performance of the trained IL model is represented as returns
+from the dynamic environment. It is improper to train the
+model once and feed image observations with fixed masks
+under such an environment.
+2.3
+Explainable Generative Adversarial Imitation
+Learning (xGAIL)
+Pan et al.
+[2020] made the first attempt to explain one of
+the state-of-the-art models Generative Adversarial Imitation
+Learning (GAIL) [Ho and Ermon, 2016], and validated their
+method xGAIL on a passenger seeking problem. xGAIL was
+designed for problems that rely on spatial-temporal data, and
+both local and global explanations were obtained separately
+from a well-trained GAIL model. However, xGAIL’s gen-
+erality is severely limited to a specific problem and model.
+In addition, xGAIL, in fact, transforms the IL problem into
+
+Demonstrations
+Mask m;
+Masked
+Environment
+Importance
+Demonstrations
+Map
+Traj o
+Traj o
+Traj 1
+Traj 1
+0
+Black-box
+Traj 2
+Traj 2
+ROm; 
+IL model
+:::
+Traj N
+Traj Nan image classification problem by extracting and analyzing
+limited frames from abundant inputs. This could cause the ab-
+sence of an overall explanation for the model’s performance
+and generate explanations with bias as most of the informa-
+tion in the demonstration was filtered by the frame extraction
+process.
+3
+R2RISE
+To overcome the above-mentioned limitations, we proposed
+a model-agnostic explanation method for imitation learning
+called R2RISE. R2RISE combines the merits of RISE and
+ROAR, and investigates the frames’ importance with respect
+to the policy’s overall performance.
+We first review how RISE formulates the problem and
+distinguishes pixels’ importance for the image classification
+problem.
+For a given image I with the size of H × W,
+RISE creates a random binary mask m with the same size
+of I, and does an element-wise multiplication between im-
+age I and mask m (denoted as I � m). The masked image
+then feeds into the black-box model (denoted as f(I � m)).
+The importance of pixels is defined as the expected score
+over all possible masks M = {m0, m1, ..., mi} conditioned
+on the event that pixel is observed (denoted as M(λ) = 1,
+if the pixel is masked, then M(λ) = 0), i.e. SI,f(λ) =
+EM[f(I � m)|M(λ) = 1]. By rewriting the above equa-
+tion as a summation over mask m and empirically estimating
+it using Monte Carlo sampling, the saliency map can be com-
+puted as a weighted sum of random masks and normalized by
+the expectation of M:
+SI,f(λ) ≈
+1
+E[M] · N
+N
+�
+i=1
+f(I
+�
+mi) · mi(λ).
+(1)
+Since RISE does not need any assumptions and informa-
+tion from the target model, RISE could be used to explain
+black-box models. The intuition behind RISE is that when
+f(I � m) is high, it indicates that the mask observes impor-
+tant pixels. With similar intuition, Hooker et al. [2019] pro-
+posed ROAR to evaluate a feature importance. An ordered set
+of feature importance is estimated, and then they replace the
+top l fraction of the ordered set with the corresponding chan-
+nel mean, where l is the pre-defined percentage of degrada-
+tion level l = [0, 10, ..., 100]. The major difference between
+RISE and ROAR is that ROAR retrains the model on the re-
+placed dataset, while RISE only trains the model once.
+Like most imitation learning methods, we assume the test-
+ing data has a similar distribution as training data, and the
+input demonstrations Dn are optimal. This could ensure eval-
+uation fairness for a wide range of IL models. The demonstra-
+tions Dn consist of multiple trajectories, and each trajectory
+could be represented as either a sequence of state-action pairs
+or observations. In this work, we represent the trajectory as
+a sequence of state-action pairs, i.e. Dn = {τ1, τ2, ..., τn},
+where τi∈[1,n] = {(s1, a1), (s2, a2), ..., (st, at)}. The black-
+box imitation learning model trains a policy (denoted as
+πDn(a|s)) on the input demonstrations Dn, then interacts
+with the environment and obtains returns R from the test-
+ing environment. For the finite horizon T, the expected return
+could be represented as the accumulation of the return at each
+time step, i.e.
+R(πDn) = E[
+T
+�
+t=0
+rt|πDn].
+(2)
+The discussion we have so far motivated us to propose a
+frame-wise explanation method for IL called R2RISE. By re-
+garding the demonstrations D as a single image, where the
+number of demonstrations is the image height H, and the
+length of the demonstration is the image width T , we could
+investigate the frame-wise importance based on the similar
+intuition of RISE. However, RISE could not be directly ap-
+plied to IL since applying a fixed mask on a dynamic envi-
+ronment frame by frame is unreasonable, and IL methods are
+commonly evaluated by the interactions with the environment
+instead of feeding the policy network with another dataset.
+In this case, we aggregate ROAR with RISE and propose
+R2RISE. It hypothesises that the importance of each frame
+is not identical and iteratively removes random frames based
+on the predefined degradation level. The modified dataset
+Dn = D � mi is used to retrain an IL model.
+The re-
+trained IL model then constantly interacts with the environ-
+ment to obtain the accumulative return, and R2RISE finally
+compute the linear combination of the returns to obtain the
+saliency map (See Figure 1). Assuming the number of gener-
+ated masks is N, and the return of each mask is the average
+return from J rounds of interaction with the environment, the
+computation of the saliency map is similar to equation (1).
+To cater to the setting of IL, we substitute the f(I � m) in
+equation (1) with equation (2):
+SDn,f(λ) ≈
+1
+E[M] · N
+N
+�
+i=1
+R(πDi) · mi(λ)
+(3)
+=
+1
+E[M] · N
+N
+�
+i=1
+E[
+T
+�
+t=0
+rt|πDi] · mi(λ)
+(4)
+=
+1
+E[M] · N · J
+N
+�
+i=1
+J
+�
+j=0
+T
+�
+t=0
+rt · mi(λ)
+(5)
+where Di = D � mi, and
+mi(λ) =
+�0,
+if the pixel is masked,
+1,
+if the pixel is observed.
+As the formula presented, R2RISE also does not require any
+information from the IL models, such that R2RISE could be
+used as a model-agnostic method to explain IL.
+To evaluate the effectiveness of R2RISE, we test two di-
+verse IL methods: Behavioural Cloning (BC) [Bain and Sam-
+mut, 1999] and Generative Adversarial Imitation Learning
+(GAIL) [Ho and Ermon, 2016]. BC directly maps the states
+to actions from the input demonstration, and the control pol-
+icy is obtained via supervised learning; GAIL, on the other
+hand, learns the policy through an iterative adversarial pro-
+cess between the generator G and discriminator D, where G
+is generating fake data distribution and D is differentiating
+the fake data distribution with the given expert distribution
+
+(a)
+Importance
+maps
+generated
+by
+R2RISE.
+(b) Important frames in BeamRider and Breakout.
+Figure 2: Importance maps and the corresponding extracted sample frames that are recognized as important.
+Algorithm 1 R2RISE
+Input: demonstrations D, target IL model f
+Parameter:
+degradation level l, number of randomized
+masks N
+Output: an importance map SD,f
+1: Initialize masks M based on degradation level l and num-
+ber of randomized masks N.
+2: Initialize blank importance map SI,f with the same
+shape as D.
+3: for mi in M do
+4:
+Randomly initializes the model f.
+5:
+Obtain masked demonstrations Dn = D � mi.
+6:
+Train model f with the masked demonstrations Dn
+and obtain policy πDn.
+7:
+Evaluate policy πDn by interacting with environment
+and obtain average return ¯R.
+8:
+Update importance map via element-wise addition,
+SD,f ← SDn,f
+�( ¯R � mi)
+9: end for
+10: return importance map SI,f
+[Zheng et al., 2022]. The ways BC and GAIL learn the pol-
+icy are far different, and we wish to validate the generality of
+R2RISE from the diverse model selection.
+4
+Experiment
+In this section, we conduct a series of experiments and ad-
+dress the following questions: (1) Is the importance between
+frames identical? (2) Can R2RISE distinguish the importance
+between frames? (3) Are there connections between the im-
+portance map obtained from different IL models?
+4.1
+Setup
+We implement experiments with GPU NVIDIA Quadro RTX
+5000, and two diverse IL models, BC and GAIL, are evalu-
+ated on two OpenAI Gym Atari tasks: Breakout and Beam-
+rider [Brockman et al., 2016].
+Similar to recent IL methods, we leverage the proximal
+policy optimization (PPO) [Schulman et al., 2017] algorithm
+from the OpenAI baselines [Dhariwal et al., 2017], utiliz-
+ing default parameters and reward function, to generate ex-
+pert demonstrations.
+The PPO training process is check-
+pointed every 20 steps, and the observations with the size of
+84 × 84 × 3 and actions between the PPO agents and the task
+environment are recorded as ”trajectories.” These trajectories,
+generated from checkpoint 1400, serve as expert demonstra-
+tions. To avoid the “causal confusion” problem (models build
+wrong causal relationships with irrelevant patterns) [De Haan
+et al., 2019b] and ensure the fairness of our evaluation, we
+mask the indicators (such as scoring broad) in frames and en-
+sure the same demonstrations as input for different IL models.
+Regarding the parameter setting, we generate 20 trajec-
+tories with a fixed length of 1000 for each IL model. Five
+random seeds and five levels of percentage degradations l =
+[10, 30, 50, 70, 90] are pre-defined for evaluation. To assess
+each given random seed and percentage degradation, we pro-
+pose to retrain 100 models with 100 randomized masks. Each
+mask contains 20*100 grids, which means that every single
+trajectory is cut into 100 snippets, and each snippet assigns
+the same importance to 10 frames. The retrained model is
+tested from 20 trials, and the average return of the trials, mul-
+tiplied element-wise with the random mask, is added to the
+final importance map.
+4.2
+Is the importance between frames identical?
+Remember that we hypothesize that the importance of frames
+is different, so we validate this hypothesis by applying sev-
+eral randomized masks on the same demonstration and com-
+paring the performance of the trained model. If the outcomes
+present noticeable deviations, then it can be inferred that the
+contribution between frames varies. To further this idea, we
+
+Importance Map of BC -- BeamRider
+# ofdemos
+0
+200
+400
+600
+800
+Importance Map of BC -- Breakout
+# of demos
+0
+200
+400
+600
+800
+Importance Map of GAIL -- BeamRider
+#of demos
+0
+200
+400
+600
+800
+Importance Map of GAlL -- Breakout
+#ofdemos
+0
+200
+400
+600
+800Sample Importance Frames of BeamRider
+800261
+000261
+000264
+000308
+000308
+ECTOL
+01
+ECTIR
+0
+01
+Sample Importance Frames of BreakoutFigure 3: Deviations in policy performance of 10 models trained by
+10 different randomized masks.
+divide each trajectory into ten segments of equal length and
+randomly assign either a value of 0 or 1 to each segment.
+Regions assigned 0 are removed, and then the preprocessed
+demonstrations are used as input to train the relevant models.
+This process iterates ten times, and we get Figure 3. Here, the
+x-axis is the number of attempts, the y-axis is the environment
+returns, and the error bars indicate the standard deviation of
+the policy performance.
+From Figure 3, we can observe the performances deviate
+from model to model. The best model achieves far better per-
+formance than the worst one, which indicates that the impor-
+tance of frames is not identical. Similarly, the importance
+maps generated by R2RISE also suggest disparities between
+frames (see Figure 2a). The x-axis is the trajectory’s length,
+the y-axis represents the trajectories, and the grayscale de-
+notes the importance. The whiter the grid is, the more impor-
+tant the grid is. Figure 2b shows the extracted frames that are
+recognized as the most important components in the demon-
+strations. For the task of BeamRider, models would put more
+weight on destroying enemy flights, whereas for the task of
+Breakout, models pay more attention to the rebounding pro-
+cess of the upper blocks, sidewalls, and paddle, which resem-
+bles the strategy we might use for these games.
+4.3
+Can R2RISE distinguish the importance
+between frames?
+This section investigates the effectiveness of R2RISE. Ac-
+cording to the abovementioned hypothesis validation, we ob-
+tain a map indicating the importance of frames. However,
+the quality of the generated maps needs to be properly eval-
+uated. In this case, we use similar causal metrics in [Petsiuk
+et al., 2018], insertion and deletion, where the availability of
+the ’cause’ will significantly influence the model’s decision-
+making and performance. Under the scenario of image clas-
+sification tasks, deleting the causal pixels will lead to a sharp
+drop in accuracy if the model gets well explained. In our
+experiment, we leverage similar intuition and expect the re-
+moval of the important frames would lead to a worse perfor-
+(a) Policy performance changes in BeamRider.
+(b) Policy performance changes in Breakout.
+Figure 4: Validations of the effectiveness of R2RISE. The lines and
+the error bars represent the mean performance and standard devi-
+ation of the model trained from a certain percentage of the most
+important (or least important) frames.
+mance while limiting the amount of input data to be the same.
+To achieve this, we transform the generated importance map
+into a mask by setting up a threshold and replacing the map
+with either 1s or 0s, depending on the threshold.
+Figure 4 shows the changes in policy performance using
+different percentages of the most important (or least impor-
+tant) frames. The x-axis is the percentage of data used to
+train the model, and the y-axis is the environment returns.
+The solid lines are the average returns from 20 trials using
+the same transformed mask and demonstrations, the error bar
+is the standard deviation of the 20 trials. From Figure 4, we
+can observe that the models trained with the most important
+frames perform well when the input data is relatively lim-
+ited for both tasks, which meets our expectations. For task
+BeamRider (see Figure 4a), the model with important frames
+always performs better than the model trained with the least
+important frames. The performance deviation at the begin-
+ning of the figure is relatively small, we think the reason is
+that the model is more sensitive to the amount of data than to
+the availability of the important frames. From the end of this
+figure, it can be seen that the performance deviation increases,
+which indicates that the missing top important frames signifi-
+cantly limit the upper bound of the model’s performance. For
+task Breakout (see Figure 4b), it can be seen that the model
+with important frames significantly outperforms the model
+without important frames until more than 60% of the total
+demonstrations are fed. When more than 60% of the data is
+given, the model’s performance starts to fluctuate. We sus-
+pect this is due to the fact that 50% of the initial input is
+sufficient to train the policy, and with more ordinary or re-
+
+BeamRider
+Returns
+1100
+900
+700
+500
+300
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+Trials
+BC
+GAILBreakout
+Environment Returns
+40
+30
+20
+10
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+Trials
++BC-GAILBeamRider
+2700
+2300
+1900
+1500
+1100
+700
+300
+10%
+20%
+30%
+40%
+50%
+60%
+70%
+80%
+90%
+Degradation Level
+withimportantframes
+withoutimportantframesBreakout
+30
+EnvironmentReturns
+25
+20
+15
+10
+5
+0
+10%
+20%
+30%
+40%
+50%
+60%
+70%
+80%
+90%
+DegradationLevel
+withimportantframes
+withoutimportantframes(a) Deviation between the importance maps in BeamRider.
+(b) Deviation between the importance maps in Breakout.
+Figure 5: Deviation between the importance maps of BC and GAIL.
+dundant frames added to the dataset, the policy performance
+is negatively influenced.
+4.4
+Are there connections between the importance
+map obtained from different IL models?
+In addition to exploring the explainability within a single IL
+model, we also attempt to examine the connections between
+IL models. In this section, we investigate the question: does
+the importance map obtained by one model have connections
+to another model? To this end, we propose two approaches to
+explore intrinsic connections. The first attempt directly com-
+pares the importance maps by projecting the values into the
+same range and calculating element-wise deviation (see Fig-
+ure 5). The larger the deviation is, the whiter the output image
+should be. From Figure 5, we can observe that most grids are
+close to black, which indicates the assigned importance for
+these grids are similar. The second attempt involves generat-
+ing a mask from an importance map generated by one model
+and applying that mask to the same demonstrations to train
+another model. The underlying assumption is that if there
+are connections between models’ importance maps, the im-
+portant frames identified by one model should work well on
+another model, leading to improved performance compared
+to those without the frames. Figure 6 displays the average
+returns of GAIL using three types of masks: the blue, orange
+and grey lines correspond to models trained with masks ex-
+tracted from the importance map obtained using BC, GAIL
+and random, respectively. It can be seen in Figure 6 that
+the model trained with BC’s mask demonstrates similar pol-
+icy performance to the model trained with GAIL’s mask, and
+both models outperform the model using a randomized mask
+under the same training epoch.
+This confirms our expec-
+tation that the importance frames recognized by BC could
+also be employed to train model GAIL with considerable per-
+formance. This could prove useful when the target model
+is time-consuming, and one could use a more time-efficient
+method like BC to obtain an alternative importance map.
+Figure 6: Average reward learning curves of GAIL trained with dif-
+ferent masks. The blue, orange and grey lines are trained with the
+masks extracted from the importance map obtained using BC, GAIL
+and random, respectively.
+5
+Limitations
+Several intriguing challenges await further exploration. Al-
+though we were able to extract promising explanations
+through a large number of masks and validations of the ob-
+tained policy, the robustness of the framework could be fur-
+ther improved through ensemble methods. As the framework
+currently relies on a single trial to train the model, ensem-
+bling IL models could reduce performance variance for each
+given mask. In addition to robustness, computation inten-
+siveness is another limitation. The time taken to obtain a sat-
+isfactory explanation is closely linked to the time spent on
+training the target model once. If the target model requires
+days to train, it would not be practical to retrain hundreds of
+times. Improving time efficiency while preserving the model-
+agnostic property remains an open challenge for R2RISE. In-
+vestigating the relationship between global explanations and
+the frames that are recognized as important is another inter-
+esting future direction. Although we observed similar pat-
+terns in the extracted frames from different trials and models,
+it is still unsafe to claim these patterns could be the global
+explanations for a specific task. Further research is needed to
+provide theoretical proof for the connections.
+6
+Conclusion
+This paper introduced a model-agnostic explaining frame-
+work for imitation learning called R2RISE. R2RISE distin-
+guishes the frames’ importance in relation to the overall pol-
+icy performance. It iteratively applies numerous randomized
+masks on the demonstrations and retrains the black-box IL
+model based on the masked demonstration. Evaluation of the
+obtained policy is conducted in a manner similar to most IL
+methods, where the policy is evaluated by the accumulated re-
+turns from the environment, and we leverage the accumulated
+returns as a coefficient to multiply with the mask to obtain the
+importance map of the frames. Experiments have shown that
+R2RISE can successfully distinguish important frames from
+the demonstrations, thus providing insight into which frames
+contribute to better performance.
+
+12
+10
+8
+AverageReturns
+4
+2
+0
+1
+51
+101
+151
+201251301351401451501551601651701
+751
+8018519019511001
+Epochs
+BC+GAIL
+GAIL+GAIL
+Random+GAILDifference between importance maps
+# of demos
+0
+200
+400
+600
+800Difference between importance maps
+# of demos
+0
+200
+400
+600
+800References
+[Bain and Sammut, 1999] Michael Bain and Claude Sam-
+mut. A framework for behavioural cloning. In Machine
+Intelligence 15, pages 103–129. Oxford University Press,
+1999.
+[Bewley et al., 2020] Tom Bewley, Jonathan Lawry, and
+Arthur Richards.
+Modelling agent policies with inter-
+pretable imitation learning. In International Workshop on
+the Foundations of Trustworthy AI Integrating Learning,
+Optimization and Reasoning, pages 180–186. Springer,
+2020.
+[Brockman et al., 2016] Greg Brockman,
+Vicki Cheung,
+Ludwig Pettersson, Jonas Schneider, John Schulman, Jie
+Tang, and Wojciech Zaremba. Openai gym. arXiv preprint
+arXiv:1606.01540, 2016.
+[Brown et al., 2019] Daniel Brown, Wonjoon Goo, Prabhat
+Nagarajan, and Scott Niekum. Extrapolating beyond sub-
+optimal demonstrations via inverse reinforcement learning
+from observations.
+In International conference on ma-
+chine learning, pages 783–792. PMLR, 2019.
+[Codevilla et al., 2019] Felipe Codevilla, Eder Santana, An-
+tonio M L´opez, and Adrien Gaidon. Exploring the lim-
+itations of behavior cloning for autonomous driving. In
+Proceedings of the IEEE/CVF International Conference
+on Computer Vision, pages 9329–9338, 2019.
+[De Haan et al., 2019a] Pim De Haan, Dinesh Jayaraman,
+and Sergey Levine. Causal confusion in imitation learn-
+ing. Advances in Neural Information Processing Systems,
+32, 2019.
+[De Haan et al., 2019b] Pim De Haan, Dinesh Jayaraman,
+and Sergey Levine. Causal confusion in imitation learn-
+ing. Advances in Neural Information Processing Systems,
+32, 2019.
+[Dhariwal et al., 2017] Prafulla
+Dhariwal,
+Christopher
+Hesse, Oleg Klimov, Alex Nichol, Matthias Plappert,
+Alec Radford, John Schulman, Szymon Sidor, Yuhuai
+Wu, and Peter Zhokhov. Openai baselines, 2017.
+[Ho and Ermon, 2016] Jonathan Ho and Stefano Ermon.
+Generative adversarial imitation learning.
+Advances in
+neural information processing systems, 29, 2016.
+[Hooker et al., 2019] Sara Hooker, Dumitru Erhan, Pieter-
+Jan Kindermans, and Been Kim. A benchmark for inter-
+pretability methods in deep neural networks. Advances in
+neural information processing systems, 32, 2019.
+[Leech, 2019] Thomas Leech. Explainable machine learn-
+ing for task planning in robotics.
+PhD thesis, Mas-
+sachusetts Institute of Technology, 2019.
+[Pan et al., 2020] Menghai Pan, Weixiao Huang, Yanhua Li,
+Xun Zhou, and Jun Luo.
+xgail:
+Explainable gener-
+ative adversarial imitation learning for explainable hu-
+man decision analysis. In Proceedings of the 26th ACM
+SIGKDD International Conference on Knowledge Discov-
+ery & Data Mining, pages 1334–1343, 2020.
+[Petsiuk et al., 2018] Vitali Petsiuk, Abir Das, and Kate
+Saenko. Rise: Randomized input sampling for explanation
+of black-box models. arXiv preprint arXiv:1806.07421,
+2018.
+[Scheller et al., 2020] Christian Scheller, Yanick Schraner,
+and Manfred Vogel. Sample efficient reinforcement learn-
+ing through learning from demonstrations in minecraft.
+In NeurIPS 2019 Competition and Demonstration Track,
+pages 67–76. PMLR, 2020.
+[Schulman et al., 2017] John Schulman, Filip Wolski, Pra-
+fulla Dhariwal, Alec Radford, and Oleg Klimov.
+Prox-
+imal policy optimization algorithms.
+arXiv preprint
+arXiv:1707.06347, 2017.
+[Yu et al., 2018] Tianhe Yu,
+Chelsea Finn,
+Annie Xie,
+Sudeep Dasari, Tianhao Zhang, Pieter Abbeel, and
+Sergey Levine.
+One-shot imitation from observing hu-
+mans via domain-adaptive meta-learning. arXiv preprint
+arXiv:1802.01557, 2018.
+[Zhang et al., 2021] Dandan Zhang, Qiang Li, Yu Zheng,
+Lei Wei, Dongsheng Zhang, and Zhengyou Zhang. Ex-
+plainable hierarchical imitation learning for robotic drink
+pouring. IEEE Transactions on Automation Science and
+Engineering, 2021.
+[Zheng et al., 2022] Boyuan Zheng, Sunny Verma, Jianlong
+Zhou, Ivor W. Tsang, and Fang Chen. Imitation learning:
+Progress, taxonomies and challenges. IEEE Transactions
+on Neural Networks and Learning Systems, pages 1–16,
+2022.
+

VNAzT4oBgHgl3EQfJ_sw/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf,len=434
+page_content='Explaining Imitation Learning through Frames  Boyuan Zheng1 , Jianlong Zhou1 , Chunjie Liu , Yiqiao Li1 and Fang Chen1  1University of Technology Sydney  Boyuan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='Zheng-1@student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='uts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='au, Jianlong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='Zhou@uts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='au  Abstract  As one of the prevalent methods to achieve au-  tomation systems, Imitation Learning (IL) presents  a promising performance in a wide range of do-  mains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' However, despite the considerable improve-  ment in policy performance, the corresponding re-  search on the explainability of IL models is still  limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Inspired by the recent approaches in ex-  plainable artificial intelligence methods, we pro-  posed a model-agnostic explaining framework for  IL models called R2RISE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' R2RISE aims to explain  the overall policy performance with respect to the  frames in demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' It iteratively retrains the  black-box IL model from the randomized masked  demonstrations and uses the conventional evalua-  tion outcome environment returns as the coefficient  to build an importance map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' We also conducted ex-  periments to investigate three major questions con-  cerning frames’ importance equality, the effective-  ness of the importance map, and connections be-  tween importance maps from different IL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  The result shows that R2RISE successfully distin-  guishes important frames from the demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  1  Introduction  Recent advances in Imitation Learning (IL), which leverages  external demonstration to reproduce the desired behaviours,  demonstrate a promising performance in fields like 3D game-  play [Scheller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', 2020], robotics [Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', 2018], and au-  tomatic driving [Codevilla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Most of the research  on IL keep applying more complex Deep Neural Network  (DNN) models, such as convolutional neural network (CNN)  and generative adversarial network (GAN), to achieve greater  performance under various condition while paying less atten-  tion to explaining what information the trained agents learned  from the external demonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' In this case, despite the suc-  cess, IL methods are becoming increasingly unexplainable,  and this problem remains an open challenge in both IL and  Explainable Artificial Intelligence (XAI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  Currently, the number of existing research that combines  XAI and IL is still limited, and they could be roughly catego-  rized into two approaches to achieve better explainability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=',  leveraging white-box models and analyzing the pixel-wise  explainability via existing computer vision techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' As  for leveraging white-box models, most existing research sub-  stitutes the prevalent neural network architecture with other  white-box models with intrinsic interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' For example,  Leech [2019] proposed a learning framework that aggregates  IL and logical automata to represent problems as compact fi-  nite state automata with human-interpretable logic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Be-  wley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' [2020] modelled the behaviour policy of a trained  black-box agent in the form of a decision tree by analyzing  its input-output statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' [2021] leveraged a hi-  erarchical structure to explain the model’s decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  On the other hand, the research related to analyzing the pixel-  wise explainability aims at the CNN structures in IL models  that are widely used to capture features from image input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  Referring to existing research in XAI and computer vision,  the explainability of the model is commonly represented as  heatmaps and analyzing the model’s decision-making process  from the heatmaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' For example, Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' [2020] proposed  a model-specific method called xGAIL that is based on Gen-  erative Adversarial Imitation Learning (GAIL) [Ho and Er-  mon, 2016] and obtains local and global explanations for the  passenger-seeking problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  In fact, before xGAIL was proposed, the research commu-  nity of IL investigated features in image inputs, but they did  not highlight the significance of explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' For example,  Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' [2019] used attention maps of the input image  frames to validate the effectiveness of the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  De Haan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  [2019a] pointed out that the IL agent could  learn wrong causal correlations between expert behaviours  and irrelevant features in the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' These methods, including  xGAIL, demonstrate what features in a single frame are sig-  nificant for models to learn the desired behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' However,  the above-mentioned methods fail to evaluate the importance  of frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Do the input image frames have identical impor-  tance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' If not, how to distinguish frames’ importance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  To tackle these problems, we attempt to explain the in-  put demonstrations as a whole by proposing a novel explain-  ing method called R2RISE, which iteratively masks random  frames in the demonstrations and evaluates the performance  of the agents trained by the masked inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' The intuition is  that the input demonstrations are regarded as a single im-  age, and frames in the demonstrations are regarded as pix-  els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' In this case, existing XAI and computer vision meth-  ods could be directly applied to investigate the importance of  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='01088v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='LG]  3 Jan 2023  Figure 1: A diagrammatic representation of a single iteration of R2RISE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' The input demonstrations are subject to element-wise multiplication  (denotes as �) with a random mask which creates a masked demonstration, with greyed frames indicating those which are masked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Sub-  sequently, the masked demonstration is used to train a black box IL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' The trained model interacts with the test environment to obtain  returns, the mean of which is element-wise multiplied with the initial mask and accumulated to the existing importance map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  frames instead of features in a single frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' R2RISE com-  bines the existing methods RISE [Petsiuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', 2018] and  ROAR [Hooker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', 2019], and achieves model-agnostic  explanations for IL models with various architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  Our main contribution is summarized as follows: 1) We  proposed a model-agnostic method to explain IL models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' 2)  We extended a novel perspective to explain IL with respect  to the whole input dataset instead of a specific frame;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' 3) We  investigated the connection between agents’ overall perfor-  mance and demonstration frames;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  2  Preliminaries  To better illustrate our approach, we first introduce the related  existing literature in the field of XAI: RISE [Petsiuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=',  2018] and ROAR [Hooker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' We then review an  insightful method xGAIL [Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', 2020] that aggregates  XAI with specific IL model GAIL [Ho and Ermon, 2016],  and discuss its limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='1  Randomized Input Sampling for Explanation  (RISE)  Randomized Input Sampling for Explanation (RISE) is one  of the state-of-the-art XAI methods proposed by Petsiuk et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  [2018] that explains black-box models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' The attractive  characteristics of RISE are its simplicity and generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Un-  like other popular XAI approaches, which calculate the gra-  dient of image classification outputs, RISE probes the target  model by randomly masking the input image and recording  the probability result with respect to the target class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' This  process is repeated multiple times, and the recorded proba-  bilities for each pixel are linearly combined to generate an  importance map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' This allows for the extraction of the most  influential region in the input image for the target decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  RISE is also significant for explaining IL, as IL typically re-  quires multiple demonstrations to train the model, which can  be regarded as a single image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' In this case, RISE can be used  to explain which frames are important for policy training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='2  RemOve And Retrain (ROAR)  RemOve And Retrain (ROAR) was proposed by Hooker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  [2019], and its name concisely summarizes the working pro-  cess of ROAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' By substituting some pixels estimated to be  important with fixed uninformative values and then retrain-  ing a new model, ROAR achieves to evaluate feature impor-  tance for a wide range of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' The motivation of ROAR  is that if the model demonstrates more sharp degradation in  performance because of the removal, then we could conclude  the proposed model is more accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' The authors also ar-  gued that the retraining process is essential as machine learn-  ing models commonly assume that the training and test dis-  tribution is similar, and repeating the training several times  could ensure a low variance in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' The intuition of  ROAR is valuable for explaining IL models, as training a sin-  gle model to determine the importance of frames is risky, and  the research community commonly uses ensemble methods to  deal with the distribution shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Retraining serval models un-  der the same removal rate could improve the reliability of the  importance along the demonstration trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' More im-  portantly, since the conventional evaluation of IL problems  is different from the classical CNN-involved XAI tasks, the  performance of the trained IL model is represented as returns  from the dynamic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' It is improper to train the  model once and feed image observations with fixed masks  under such an environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='3  Explainable Generative Adversarial Imitation  Learning (xGAIL)  Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  [2020] made the first attempt to explain one of  the state-of-the-art models Generative Adversarial Imitation  Learning (GAIL) [Ho and Ermon, 2016], and validated their  method xGAIL on a passenger seeking problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' xGAIL was  designed for problems that rely on spatial-temporal data, and  both local and global explanations were obtained separately  from a well-trained GAIL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' However, xGAIL’s gen-  erality is severely limited to a specific problem and model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  In addition, xGAIL, in fact, transforms the IL problem into  Demonstrations  Mask m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  Masked  Environment  Importance  Demonstrations  Map  Traj o  Traj o  Traj 1  Traj 1  0  Black-box  Traj 2  Traj 2  ROm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  IL model  :::  Traj N  Traj Nan image classification problem by extracting and analyzing  limited frames from abundant inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' This could cause the ab-  sence of an overall explanation for the model’s performance  and generate explanations with bias as most of the informa-  tion in the demonstration was filtered by the frame extraction  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  3  R2RISE  To overcome the above-mentioned limitations, we proposed  a model-agnostic explanation method for imitation learning  called R2RISE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' R2RISE combines the merits of RISE and  ROAR, and investigates the frames’ importance with respect  to the policy’s overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  We first review how RISE formulates the problem and  distinguishes pixels’ importance for the image classification  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  For a given image I with the size of H × W,  RISE creates a random binary mask m with the same size  of I, and does an element-wise multiplication between im-  age I and mask m (denoted as I � m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' The masked image  then feeds into the black-box model (denoted as f(I � m)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  The importance of pixels is defined as the expected score  over all possible masks M = {m0, m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', mi} conditioned  on the event that pixel is observed (denoted as M(λ) = 1,  if the pixel is masked, then M(λ) = 0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' SI,f(λ) =  EM[f(I � m)|M(λ) = 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' By rewriting the above equa-  tion as a summation over mask m and empirically estimating  it using Monte Carlo sampling, the saliency map can be com-  puted as a weighted sum of random masks and normalized by  the expectation of M:  SI,f(λ) ≈  1  E[M] · N  N  �  i=1  f(I  �  mi) · mi(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  (1)  Since RISE does not need any assumptions and informa-  tion from the target model, RISE could be used to explain  black-box models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' The intuition behind RISE is that when  f(I � m) is high, it indicates that the mask observes impor-  tant pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' With similar intuition, Hooker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' [2019] pro-  posed ROAR to evaluate a feature importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' An ordered set  of feature importance is estimated, and then they replace the  top l fraction of the ordered set with the corresponding chan-  nel mean, where l is the pre-defined percentage of degrada-  tion level l = [0, 10, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', 100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' The major difference between  RISE and ROAR is that ROAR retrains the model on the re-  placed dataset, while RISE only trains the model once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  Like most imitation learning methods, we assume the test-  ing data has a similar distribution as training data, and the  input demonstrations Dn are optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' This could ensure eval-  uation fairness for a wide range of IL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' The demonstra-  tions Dn consist of multiple trajectories, and each trajectory  could be represented as either a sequence of state-action pairs  or observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' In this work, we represent the trajectory as  a sequence of state-action pairs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Dn = {τ1, τ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', τn},  where τi∈[1,n] = {(s1, a1), (s2, a2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', (st, at)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' The black-  box imitation learning model trains a policy (denoted as  πDn(a|s)) on the input demonstrations Dn, then interacts  with the environment and obtains returns R from the test-  ing environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' For the finite horizon T, the expected return  could be represented as the accumulation of the return at each  time step, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  R(πDn) = E[  T  �  t=0  rt|πDn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  (2)  The discussion we have so far motivated us to propose a  frame-wise explanation method for IL called R2RISE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' By re-  garding the demonstrations D as a single image, where the  number of demonstrations is the image height H, and the  length of the demonstration is the image width T , we could  investigate the frame-wise importance based on the similar  intuition of RISE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' However, RISE could not be directly ap-  plied to IL since applying a fixed mask on a dynamic envi-  ronment frame by frame is unreasonable, and IL methods are  commonly evaluated by the interactions with the environment  instead of feeding the policy network with another dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  In this case, we aggregate ROAR with RISE and propose  R2RISE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' It hypothesises that the importance of each frame  is not identical and iteratively removes random frames based  on the predefined degradation level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' The modified dataset  Dn = D � mi is used to retrain an IL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  The re-  trained IL model then constantly interacts with the environ-  ment to obtain the accumulative return, and R2RISE finally  compute the linear combination of the returns to obtain the  saliency map (See Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Assuming the number of gener-  ated masks is N, and the return of each mask is the average  return from J rounds of interaction with the environment, the  computation of the saliency map is similar to equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  To cater to the setting of IL, we substitute the f(I � m) in  equation (1) with equation (2):  SDn,f(λ) ≈  1  E[M] · N  N  �  i=1  R(πDi) · mi(λ)  (3)  =  1  E[M] · N  N  �  i=1  E[  T  �  t=0  rt|πDi] · mi(λ)  (4)  =  1  E[M] · N · J  N  �  i=1  J  �  j=0  T  �  t=0  rt · mi(λ)  (5)  where Di = D � mi, and  mi(λ) =  �0,  if the pixel is masked,  1,  if the pixel is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  As the formula presented, R2RISE also does not require any  information from the IL models, such that R2RISE could be  used as a model-agnostic method to explain IL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  To evaluate the effectiveness of R2RISE, we test two di-  verse IL methods: Behavioural Cloning (BC) [Bain and Sam-  mut, 1999] and Generative Adversarial Imitation Learning  (GAIL) [Ho and Ermon, 2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' BC directly maps the states  to actions from the input demonstration, and the control pol-  icy is obtained via supervised learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' GAIL, on the other  hand, learns the policy through an iterative adversarial pro-  cess between the generator G and discriminator D, where G  is generating fake data distribution and D is differentiating  the fake data distribution with the given expert distribution  (a)  Importance  maps  generated  by  R2RISE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  (b) Important frames in BeamRider and Breakout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  Figure 2: Importance maps and the corresponding extracted sample frames that are recognized as important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  Algorithm 1 R2RISE  Input: demonstrations D, target IL model f  Parameter:  degradation level l, number of randomized  masks N  Output: an importance map SD,f  1: Initialize masks M based on degradation level l and num-  ber of randomized masks N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  2: Initialize blank importance map SI,f with the same  shape as D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  3: for mi in M do  4:  Randomly initializes the model f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  5:  Obtain masked demonstrations Dn = D � mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  6:  Train model f with the masked demonstrations Dn  and obtain policy πDn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  7:  Evaluate policy πDn by interacting with environment  and obtain average return ¯R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  8:  Update importance map via element-wise addition,  SD,f ← SDn,f  �( ¯R � mi)  9: end for  10: return importance map SI,f  [Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' The ways BC and GAIL learn the pol-  icy are far different, and we wish to validate the generality of  R2RISE from the diverse model selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  4  Experiment  In this section, we conduct a series of experiments and ad-  dress the following questions: (1) Is the importance between  frames identical?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' (2) Can R2RISE distinguish the importance  between frames?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' (3) Are there connections between the im-  portance map obtained from different IL models?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='1  Setup  We implement experiments with GPU NVIDIA Quadro RTX  5000, and two diverse IL models, BC and GAIL, are evalu-  ated on two OpenAI Gym Atari tasks: Breakout and Beam-  rider [Brockman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', 2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  Similar to recent IL methods, we leverage the proximal  policy optimization (PPO) [Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', 2017] algorithm  from the OpenAI baselines [Dhariwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', 2017], utiliz-  ing default parameters and reward function, to generate ex-  pert demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  The PPO training process is check-  pointed every 20 steps, and the observations with the size of  84 × 84 × 3 and actions between the PPO agents and the task  environment are recorded as ”trajectories.” These trajectories,  generated from checkpoint 1400, serve as expert demonstra-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' To avoid the “causal confusion” problem (models build  wrong causal relationships with irrelevant patterns) [De Haan  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', 2019b] and ensure the fairness of our evaluation, we  mask the indicators (such as scoring broad) in frames and en-  sure the same demonstrations as input for different IL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  Regarding the parameter setting, we generate 20 trajec-  tories with a fixed length of 1000 for each IL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Five  random seeds and five levels of percentage degradations l =  [10, 30, 50, 70, 90] are pre-defined for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' To assess  each given random seed and percentage degradation, we pro-  pose to retrain 100 models with 100 randomized masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Each  mask contains 20*100 grids, which means that every single  trajectory is cut into 100 snippets, and each snippet assigns  the same importance to 10 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' The retrained model is  tested from 20 trials, and the average return of the trials, mul-  tiplied element-wise with the random mask, is added to the  final importance map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='2  Is the importance between frames identical?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  Remember that we hypothesize that the importance of frames  is different, so we validate this hypothesis by applying sev-  eral randomized masks on the same demonstration and com-  paring the performance of the trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' If the outcomes  present noticeable deviations, then it can be inferred that the  contribution between frames varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' To further this idea,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' we  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='Importance Map of BC -- BeamRider  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='# ofdemos  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='400  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='600  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='800  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='Importance Map of BC -- Breakout  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='# of demos  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='400  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='600  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='800  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='Importance Map of GAIL -- BeamRider  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='#of demos  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='400  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='600  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='800  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='Importance Map of GAlL -- Breakout  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='#ofdemos  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='400  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='600  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='800Sample Importance Frames of BeamRider  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='800261  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='000261  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='000264  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='000308  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='000308  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='ECTOL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='01  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='ECTIR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='01  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='Sample Importance Frames of BreakoutFigure 3: Deviations in policy performance of 10 models trained by  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='10 different randomized masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  divide each trajectory into ten segments of equal length and  randomly assign either a value of 0 or 1 to each segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  Regions assigned 0 are removed, and then the preprocessed  demonstrations are used as input to train the relevant models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  This process iterates ten times, and we get Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Here, the  x-axis is the number of attempts, the y-axis is the environment  returns, and the error bars indicate the standard deviation of  the policy performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  From Figure 3, we can observe the performances deviate  from model to model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' The best model achieves far better per-  formance than the worst one, which indicates that the impor-  tance of frames is not identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Similarly, the importance  maps generated by R2RISE also suggest disparities between  frames (see Figure 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' The x-axis is the trajectory’s length,  the y-axis represents the trajectories, and the grayscale de-  notes the importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' The whiter the grid is, the more impor-  tant the grid is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Figure 2b shows the extracted frames that are  recognized as the most important components in the demon-  strations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' For the task of BeamRider, models would put more  weight on destroying enemy flights, whereas for the task of  Breakout, models pay more attention to the rebounding pro-  cess of the upper blocks, sidewalls, and paddle, which resem-  bles the strategy we might use for these games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='3  Can R2RISE distinguish the importance  between frames?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  This section investigates the effectiveness of R2RISE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Ac-  cording to the abovementioned hypothesis validation, we ob-  tain a map indicating the importance of frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' However,  the quality of the generated maps needs to be properly eval-  uated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' In this case, we use similar causal metrics in [Petsiuk  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', 2018], insertion and deletion, where the availability of  the ’cause’ will significantly influence the model’s decision-  making and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Under the scenario of image clas-  sification tasks, deleting the causal pixels will lead to a sharp  drop in accuracy if the model gets well explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' In our  experiment, we leverage similar intuition and expect the re-  moval of the important frames would lead to a worse perfor-  (a) Policy performance changes in BeamRider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  (b) Policy performance changes in Breakout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  Figure 4: Validations of the effectiveness of R2RISE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' The lines and  the error bars represent the mean performance and standard devi-  ation of the model trained from a certain percentage of the most  important (or least important) frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  mance while limiting the amount of input data to be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  To achieve this, we transform the generated importance map  into a mask by setting up a threshold and replacing the map  with either 1s or 0s, depending on the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  Figure 4 shows the changes in policy performance using  different percentages of the most important (or least impor-  tant) frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' The x-axis is the percentage of data used to  train the model, and the y-axis is the environment returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  The solid lines are the average returns from 20 trials using  the same transformed mask and demonstrations, the error bar  is the standard deviation of the 20 trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' From Figure 4, we  can observe that the models trained with the most important  frames perform well when the input data is relatively lim-  ited for both tasks, which meets our expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' For task  BeamRider (see Figure 4a), the model with important frames  always performs better than the model trained with the least  important frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' The performance deviation at the begin-  ning of the figure is relatively small, we think the reason is  that the model is more sensitive to the amount of data than to  the availability of the important frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' From the end of this  figure, it can be seen that the performance deviation increases,  which indicates that the missing top important frames signifi-  cantly limit the upper bound of the model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' For  task Breakout (see Figure 4b), it can be seen that the model  with important frames significantly outperforms the model  without important frames until more than 60% of the total  demonstrations are fed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' When more than 60% of the data is  given, the model’s performance starts to fluctuate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' We sus-  pect this is due to the fact that 50% of the initial input is  sufficient to train the policy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
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+page_content='withoutimportantframes(a) Deviation between the importance maps in BeamRider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  (b) Deviation between the importance maps in Breakout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  Figure 5: Deviation between the importance maps of BC and GAIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  dundant frames added to the dataset, the policy performance  is negatively influenced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='4  Are there connections between the importance  map obtained from different IL models?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  In addition to exploring the explainability within a single IL  model, we also attempt to examine the connections between  IL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' In this section, we investigate the question: does  the importance map obtained by one model have connections  to another model?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' To this end, we propose two approaches to  explore intrinsic connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' The first attempt directly com-  pares the importance maps by projecting the values into the  same range and calculating element-wise deviation (see Fig-  ure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' The larger the deviation is, the whiter the output image  should be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' From Figure 5, we can observe that most grids are  close to black, which indicates the assigned importance for  these grids are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' The second attempt involves generat-  ing a mask from an importance map generated by one model  and applying that mask to the same demonstrations to train  another model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' The underlying assumption is that if there  are connections between models’ importance maps, the im-  portant frames identified by one model should work well on  another model, leading to improved performance compared  to those without the frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Figure 6 displays the average  returns of GAIL using three types of masks: the blue, orange  and grey lines correspond to models trained with masks ex-  tracted from the importance map obtained using BC, GAIL  and random, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' It can be seen in Figure 6 that  the model trained with BC’s mask demonstrates similar pol-  icy performance to the model trained with GAIL’s mask, and  both models outperform the model using a randomized mask  under the same training epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  This confirms our expec-  tation that the importance frames recognized by BC could  also be employed to train model GAIL with considerable per-  formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' This could prove useful when the target model  is time-consuming, and one could use a more time-efficient  method like BC to obtain an alternative importance map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  Figure 6: Average reward learning curves of GAIL trained with dif-  ferent masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' The blue, orange and grey lines are trained with the  masks extracted from the importance map obtained using BC, GAIL  and random, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  5  Limitations  Several intriguing challenges await further exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Al-  though we were able to extract promising explanations  through a large number of masks and validations of the ob-  tained policy, the robustness of the framework could be fur-  ther improved through ensemble methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' As the framework  currently relies on a single trial to train the model, ensem-  bling IL models could reduce performance variance for each  given mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' In addition to robustness, computation inten-  siveness is another limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' The time taken to obtain a sat-  isfactory explanation is closely linked to the time spent on  training the target model once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' If the target model requires  days to train, it would not be practical to retrain hundreds of  times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Improving time efficiency while preserving the model-  agnostic property remains an open challenge for R2RISE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' In-  vestigating the relationship between global explanations and  the frames that are recognized as important is another inter-  esting future direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Although we observed similar pat-  terns in the extracted frames from different trials and models,  it is still unsafe to claim these patterns could be the global  explanations for a specific task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Further research is needed to  provide theoretical proof for the connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  6  Conclusion  This paper introduced a model-agnostic explaining frame-  work for imitation learning called R2RISE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' R2RISE distin-  guishes the frames’ importance in relation to the overall pol-  icy performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' It iteratively applies numerous randomized  masks on the demonstrations and retrains the black-box IL  model based on the masked demonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Evaluation of the  obtained policy is conducted in a manner similar to most IL  methods, where the policy is evaluated by the accumulated re-  turns from the environment, and we leverage the accumulated  returns as a coefficient to multiply with the mask to obtain the  importance map of the frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Experiments have shown that  R2RISE can successfully distinguish important frames from  the demonstrations, thus providing insight into which frames  contribute to better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  12  10  8  AverageReturns  4  2  0  1  51  101  151  201251301351401451501551601651701  751  8018519019511001  Epochs  BC+GAIL  GAIL+GAIL  Random+GAILDifference between importance maps  # of demos  0  200  400  600  800Difference between importance maps  # of demos  0  200  400  600  800References  [Bain and Sammut, 1999] Michael Bain and Claude Sam-  mut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' A framework for behavioural cloning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' In Machine  Intelligence 15, pages 103–129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Oxford University Press,  1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  [Bewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', 2020] Tom Bewley, Jonathan Lawry, and  Arthur Richards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  Modelling agent policies with inter-  pretable imitation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' In International Workshop on  the Foundations of Trustworthy AI Integrating Learning,  Optimization and Reasoning, pages 180–186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Springer,  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  [Brockman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', 2016] Greg Brockman,  Vicki Cheung,  Ludwig Pettersson, Jonas Schneider, John Schulman, Jie  Tang, and Wojciech Zaremba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Openai gym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' arXiv preprint  arXiv:1606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='01540, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  [Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', 2019] Daniel Brown, Wonjoon Goo, Prabhat  Nagarajan, and Scott Niekum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Extrapolating beyond sub-  optimal demonstrations via inverse reinforcement learning  from observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  In International conference on ma-  chine learning, pages 783–792.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' PMLR, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  [Codevilla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', 2019] Felipe Codevilla, Eder Santana, An-  tonio M L´opez, and Adrien Gaidon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Exploring the lim-  itations of behavior cloning for autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' In  Proceedings of the IEEE/CVF International Conference  on Computer Vision, pages 9329–9338, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  [De Haan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', 2019a] Pim De Haan, Dinesh Jayaraman,  and Sergey Levine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Causal confusion in imitation learn-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Advances in Neural Information Processing Systems,  32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  [De Haan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', 2019b] Pim De Haan, Dinesh Jayaraman,  and Sergey Levine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Causal confusion in imitation learn-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Advances in Neural Information Processing Systems,  32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  [Dhariwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', 2017] Prafulla  Dhariwal,  Christopher  Hesse, Oleg Klimov, Alex Nichol, Matthias Plappert,  Alec Radford, John Schulman, Szymon Sidor, Yuhuai  Wu, and Peter Zhokhov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Openai baselines, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  [Ho and Ermon, 2016] Jonathan Ho and Stefano Ermon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  Generative adversarial imitation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  Advances in  neural information processing systems, 29, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  [Hooker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', 2019] Sara Hooker, Dumitru Erhan, Pieter-  Jan Kindermans, and Been Kim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' A benchmark for inter-  pretability methods in deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Advances in  neural information processing systems, 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  [Leech, 2019] Thomas Leech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Explainable machine learn-  ing for task planning in robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  PhD thesis, Mas-  sachusetts Institute of Technology, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  [Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', 2020] Menghai Pan, Weixiao Huang, Yanhua Li,  Xun Zhou, and Jun Luo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  xgail:  Explainable gener-  ative adversarial imitation learning for explainable hu-  man decision analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' In Proceedings of the 26th ACM  SIGKDD International Conference on Knowledge Discov-  ery & Data Mining, pages 1334–1343, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  [Petsiuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', 2018] Vitali Petsiuk, Abir Das, and Kate  Saenko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Rise: Randomized input sampling for explanation  of black-box models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' arXiv preprint arXiv:1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='07421,  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  [Scheller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', 2020] Christian Scheller, Yanick Schraner,  and Manfred Vogel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Sample efficient reinforcement learn-  ing through learning from demonstrations in minecraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  In NeurIPS 2019 Competition and Demonstration Track,  pages 67–76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' PMLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  [Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', 2017] John Schulman, Filip Wolski, Pra-  fulla Dhariwal, Alec Radford, and Oleg Klimov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  Prox-  imal policy optimization algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  arXiv preprint  arXiv:1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='06347, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  [Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', 2018] Tianhe Yu,  Chelsea Finn,  Annie Xie,  Sudeep Dasari, Tianhao Zhang, Pieter Abbeel, and  Sergey Levine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  One-shot imitation from observing hu-  mans via domain-adaptive meta-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' arXiv preprint  arXiv:1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='01557, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', 2021] Dandan Zhang, Qiang Li, Yu Zheng,  Lei Wei, Dongsheng Zhang, and Zhengyou Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Ex-  plainable hierarchical imitation learning for robotic drink  pouring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' IEEE Transactions on Automation Science and  Engineering, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content='  [Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=', 2022] Boyuan Zheng, Sunny Verma, Jianlong  Zhou, Ivor W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Tsang, and Fang Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' Imitation learning:  Progress, taxonomies and challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}
+page_content=' IEEE Transactions  on Neural Networks and Learning Systems, pages 1–16,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf'}

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@@ -0,0 +1,990 @@
+On the saturation stress of deformed metals
+S. Queyreau1∗
+1 Universit´e Sorbonne Paris Nord, LSPM-CNRS,
+UPR 3407, 93430 Villetaneuse, France
+1
+arXiv:2301.00625v1  [cond-mat.mtrl-sci]  2 Jan 2023
+
+Abstract
+Crystalline materials exhibit an hysteresis behaviour when deformed cyclically. The origins of
+this tension-compression asymmetry have been fully understood only recently as being caused by
+an asymmetry in the junction strength and a reduced mean free path of dislocations inherited
+from previous deformation stage. Here, we investigate the saturation stress in fcc single- and poly-
+crystals using a Crystal Plasticity framework derived from dislocation dynamics simulations. In
+the absence of plastic localization and damage mechanism, the single-crystal mechanical response
+eventually saturates. We show that the cyclic saturation stress converges asymptotically to the
+monotonic saturation stress as the cycle plastic increment increases, and this convergence can be
+observed for some experimental conditions. The analysis of the experimental literature suggests
+that the mechanisms controlling the saturation in single crystals are the same controlling the cyclic
+response of polycrystals with large grains. We propose also analytical and approximated models to
+predict the saturation stress over the considered loading conditions. The saturation stress appears
+as a fundamental property of dislocations, explaining the consistency observed in the experimental
+literature. This work provides a unified view on the monotonous and cyclic responses of fcc single
+and poly-crystals, which may help in interpreting experimental data.
+I.
+INTRODUCTION
+Deformation of crystalline materials depends upon the deformation history undergone by
+the materials. This is particularly apparent through the existence of hysteresis curves when
+alternating the loading direction in tension and compression as in cyclic deformation [1–3].
+A part of the deformation is reversible and the resulting hardening is much smaller that in
+continuous monotonic deformation, explaining why deformation can be easily accumulated
+this way. At every two cycles, the flow stress may increase usually in a continuous manner,
+until reaching a saturation shear stress [4–10].
+In single crystals, this saturation stress
+depends upon the loading direction and the materials as it not always scales with the shear
+modulus [10]. For a given loading direction and material, the saturation stress typically
+increases with the cycle shear increment γp,cy, and the saturation stress may well be below
+the stress values observed in monotonous deformation. For example, in Cu -certainly the
+∗ sylvain.queyreau@cnrs.fr
+2
+
+most studied system- deformed in single glide condition, the cyclic saturation stress ranges
+from 16 to 40 MPa exhibits a three stage curve with a central plateau value of 28 MPa.
+In stable multislip conditions, the cyclic saturation stress typically increases monotonously
+with an apparent slope that depends upon the loading direction and the material [7–10]
+and may reach 50 or 60 MPa. In comparison the maximum stress reached by monotonically
+deformed single crystal of Cu is in the range of 80-100 MPa.
+Until recently, this tension-compression asymmetry -also known as the Bauschinger Ef-
+fect (BE)- was commonly thought to be related to the building up of Long-Range Internal
+Stresses (LRIS) or backstress associated to the formation of dislocation patterns according
+to the so-called composite model [3, 11, 12]. However, dislocation patterns are rather weak
+at smaller strain, they may well be different from the archetype of the composite model, and
+no LRIS was found in recent large scale mesoscale simulations [13] or by X-ray microdiffrac-
+tion [14, 15] (except at very large strain [16]). Another explanation consisted in the partial
+dissolution of the microstructure [17–20], as a mechanistic way to reduce dislocation den-
+sity and thus flow stress. However, no clear dislocation elementary mechanism was clearly
+identified as there is no in-situ observation of dislocation motion during cyclic deformation.
+Ultimately, these explanations explain neither the transient nature of the Bauschinger effect
+nor its reversibility component.
+The present authors [13, 21] recently proposed a systematic study of cyclic deformations
+in single crystals by means of Discrete Dislocations Dynamic (DDD) simulations. Despite
+of the present of a pronounced Bauschinger effect in the simulations, no LRIS was measured
+in the simulations. Statistical analysis of the DDD simulations showed that the tension-
+compression asymmetry is caused by two original elementary mechanisms of dislocations.
+The first mechanism i) is related to easy destruction of binary junctions as their stability is
+asymmetrical as they formed from mobile segments whose curvature is driven by the applied
+loading. Junctions formed is tension are thus more stable in tension than in compression, and
+are easily destroyed during the backward motion of dislocation in compression. The second
+elementary mechanism ii) is related to the reduction of the mean free path of dislocations
+as they glide in regions of the crystal that they already explored. In the backward motion,
+the mobile segments unwind stored segments on the edge of the swept area, leading to a
+reduction in the storage rate. These mechanisms naturally explain the transient nature of
+the tension-compression asymmetry and the reversible component of plastic deformation.
+3
+
+From these results, the authors proposed a modified Crystal Plasticity framework, that
+was implemented into a FEM to include the physics highlighted in the DDD. As a results
+this multiscale approach was successful in predicting the Bauschinger effect, hysteresis and
+saturation stress reported in most of the experimental literature on cyclically deformed fcc
+single crystals.
+Now that we have a CP framework justified from mesoscale physics and validated by
+confrontation with the existing single-crystal literature, we can now assess the full implica-
+tions of these results and of the model. In particular, when comparing the literature data
+obtained though out decades on cyclic deformation of single crystals - the saturation stresses
+- observed in cyclic deformation are clearly more reproducible that the Bauschinger effect
+observed on a single tension-compression experiment. This fact suggests that the saturation
+stress in cyclic deformation is the sole result of the average of some basic dislocation mech-
+anisms, while the Bauschinger effect is impacted by the initial state of the material, such as
+the impurity content, the initial dislocation density or the microstructure, but also by the
+loading conditions. In other words, the saturation stress seems to be a basic property of
+the material and dislocations. Simulation results could also be processed in order to provide
+simple equations to help in interpreting and/or fitting the experimental data or to trace
+back to some of the fundamental constants of the CP model e.g. in an inverse approach.
+Finally, the single crystal is an integral part of the more complex polycrystal system. The
+elementary mechanisms described above are general enough to be operative in a polycrystal.
+The insertion of GB adds a new lengthscale, and a competition between this lengthscale and
+the ones associated with elementary mechanisms i and ii. These ideas will act as motivations
+of the present paper.
+The objectives of the present work are thus to extend and analyse the CP FEM results
+obtained for the cyclic deformation of fcc single-crystals. We will see that the saturation
+stress of cyclic loading is actually related to the theoretical saturation stress obtained in
+monotonous conditions. Analytical solutions to the constituting differential equations of the
+CP framework will be provided and can be employed to help in interpreting experimental
+data.
+Finally, the implication of the mechanisms i and ii on the cyclic deformation of
+polycrystal will be assessed.
+4
+
+II.
+METHODOLOGY
+This section presents the Crystal Plasticity framework derived from the DDD results in
+[13]. Readers interested in the presentation of the technical details of the DDD technique
+along with a review of recent progresses are referred to [22, 23].
+This CP model is an
+extension of the dislocation density based descriptions proposed in the seminal work [24, 25]
+and subsequent works inspired from DDD [26] for monotonous loadings.
+In fcc metals
+at intermediate and high temperature, the flow stress is controlled by the formation and
+destruction of reactions -junctions- between dislocations belong to different slip systems.
+The critical shear stress τ i
+c on the active slip system ’i’ is thus related to the forest obstacle
+densities ρj and expressed as :
+τ i
+c = µb
+��
+j
+aijρj
+(1)
+Where µ is the isotropic shear modulus, b the norm of the Burgers vector and aij the
+components of the interaction matrix. These last interaction coefficients measure the average
+strength of the two interacting slip systems and can be determined in a straigthforward
+manner from DDD [26–29].
+For reasons of symmetry, only six different interactions of
+different nature exist among the twelve a0/2 < 110 > 111 slip systems existing in fcc metals.
+The interactions are the self-interaction, the coplanar interaction, three junctions: Lomer,
+Hirth and glissile reactions, and the collinear interaction. The interaction matrix has to
+have non identical components in order to reproduce fully the anisotropy of the plastic
+deformation of fcc single-crystals.
+A first extension of the CP framework was introduced to capture the asymmetry of
+dislocation junctions (mechanism ’i’ in the introduction) and to reproduce the tension-
+compression asymmetry observed in DDD [13]. Junctions formed during the forward loading
+from curved segments inherit a mechanical stability asymmetry. The junctions formed during
+prestrain or forward tension loading, are stronger when stressed in tension and statistically
+weaker when now stressed in compression, and this asymmetry in mechanical stability can be
+considered as a line tension effect, with the parent moving segments colliding when they were
+in extension. As a results, while plastic flow is still controlled by junctions destruction, the
+interaction coefficients during the backward loading are effectively weakened over a transient
+that depends on the amount of the initial forward deformation. The pool of weak junctions
+gets destroyed as backward deformation proceeds, until mobile segments explore new area
+5
+
+of the crystal and form junctions polarized in compression. A reversibility function ra was
+introduced to reproduce these effects as:
+abck
+ij = (1 − ra) × aij, with ra = exp
+�
+�−
+γi
+bck
+Cab
+�
+∆ρi
+pr
+�
+�
+(2)
+with aij the reference and constant interaction coefficient measured in continued tension.
+the subscripts ’pr’ and ’bck’ refer to prestrain and backward loading, respectively. In a
+cyclic loading, the prestrain becomes the previous cycle, while the backward loading cor-
+responds to the current cycle. The amount of backward strain impacted by this transient
+on the junction stability is approximated as Cab
+�
+∆ρi
+pr. This last equation can provide a
+lengthscale estimate to this effect. The density increase during previous deformation cy-
+cle is ∆ρi
+pr = max(ρi
+pr) − min(ρi
+pr) and corresponds to the potentially impacted dislocation
+density polarized according to the initial loading. The constant Ca has been determined as
+Ca = 0.6 ± 0.1 through a statistical analysis of the interaction coefficients over a large panel
+of relevant DDD simulations [13]. The γi
+bck/Cab
+�
+∆ρi
+pr term within the exponential of ra
+states the competition between the easy destruction of junctions formed in tension, and the
+formation of new junctions in never-explored regions of the crystal.
+A second fundamental equation in the CP framework describes the evolution of the
+dislocation density on active slip system ’i’ with the system shear γi.
+The evolution of
+the dislocation density is related to the kinetics of plastic activity that occurs through
+intermittent busts or avalanches of dislocation motion. In principle, dislocations are stored
+at the end of an avalanche as dislocation segments left at the edge of the swept area. This
+being said the exact theoretical connection between dislocation avalanches at the mesoscale
+and the observed continuous macroscale storage of dislocations has still to be formulated.
+For the present work and in monotonic loading conditions, the density evolution takes the
+following simple form:
+dρi
+dγi = 1
+b
+� 1
+Lhkl
++ 1
+LI
+− yρi
+�
+= 1
+b
+�
+�
+��
+i̸=j aijρj
+Khkl
++
+�
+a′
+0ρi
+KI
+− yρi
+�
+�
+(3)
+where the first two terms relate to the dislocation storage and the last term represents the
+dynamic recovery. The storage rate is commonly related to the Mean Free Path (MFP)
+of dislocations Li, which represents the average distance covered by dislocations before the
+temporary or permanent storage.
+The MFP of dislocations typically depends upon the
+6
+
+loading axis and thus the number of active slip systems. In [24, 26], Kubin et al. proposed
+a simple decomposition of the MFP into elementary ingredients: i) the rate p0 of forming
+a junction that will ultimately store a dislocation segment, and ii) the average length of
+stored dislocation < l > and iii) junction segments and < lj >, both of which scale with
+the dislocation density < l >= k0/√¯aρ and < lj >= κ < l >. ¯a is the average interaction
+coefficient for the considered orientation. Statistical analysis of DDD results showed that
+parameters k0, p0, κ to be constants independent of the loading direction. The MFP can
+thus be written:
+Li = µb2
+τ i
+c
+� √¯an(1 + κ)3/2
+p0k0(n − 1 − κ)
+�
+(4)
+the third term is associated to the dynamic recovery occurring at large dislocation density.
+y is related to critical distance at which two dislocations can easily annihilate, which may be
+measured from atomistic data. Here, y takes different values depending on the loading axis
+to reproduce the anisotropy of the onset of stage III observed on experimental deformation
+curves on single-crystal [30, 31].
+Table I. List of physical parameters employed in the CP simulations of the cyclic deformation of fcc
+single crystals. Most of these parameters are coming from DDD simulations for monotonic or cyclic
+deformation, few parameters are coming from the experiemental litterature but for monotonic
+loading only. No parameters or equations were fitted to the cyclic deformation literature. The
+resulting FEM simulations are therefore truly predictive of cyclic deformation.
+a′
+0 (self) aortho (Hirth) a2 (glissile)
+a3 (Lomer)
+acolli (Collinear)
+ρref
+ρ0
+0.122
+0.07
+0.137
+0.122
+0.625
+1012 m−2
+1012/nsys
+KI
+K112
+K111
+K001
+Ca
+Ap
+Cp
+90
+10.42
+7.29
+5
+0.6 ± 0.1
+2 ± 0.6
+2.3 ± 0.3
+˙γapp
+˙γi
+0
+m
+yI (SG, [112])
+y001 (nm)
+y111 (nm)
+nsys × ˙γi
+0
+10−4 s−1
+35
+0.5 nm
+3.6 (Ni), 3.4 (Cu) 2 (Ni), 1.5 (Cu)
+When considering the cyclic deformation, DDD simulations [13, 21] showed that dislo-
+cation evolution is still associated to the storage of segments in the wake of avalanches at
+the fringe of the area swept by dislocations, with some similarities to what was observed
+in alloys [32]. Equation 3 is thus still valid. However, the reduced storage (mechanism ii
+7
+
+in the introduction) observed at the loading reversal where the stored segments are simply
+unwinded, needs to be taken into account through a modified MFP (through a change of
+rate of locking dislocation rate p0). A second reversibility function rp is thus introduced
+[13, 21] as:
+pbck
+0
+= (1 − rp) × p0, with rp = Ap × exp
+�
+�−
+γi
+bck
+Cpb
+�
+∆ρi
+pr
+�
+�
+(5)
+Where Ap and Cp are two additional constants measuring the initial MFP drop and the
+transient length of the reduced MFP, respectively. Statistical analysis over the transient
+observed in DD showed that Aa = 2±0.6 and Cp = 2.3±0.3. Interestingly, the transient on
+the reduced MFP (ii) is much larger that the one on the reduced junction stability (i). These
+different transients explain the non-monotonic response at the beginning of the backward
+deformation that is sometimes observed on experimental deformation curves [33–35].
+Finally, the flow rule provides a close form to the CP framework in connecting the plastic
+activity ˙γi on the active system to the critical resolved shear stress τ i
+c and the applied shear
+stress τ i:
+˙γi = ˙γi
+0
+�τ i
+τ i
+c
+�m
+(6)
+In the case of fcc metals at room temperature, the strain rate sensitivity is related to the
+formation and dragging of jogs along dislocation lines. Here, the constant γi
+0 is taken as
+γapp/nact, with γapp the applied shear rate and nact the number of active slip system. The
+sensitivity exponent is chosen as m > 35 to stay as close as possible to the Schmid criterion.
+The proposed CP modeling focuses on the some fundamental mechanisms of dislocations
+that are described by some average constants obtained from statistical analysis of DDD
+results. The resulting FEM simulations are expected to be general and representative of
+fcc pure systems. The simulations will not include plastic localization nor finite geometry
+effects for now.
+Great care has been paid to the numerical resolution of the set of ODE presented above
+using the Z-Set Finite Element software and Matlab. We employ a double nested Newton-
+Raphson implicit scheme to solve the corresponding non-linear equations and obtain ρi and τ i
+for a given time step. Gradients required in the Newton-Raphson are expressed analytically,
+and a relative convergence criterion was set to 10−7. The representative volume was set to
+a single point of integration in order to simulate several thousand of cycles of deformation
+and finite geometry effects are left for a future work. The parametrization and resolution
+8
+
+10-2
+10-1
+100
+1012
+1013
+1014
+1015
+0
+0.2
+0.4
+0.6
+0.8
+0.9
+0
+2
+4
+6
+8
+10
+107
+0
+1
+2
+3
+4
+5
+0
+2
+4
+6
+8
+10
+12
+14
+107
+τ (MPa)
+s.g.
+τ (MPa)
+ρ (m−2)
+s.g.
+[001]
+[111]
+[112]
+[001]
+[112]
+[111]
+[001]
+[111]
+[112]
+a
+b
+c
+γ
+γ
+γ
+τI
+sat,mo
+τ001
+sat,mo
+τ111
+sat,mo
+ρ111
+sat,mo
+a
+a
+ρI
+sat,mo
+Figure 1. a. One-to-one comparison between the CP prediction (thick continuous lines) and the
+reference data from Takeuchi [30] (circles) on the monotonous deformation of Cu single-crystals
+at RT. Key single-crystal orientations are chosen. b. Extension of the CP prediction until true
+saturation of the stress is obtained, in the absence of finite geometry and plastic localization effects.
+c. Corresponding total dislocation density evolutions in log-log scale. Note that the saturation
+density for [001] and [111] simulations are of the order of > 1015 m−2, which is still lower than the
+saturation density obtained in recent large scale MD simulations of single-crystals of Al [36].
+strategy were validated in [13] by the one-to-one comparison of CP predictions with the
+reference DDD simulations.
+III.
+RESULTS
+A.
+Saturation stress in monotonic deformation in absence of damage
+In this first section, we focus on the basic case of the single-crystal deformed monotonically
+in tension. This constitutes as well-pose problem, for which reference experimental data
+exists. The plastic response is mostly homogeneous during most of the deformation, in the
+sense that the slip activity is similar in all part of the single-crystalline sample [37]. We will
+see later that this basic plastic response can be connected to cyclic deformation.
+First, to demonstrate the validity of the CP framework and its parametrization from
+DDD results, we start with a one-to-one confrontation of the CP prediction with some of
+the reference data from the experimental literature on single crystal deformation [30]. Such
+experiments are rather delicate to perform as the materials state and experimental conditions
+can have a dramatic impact on the mechanical response, such as the impurity content, the
+9
+
+precise orientation of the sample, or whether the jaws can rotate to accommodate crystalline
+rotation and ensure uniaxial deformation. Here, we compare our CP simulations with the
+reference work from Takeuchi on Cu single-crystals for selected loading directions: [112],
+[111] and [001] directions. The initial density was slightly adjusted in the simulations about
+1012 m−2 to capture the initial flow stress.
+This comparison is shown in Figure 1.a, where a nice quantitative agreement is found
+between the simulations and the experimental reference data. The classical picture of the
+response of the single crystal is recovered here. The initial hardening rate depends upon
+the orientation and the number of slip systems activated simultaneously, e.g. in a increasing
+order: [112], [111] and [001] having two, three and four slip systems activated simultaneously.
+For [111] and [001] loading directions, these numbers of slips correspond to only half of the
+six or eight possible slip systems as these are pairs of colinear interactions. The colinear
+interaction is known to be a specific reaction among junctions, as it leads to the annihilation
+of the intersecting segments of dislocations and is associated to a very large hardening due
+to the shortening of mobile segments. DDD simulations have shown that when starting
+with pairs of colinear systems, one of the colinear system will take over the other for [111] or
+[001] simulations [38]. Another notable aspect of the single-crystal response is the anisotropic
+onset of the stage III, when the dynamic recovery becomes noticeable. The dynamic recovery
+of the [111] in particular is weaker, so much so that the [111] curve eventually crosses the
+initially steepest [001] curve at a deformation of about 40% of plastic strain.
+In agreement with the response of ductile materials, prior to the fracture of the sample,
+the experimental flow stress decreases after a maximum corresponding to the striction of
+the sample. This decrease is expectedly absent in the CP simulations as plastic localization
+and fracture mechanics are not included. Fracture mechanics may be included in different
+manner in FEM, for example using a cohesive zone framework [39, 40], in connection to
+atomistic mechanisms controlling the debonding of matter [41–43]. For now, let us focus on
+the plastic response in the absence of fracture mechanism. For a deformation corresponding
+to the experimental fracture, the hardening rate in the simulations has still a non-zero value.
+When recalling that equation 3 was proposed initially to saturate, the CP deformation curves
+are thus expected to saturate at a larger deformation.
+We thus expanded the deformation range until saturation of the stress is obtained, and
+this is shown in figure 1.b for Cu. All curves saturate eventually, the corresponding deforma-
+10
+
+tion at saturation is however rather large, even beyond 100% for the [001] and [111] curves.
+The saturation stress in monotonous condition τsat,mo follows the following hierarchy, with
+the single glide condition [135] having the smallest saturation stress of 28 MPa, [112], [001]
+and [111] having the largest τsat,mo = 140 MPa. Since yc is material independent for the
+[135] and [112] curves, the τsat,mo for these curves scales nicely with the shear modulus of
+the considered materials. This saturation stress values will be exploited a bit further.
+From a theoretical point of view, this saturation stress observed in the simulation can
+be predicted from the set of constituting ODE of the CP framework. The starting point is
+obviously the dislocation density evolution that has to be set to zero. In the case of single
+glide the resolution is straigthforward leading to:
+ρsat,I =
+�
+a′
+0
+K2
+I y2
+c,I
+(7)
+The stable multislip conditions are a bit less straightforward to solve as the interaction
+coefficient exhibit a dependence upon the dislocation density inherited from the simplified
+line tension that is assumed in the critical stress equation [28, 44]. When the dislocation
+density is large this correction becomes important, which is the case at the large satura-
+tion stresses under consideration here.
+The logarithmic correction is defined as c(ρi) =
+ln
+�
+1/b
+�
+aijρj
+�
+/ ln
+�
+1/b√aijρref
+�
+, where ρref is the reference dislocation density used to de-
+termine the interaction coefficient aij. The saturation density is now:
+ρsat,hkl =
+�
+�
+�
+a′
+0
+KIyhkl
++
+�
+(nact − 1)¯ac(ρi)
+Khklyhkl
+�
+�
+2
+(8)
+where nact is the number of activated systems. This equation is an implicit equation for ρi,
+but since the density is contained in a logarithm function, this equation converges after only
+few iterations to evaluate ρsat. Finally, the saturation stress is obtained in inserting the ρsat
+into the critical stress:
+τsat = µb√¯anactρsat
+(9)
+In the past equations, the saturation stresses are solely function of interaction coefficients,
+MFP and y. In the model, these quantities are defined from statistical averages over evolv-
+ing dislocation microstructures, and are weakly impacted by the nature of the fcc system
+under consideration (at least not as a first order approximation). The initial dislocation
+microstructure or density are absent from these equations. The saturation stresses appear
+11
+
+10-4
+10-3
+10-2
+10
+20
+30
+50
+100
+150
+10-4
+10-3
+10-2
+20
+40
+60
+80
+100
+120
+140
+10-5
+10-4
+10-3
+10-2
+10-1
+20
+30
+40
+50
+60
+70
+τ001
+sat,mono
+τ111
+sat,mono
+τI
+sat,mo
+[001]
+[111]
+Cu
+Cu
+Ni
+Ni
+s.g.
+Ni
+Cu
+Ag
+τ112
+sat,mo
+τI
+sat,mo
+τI
+sat,mo
+τI
+sat,cy (MPa)
+τ001
+sat,cy (MPa)
+τ111
+sat,cy (MPa)
+γp,cy
+γp,cy
+γp,cy
+τ112
+sat,mo
+Figure 2.
+The saturation shear stress τsat,cy for cyclic deformation in fcc metals measured in a
+number of experimental studies [4–10] compared to the prediction of the CP model in [13, 21] and
+to the theoretical saturation stress in monotonous conditions from previous section. Data is shown
+as function of the loading direction: a) initially single glide condition, b) [001] and c) [111] for Cu,
+Ni and Ag and the plastic strain increment γp,cy
+thus as a fundamental dislocation property representing the balance of storage and recovery
+processes among dislocations in fcc crystals.
+An approximate solution for the evolution ρi(γi) is provided in the appendixes for single
+and stable multislip conditions. Theses functions can be used to interpret or fit experimen-
+tal data. In monotonous condition, the dislocation density increases monotonically until
+saturation. The strain corresponding to the saturation density can be estimated from the
+density evolution as an integral:
+∆γi =
+� ρsat
+ρ0
+�
+�1
+b
+�
+�
+�� aijρj
+Khkl
++
+�
+a′
+0ρi
+KI
+− yρi
+�
+�
+�
+�
+−1
+dρi
+(10)
+This only provides an estimation as the saturation density is only reached as γi tends to
++∞. We now recover a dependence upon the initial dislocation density ρ0 that affects the
+amount of deformation required before saturation.
+B.
+Saturation stress in cyclic deformation in single crystals
+The CP framework can be applied to alternating loadings and the simulations capture
+most of the features of cyclic deformation [10, 45]. The maximum stress reached at each
+cycle increases monotonically until spontaneously saturating. For example in single glide
+12
+
+conditions, we recover the so-called three stage curves of the now saturation stress in cyclic
+deformation τsat,cy with respect to the shear increment per cycle γp,cy. This is shown in figure
+2.a for some fcc metals in comparison with most of the existing experimental literature [4–
+10]. In the central plateau (or pseudo plateau) the microstructure typically transform from
+no Persistent Slip Band (PSB) to progressively fully constituted of PSB [10, 45]. The plateau
+ends for strain increments of the order of 1% when a secondary slip system activates. In the
+CP simulations, the center region of the curve is not exactly a plateau as it exhibits a slight
+slope.
+Figure 2.b and 2.c show the saturation stress for cyclic deformation of Cu and Ni single
+crystals oriented along [001] and [111] in stable multislip conditions. The saturation stress
+τsat,cy increases here monotonically without the presence of stages. The slope of the exper-
+imental curves seems however to decrease as the strain increment per cycle γp,cy increases.
+This effect is more apparent on the data concerning Ni single crystals. The CP predictions of
+the τsat,cy are in nice qualitative and quantitative agreement with the existing experimental
+data. The CP predictions seem however to overestimate the saturation stress at larger γp,cy,
+and this was suggested to be a consequence of the absence of dislocation microstructure in
+the CP simulations [13].
+The saturation stress τsat,cy can be also obtained from the analytical resolution of the
+set of ODE presented in section 2. The resolution is however a bit more delicate with the
+presence of the two reversible functions ra and rp that are themselves function of γi and the
+dislocation density ∆ρi stored on the previous cycle. In cyclic deformation, we highlight that
+the saturation stress does not mecessary correspond to an horizontal tangent of the shear
+strain-shear stress curves of the considered cycle (except in the large γp,cy limit discussed
+a bit further). At the end of a cycle at saturation, the hardening may well be non zero,
+but wince the cycle starts with a large dislocation density decrease, the increment over the
+entire cycle is null. To obtain the saturation density, one has to solve the integrated density
+evolution function:
+∆ρi = 0 =
+� γp,cy/nact
+0
+1
+b
+�
+�
+�� aijρj
+Khkl
++
+�
+a′
+0ρi
+KI
+− yρi
+�
+� dγi
+(11)
+Therefore, the dislocation evolution ρi(γi) must be solved first contrary to the monotonous
+case. This can be done if the logarithm correction is approximated as a Taylor series where
+the dislocation ρi in the current cycle is in the proximity of ρi
+sat. The details are given in
+13
+
+the Appendixes. The resolution leads once more to an implicit equation to obtain ρi
+sat.
+Next, we connect the saturation stresses obtained in monotonous and cyclic deformation.
+Previous section has shown that the saturation stress in monotonous conditions can be
+obtained for rather large deformation amount. These saturation stress results could thus be
+drawn as well on the figure 2 for one cycle of the corresponding amount γp,cy. One would
+expect that the τsat,mono would act as an asymptotic limit to the τsat,cy curves. This is exactly
+what is obtained when considering the case of single glide condition. The saturation stresses
+obtained in single glide and monotonous conditions in previous section are strikingly close
+to the saturation stress at the plateau in cyclic conditions. The τsat,cy curves obtained from
+experiments or from the CP simulations asymptotically converge towards τsat,mono.
+When now considering the third stage of the curves in figure 2.a or the multislip condi-
+tions in figure 2.b and 2.c, the agreement with experiments is a bit less obvious. The CP
+simulations are carried out past the range of γp,cy considered in the experiments. Similarly
+to the single glide condition, the simulated τsat,cy curves converges towards the τsat,mono limit
+defined in previous section. In figure 2.a, the third stage converges certainly toward the
+τsat,mono in secondary double glide, that is probably out of reach. The agreement with the
+experimental data is rather good for the case of Cu. However, the quantitative agreement
+between with the model is a bit less good for Ni single crystal data, nonetheless the exper-
+imental data seem to enter a plateau for the largest γp,cy considered experimentally, which
+could be well agree with the analysis proposed here, but not the quantitative values of the
+saturation stress. The τsat,mono being here overestimated in the model due maybe because
+of some specific parameters less well documented for Ni, such as yhkl or the stronger impact
+of microstructures that are absent in the model.
+C.
+Saturation stress in polycrystals
+This last section aims at connecting our results on the single crystal to the more complex
+and general polycrystalline system. What comes next is simply an illustration as the plastic
+response of the polycrystals is the average of many different features or elementary mech-
+anisms that are not addressed here, ranging from the crystallographic texture, grain size
+and morphology, and dislocation-grain boundary interactions. Besides, simulating using CP
+FEM the cyclic deformation over more than 10,000 cycles while accounting for the a rep-
+14
+
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+0
+50
+100
+150
+200
+250
+σsat,cy (MPa)
+Δϵp,cy/2
+Figure 3.
+compilation of saturation stresses obtained experimentally on polycrystalline Cu with
+large grain size [46] as function of strain increment ∆ϵp,cy. The data covers the work from Sax-
+ena and Antolovich (1975, squares), Figueroa et al. (1981, circles), Mughrabi and Wang (1981,
+triangles) and Rasmussen and Pederson (1980, diamonds)
+resentative polycrystalline microstructure seems computationally out of reach, still to date.
+This being said we reprise a part of the analysis initiated by Magnin et al. in [46], where
+they compiled the cyclic response of polycrystalline Cu with relatively large grains. The
+collection of saturation stresses σsat from the literature is displayed in figure 3. These curves
+typically show an increase in the σsat with the plastic increment per cycle ϵp,cy, some curves
+may suggest a quasi plateau regime starting for ϵp,cy > 10−5 and ending around ϵp,cy ≈ 10−3.
+The deformation microstructures transform from veines to PSB that are typical of the pat-
+terns observed in single glide conditions in single crystals. For ϵp,cy >≈ 10−3, the saturation
+stress increases more rapidly. The deformation microstructures now correspond to mazes,
+that are typical of the patterns observed in single-crystals deformed in multislip conditions.
+The polycrystalline plastic response can be understood in terms of effective single-crystal
+response through the so-called Taylor coefficient M, which depends upon the crystallographic
+texture. The so-called Hall-Petch effect is neglected and this is a valid assumption in the
+case of large grain polycrystals. In the absence of precise M measures, the Taylor or Sachs
+hypothesises are commonly considered as bounding limits. The figure 4 shows theses two
+transformations of the polycrystal saturation stress into a single crystal approximation as
+15
+
+10-5
+10-4
+10-3
+10-2
+10-1
+10
+20
+30
+40
+50
+60
+70
+Cu
+s.g.
+[001]
+[111]
+PolyX Sachs
+PolyX Taylor
+γp,cy
+τsat,cy (MPa)
+Figure 4.
+Comparison of the cyclic saturation shear stress of single crystals and polycrystals in
+pure Cu from Mughrabi [47] and Wang using the Taylor coefficient following the procedure from
+Magnin et al. [46].
+τsat = σsat/M. The Sachs is typically a good approximation at small strain, while the Taylor
+approximation works well at large strain. The average single crystal behaviour is expected
+to follow Sacks approximation first and transition to the Taylor ones at larger deformation.
+These curves will be compared with the saturation stress obtained experimentally on single
+crystals. At small strain, one might expect that grains in the polycrystal deform in sin-
+gle glide condition as suggested by the dislocation microstructures, and we see a sticking
+agreement between the single crystal τsat and the Sachs approximation for ϵp,cy < 5.10−4.
+For very small strain, so grains may remain in the elastic regime. For large deformation,
+one might expect that all grains to deform in multislip conditions, in accordance with the
+maze microstructure observed experimentally, and the [001] curve in particular remains be-
+tween the two approximations. Between these two extreme situations, the polycrystalline
+behaviour is certainly a composite response of the single glide and multislip grain activity,
+with more and more grains transitioning into multislip as γp,cy is increased.
+These correlations may thus explain the shape of the raw σsat curves from the previous
+figures. Most grains of the polycrystals are thus following the single crystal response, which
+will eventually reach a saturation stress corresponding to τsat,mono of 28 MPa. The plateau in
+16
+
+the single-crystal system is transformed in a quasi-plateau in the polycrystalline system as
+more grains transition into multislip conditions. At large ϵp,cy = ϵIII ≈ 10−3 the polycrystal
+enters stage III where most of the grains are deformed in multislip conditions. Interestingly,
+the curves for single slip and [001] obtained for the single crystal cross for a deformation
+≈ MγIII = M ∗ 3e−3 that corresponds well to the onset of the third stage ϵIII observed
+in the figure 4. The cyclic behaviour of the polycrystal with large grains seems thus to
+correspond simply to the average of the individual single grain behaviour, similar to what
+is accepted for polycrystals in monotonic deformations.
+Finally going back to our CP framework, we showed that its predictions were in quan-
+titative agreement with the single crystal cyclic response, and we proposed expressions to
+predict the saturation stress in these conditions. The previous qualitative analysis means
+that we can use these results to estimate the saturation stress of the polycrystal with large
+grains as well.
+IV.
+CONCLUSION
+In this paper, we employ a physically based CP FE model derived from our recent DDD
+analysis [13, 21] to analyse the saturation stresses obtained in monotonic and cyclic defor-
+mations of single-crystals and polycrystals of fcc metals.
+• First, we compare the model results to reference strain-stress deformation curves from
+the experimental literature on Cu single-crystals. In the absence of plastic localization
+and fracture mechanism, the CP model expectedly saturates at large strain.
+• For cyclic deformations, the CP model reproduces the cyclic saturation over various
+loading conditions and fcc metals. The saturation stress observed in monotonic con-
+ditions acts as an asymptotic behaviour at large cycle increment strain γp,cy to the
+cyclic deformation. This asymptotic behaviour is clearly reached in the plateau stage
+of single-glide conditions for several fcc metals and for Ni single crystals in multi-slip
+conditions at large strain increment.
+• From an analysis of the experimental literature on large grain polycrystals, we have
+shown that the polycrystalline response to cyclic deformation corresponds to the re-
+sponse of an aggregate of effective single-crystals. At small strain increment, grains
+17
+
+are deformed elastically or in single slip, a quasi plateau can thus be sometimes seen
+as in single-crystals. Then at larger strain, grains transition into multislip conditions
+with larger saturation stresses.
+• We proposed analytical or approximated solutions of the CP ODE to predict the
+saturation stresses in single or multislip conditions for single crystals. These models
+can be employed to interpret experimental data.
+• These results were obtained in the absence of plastic localisation and dislocation mi-
+crostructure, which means that these features have only a secondary order impact on
+the macroscopic behaviour.
+• The saturation stress in monotonic or cyclic conditions appear as a fundamental prop-
+erty of dislocations mechanism as it is simply related to averages of dislocations inter-
+actions (through the interaction coefficients), dynamical recovery mechanism (through
+y), and reversibility part of the microstructure.
+• These quantities are phenomenological averages from DDD, where large number of
+binary interactions are occurring in absence of patterns.
+• Experimental data on the saturation stress can thus now be used to define these
+physical quantities and the procedure can be applied to other bcc or hcp crystalline
+systems.
+[1] J. Bauschinger, ¨Uber die ver¨anderung der elastizit¨atsgrenze und der festigkeit des eisens und
+stahls durch strecken und quetschen, durch erw¨armen und abk¨uhlen und durch oftmals wieder-
+holte beanspruchung, Mitteilungen des mechanisch-technischen Laboratoriums der K¨oniglich
+Technischen Hochschule M¨unchen 13 (1886).
+[2] L. Brown and W. Stobbs, The work-hardening of copper-silica, Philosophical Magazine 23,
+1201 (1971).
+[3] A. S. Argon, Strengthening mechanisms in crystal plasticity, Oxford series on materials mod-
+elling No. 4 (Oxford Univ. Press, Oxford, 2008) oCLC: 255673019.
+18
+
+[4] A. S. Cheng and C. Laird, Mechanisms of fatigue hardening in copper single crystals: The
+effects of strain amplitude and orientation, Materials Science and Engineering 51, 111 (1981).
+[5] H. Mughrabi, The cyclic hardening and saturation behaviour of copper single crystals, Mate-
+rials Science and Engineering 33, 207 (1978).
+[6] T. Lepist¨o, V.-T. Kuokkala, and P. Kettunen, Dislocation arrangements in cyclically deformed
+copper single crystals, Materials Science and Engineering 81, 457 (1986).
+[7] J. Bretschneider, C. Holste, and B. Tippelt, Cyclic plasticity of nickel single crystals at elevated
+temperatures, Acta Materialia 45, 3775 (1997).
+[8] B. Gong, Z. Wang, and Z. Wang, Cyclic deformation behavior and dislocation structures of
+[001] copper single crystals—I Cyclic stress-strain response and surface feature, Acta Materi-
+alia 45, 1365 (1997).
+[9] P. Li, Z. Zhang, X. Li, S. Li, and Z. Wang, Effect of orientation on the cyclic deformation
+behavior of silver single crystals: Comparison with the behavior of copper and nickel single
+crystals, Acta Materialia 57, 4845 (2009).
+[10] P. Li, S. Li, Z. Wang, and Z. Zhang, Fundamental factors on formation mechanism of disloca-
+tion arrangements in cyclically deformed fcc single crystals, Progress in Materials Science 56,
+328 (2011).
+[11] R. J. Asaro, Elastic-plastic memory and kinematic-type hardening, Acta Metallurgica 23,
+1255 (1975).
+[12] H. Mughrabi, Dislocation clustering and long-range internal stresses in monotonically and
+cyclically deformed metal crystals, Revue de Physique Appliquee 23, 367 (1988).
+[13] S. Queyreau and B. Devincre, A Multiscale Investigation of the Physical Origins of Ten-
+sion–Compression Asymmetry in Crystals and their Implications for Cyclic Behavior (2021),
+working paper or preprint.
+[14] M. E. Kassner, P. Geantil, L. E. Levine, and B. C. Larson, Mapping mesoscale heterogeneity
+in the plastic deformation of a copper single crystal, Int. J. Mech. Sci. 100, 333 (2009).
+[15] M. E. Kassner, P. Geantil, and L. E. Levine, Long range internal stresses in single-phase
+crystalline materials, In Honor of Rob Wagoner, International Journal of Plasticity 45, 44
+(2013).
+[16] L. E. Levine, M. R. Stoudt, A. Creuziger, T. Q. Phan, R. Xu, and M. E. Kassner, Basis for
+the Bauschinger effect in copper single crystals: changes in the long-range internal stress with
+19
+
+reverse deformation, Journal of Materials Science 54, 6579 (2019).
+[17] S. Buckley and Entwistle, K.M., The bauschinger effect in super-pure aluminum single crystals
+and polycrystals, Acta Metallurgica 4, 352 (1956), publisher: Pergamon.
+[18] A. Sleeswyk, M. James, D. Plantinga, and W. Maathuis, Reversible strain in cyclic plastic
+deformation, Acta Metallurgica 26, 1265 (1978).
+[19] C. D´epr´es, M. Fivel, and L. Tabourot, A dislocation-based model for low-amplitude fatigue
+behaviour of face-centred cubic single crystals, Scripta Materialia 58, 1086 (2008).
+[20] E. Rauch, J. J. Gracio, F. Barlat, and G. Vincze, Modelling the plastic behaviour of met-
+als under complex loading conditions, Modelling and Simulation in Materials Science and
+Engineering 19 (2011).
+[21] S. Queyreau and B. Devincre, On the Origins of Tension–Compression Asymmetry in Crystals
+and Implications for Cyclic Behavior (2020), working paper or preprint.
+[22] B. Devincre, R. Madec, G. Monnet, S. Queyreau, R. Gatti, and L. Kubin, Mechanics of nano-
+objects (Presses de l’Ecole des Mines de Paris, 2011) Chap. Modeling crystal plasticity with
+dislocation dynamics simulations: The ’microMegas’ code.
+[23] S. Queyreau, Dislocation based mechanics: the various contributions of dislocation dynamics
+simulations (2020).
+[24] L. Kubin, B. Devincre, and T. Hoc, Modeling dislocation storage rates and mean free paths
+in face-centered cubic crystals, Acta Materialia 56, 6040 (2008).
+[25] U. Kocks and H. Mecking, Physics and phenomenology of strain hardening: the fcc case,
+Progress in Materials Science 48, 171 (2003).
+[26] B. Devincre, L. Kubin, and T. Hoc, Physical analyses of crystal plasticity by dd simulations,
+Scripta Materialia 54, 741 (2006).
+[27] S. Queyreau and B. Devincre, Bauschinger effect in precipitation-strengthened materials: A
+dislocation dynamics investigation, Philosophical Magazine Letters 89, 419 (2009).
+[28] S. Queyreau, G. Monnet, and B. Devincre, Orowan strengthening and forest hardening super-
+position examined by dislocation dynamics simulations, Acta Materialia 58, 5586 (2010).
+[29] R. Madec and L. P. Kubin, Dislocation strengthening in fcc metals and in bcc metals at high
+temperatures, Acta Materialia 126, 166 (2017).
+[30] T. Takeuchi, Work hardening of copper single crystals with multiple glide orientations, Trans-
+actions of the Japan Institute of Metals 16, 629 (1975).
+20
+
+[31] L. Kubin, Dislocations, mesoscale simulations and plastic flow, in Oxford Series On Materials
+Modelling, Vol. 5, edited by A. Sutton and R. Rudd (Oxford University Press, 2013).
+[32] S. Queyreau, G. Monnet, and B. Devincre, Slip systems interactions in [alpha]-iron determined
+by dislocation dynamics simulations, International Journal of Plasticity 25, 361 (2009).
+[33] H. Ebener, Acoustic emission and the Bauschinger effect in Cu single crystals, Scripta Metal-
+lurgica et Materialia 25, 2035 (1991).
+[34] R. C. Daniel and G. T. Horne, The Bauschinger effect and cyclic hardening in copper, Met-
+allurgical Transactions 2, 1161 (1971).
+[35] Y. Nasu, T. Takeda, T. Tominaga, and O. Kato, The Bauschinger Effect of Aluminum Single
+Crystals Tested under Plane-Strain Compression, Bulletin of JSME 27, 145 (1984).
+[36] L. A. Zepeda-Ruiz, A. Stukowski, T. Oppelstrup, N. Bertin, N. R. Barton, R. Freitas, and
+V. V. Bulatov, Atomistic insights into metal hardening, Nature Materials 20, 315 (2021).
+[37] C. Kahloun, G. Monnet, S. Queyreau, L. Le, and P. Franciosi, A comparison of collective
+dislocation motion from single slip quantitative topographic analysis during in-situ AFM room
+temperature tensile tests on Cu and Fe alpha crystals, International Journal of Plasticity 84,
+277 (2016).
+[38] B. Devincre, L. Kubin, and T. Hoc, Collinear superjogs and the low-stress response of fcc
+crystals, Scripta Materialia 57, 905 (2007).
+[39] Y. Mishin, M. Asta, and J. Li, Atomistic modeling of interfaces and their impact on mi-
+crostructure and properties, Acta Materialia 58, 1117 (2010).
+[40] J. R. Rice and J.-S. Wang, Embrittlement of interfaces by solute segregation, Materials Science
+and Engineering: A Poceedings of the Symposium on Interfacial Phenomena in Composites:
+Processing Characterization and Mechanical Properties, 107, 23 (1989).
+[41] A. Van der Ven, The thermodynamics of decohesion, Acta Materialia 52, 1223 (2004).
+[42] F. J. H. Ehlers, M. Seydou, D. Tingaud, F. Maurel, Y. Charles, and S. Queyreau, Ab initio
+determination of the traction–separation curve for a metal grain boundary: a critical assess-
+ment of strategies, Modelling and Simulation in Materials Science and Engineering 24, 085014
+(2016).
+[43] F. J. H. Ehlers, M. Seydou, D. Tingaud, F. Maurel, S. Queyreau, and Y. Charles, Ab initio
+studies of two Al grain boundaries subjected to mixed tension/shear mode loading: how shear
+may promote breakage, Modelling and Simulation in Materials Science and Engineering 25,
+21
+
+064001 (2017).
+[44] B. Devincre, T. Hoc, and L. Kubin, Dislocation mean free paths and strain hardening of
+crystals, Science 320, 1745 (2008).
+[45] H. Mughrabi, Fatigue, an everlasting materials problem - still en vogue, Procedia Engineering
+2, 3 (2010).
+[46] T. Magnin, J. Driver, J. Lepinoux, and L. Kubin, Aspects microstructuraux de la d´eformation
+cyclique dans les m´etaux et alliages C.C. et C.F.C. II. — Saturation cyclique et localisation
+de la d´eformation, Revue de Physique Appliqu´ee 19, 483 (1984).
+[47] H. Mughrabi and R. Wang, Cyclic deformation of face-centred cubic polycrystals: a compar-
+ison with observations on single crystals, in Deformation of Polycrystals: Mechanisms and
+Microstructures. 2 nd Riso Int. Symposium on Metallurgy and Materials Science (1981) pp.
+87–98.
+22
+

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+GEOMETRY OF SET FUNCTIONS IN COOPERATIVE GAME THEORY
+DYLAN LAPLACE MERMOUD
+This job market paper will be updated regularly, please check my
+personal webpage for the last version.
+Abstract. They are many unexplored links between cooperative transferable utility
+games and convex, discrete or combinatorial geometry. In this paper, we investigate
+some of these links, such as the ones between cores of convex games and generalised
+permutohedra, also named polymatroids or base polyhedra. Another link that we
+investigate is the one between the resonance hyperplane arrangement and the set of
+sets of preimputations which are effective for a given coalition. These bridges can
+give interpretation and intuition to cooperative game theory, as well as bring new
+results and tools from other fields into the study of cooperative games.
+1. Introduction
+In his pioneer work on game theory, von Neumann (1928) described a n-person
+game as a set of ‘2n − 1 constants’, representing what one coalition can obtain by
+itself during the game. In the next century, Postnikov (2009) characterised his newly
+defined generalised permutohedra by ‘collections {zI} of the 2n − 1 coordinates zI, for
+nonempty subsets I ⊆ [n], that belong to a certain deformation cone’. Furthermore,
+the polyhedral complex known as the braid (arrangement) fan is determined by a set
+of 2n − 1 n-dimensional (0, 1)-vectors defining its rays (Barvinok, 2002). As a final ex-
+ample, let us talk about the resonance arrangements (K¨uhne, 2020), previously called
+all-subsets arrangements (Kamiya et al., 2012), which are hyperplane arrangements
+where each hyperplane is defined as the orthogonal complement of the elements of a set
+of 2n−1 n-dimensional (0, 1)-vectors. As we can expect, all of these mathematical con-
+cepts are closely related and bring more understanding to one another. For instance,
+a generalised permutohedron is the core of a convex game (Shapley, 1971; Castillo &
+Liu, 2022) and a hyperplane of a generic resonance arrangement is the set of preim-
+putations for which a specific coalition is effective. Also, the resonance arrangement
+cuts the space of preimputations in several polyhedral sets, corresponding to what is
+called feasible regions (Grabisch & Sudh¨olter, 2021; Laplace Mermoud et al., 2022).
+Likewise, the cones of a braid fan correspond to the polar regions of a game, a subset
+of the set of preimputations that is important in the study of domination between
+preimputations.
+Moreover, the braid arrangement and the resonance arrangement
+are adjoints (Aguiar & Mahajan, 2017). The maximal cones of the standard braid
+arrangement are in bijection with the set of maximal unbalanced sets (Billera et al.,
+Date: January 7, 2023.
+2020 Mathematics Subject Classification. MSC Primary 91A12; Secondary 52C35, 28A10.
+1
+arXiv:2301.02950v1  [cs.GT]  8 Jan 2023
+
+2
+DYLAN LAPLACE MERMOUD
+2012), the dual counterpart of the minimal balanced set defined by Shapley (1965)
+which are the cornerstone of the study of non-emptiness of the core.
+All these concepts have their standard objects, which are related either to the game
+with a characteristic function v defined for all coalitions S by v(S) = |S|(|S| − 1)/2
+or to the game assigning 0 to every coalition. Then, all other games define deforma-
+tions of these standard objects. In this paper, I present the formal definitions of the
+aforementioned concepts, and then I discuss the link between them and cooperative
+games. The work on projection together with the study of deformed braid fans and
+facial configuration of cores can lead to a resolution of the problem of knowing whether
+a game is convergent, i.e., if it is balanced and its core is a von Neumann-Morgenstern
+stable set.
+2. Definitions and notation
+Let N be a finite set of cardinality n, and denote by 2N its power set, i.e., the set
+of all subsets of N. A set function on N is a mapping ξ : 2N → R, assigning a real
+number ξ(S) to any subset S of N. A set function ξ is called grounded if ξ(∅) = 0.
+Definition 1 (von Neumann, 1928).
+A (cooperative transferable utility) game is an ordered pair (N, v), with N a finite
+set whose elements are called the players, and v a grounded set function called the
+characteristic function of the game.
+We denote by N the set of nonempty subsets of N, called coalitions. Throughout
+this paper, we denote by ξ the set functions and by v the characteristic functions of
+games. The interpretation of the characteristic function v that we will follow is that
+v(S) is the quantity of resources acquired by the players in S ∈ N after one unit of
+time if they cooperate to form the coalition S. One of the two main parts of coopera-
+tive game theory is to study how the coalitions share the resources acquired by their
+players, according to the value of the subcoalitions (including the singletons). Another
+question in cooperative game theory, which interests us in this paper, is to know at
+which scale the cooperation may occur. More specifically, we want to know at which
+level the commonly acquired resources can benefit any subcoalition, and subsequently
+whether the subcoalitions can threaten to leave the current state of cooperation to
+ensure a quantity that satisfies them. These two properties, called balancedness and
+stability, are defined in the sequel.
+In this paper, we denote by RN the Euclidean vector space built as the Cartesian
+product of n copies of R, one for each player in N. We use the same notation RS
+which only considers the players belonging to the coalition S. An element x ∈ RN is
+called payoff vector and associates a real value xi to any player i ∈ N. We denote by
+x(S) the sum of the values associated with the players of S: x(S) = �
+i∈S xi.
+Definition 2. A payoff vector x ∈ RN is called
+• a preimputation if N is effective for x, i.e., if x(N) = v(N). We denote the
+set of preimputations by X(N, v).
+
+GEOMETRY OF SET FUNCTIONS IN COOPERATIVE GAME THEORY
+3
+• an upper vector if, for all S ∈ N, we have x(S) ≥ v(S). We denote the set of
+upper vectors by U(N, v).
+The set X(N, v) forms a hyperplane of RN, and the set U(N, v) (van Gellekom et
+al., 1999) is a polyhedral set unbounded from above. A preimputation represents an
+allocation of the value of the grand coalition N among its players, while an upper
+vector is an allocation that benefits any coalition. A preimputation is Pareto optimal:
+increasing some coordinates without changing their total sum implies decreasing some
+other coordinates. We denote by AS(N, v) the set of preimputations for which S is
+effective, i.e.,
+AS(N, v) := {x ∈ X(N, v) | x(S) = v(S)}.
+If there is no confusion risk, we denote these sets by AS. Notice that these sets are
+affine subspaces of codimension 2, and have the same normals for any game (N, v),
+depending only on S and not on the characteristic function. The latter is only respon-
+sible for the relative positions of these affine subspaces, as increasing the value v(S)
+of a coalition will solely shift the subspace along their normals.
+Each AS cuts the hyperplane X(N, v) in two halves, denoted by A≥
+S := {x ∈ X(N, v) |
+x(S) ≥ v(S)} and A≤
+S := {x ∈ X(N, v) | x(S) ≤ v(S)}.
+Each preimputation in
+A≥
+S benefits the coalition S.
+The preimputations that benefit every coalition, i.e.,
+preimputations being also upper vectors, are said to be coalitionally rational, and
+their set, denoted by
+C(N, v) :=
+�
+S∈N
+A≥
+S = X(N, v) ∩ U(N, v),
+is called the core (Gillies, 1959). When nonempty, the core is a bounded convex poly-
+hedral set, i.e. a polytope, lying in the hyperplane X(N, v).
+A game with a nonempty core, i.e., for which there exists a preimputation benefiting
+every coalition, is called a balanced game. The reason for this denomination will be
+discussed in Section 4. A balanced game (N, v) is a game for which there is no better
+state of cooperation for the players than the grand coalition N. It implies in particular
+that there exists no partition of N such that the sum of the values of the partition’s
+blocks is strictly greater than the value of N. For illustration, let (N, v) be a game
+with N = {a, b, c} and the values:
+S
+{a}
+{b}
+{c}
+{a, b}
+{a, c}
+{b, c}
+N
+v(S)
+5
+0
+0
+8
+8
+8
+10
+.
+This game is not balanced because the players will prefer to be organised as {{a}, {b, c}}
+and get 13 rather than form the grand coalition N and get 10. But it is not sufficient
+to look at the partition of N to know whether the core is nonempty. Players can spend
+fractions of their time in different coalitions, as we can see in the following example.
+
+4
+DYLAN LAPLACE MERMOUD
+Let (N, v) be a game with N = {a, b, c} and the values:
+(1)
+S
+{a}
+{b}
+{c}
+{a, b}
+{a, c}
+{b, c}
+N
+v(S)
+0
+0
+0
+8
+8
+8
+10
+.
+The only difference between these games is v({a}), but it is sufficient for the players
+to prefer the grand coalition to the organisation {{a}, {b, c}}. But what if they spend
+half of their time with one player and the other half with the other player? They
+obtain
+1
+2v({a, b}) + 1
+2v({a, c}) + 1
+2v({b, c}) = 3 · 8
+2
+= 12,
+that is still greater than 10 that they can get if they form the grand coalition. Therefore
+this game is also not balanced.
+The set B = {{a, b}, {a, c}, {b, c}} together with
+the systems of weights λB = (λ{a,b} =
+1
+2, λ{a,c} =
+1
+2, λ{b,c} =
+1
+2), as well as the set
+B′ = {{a}, {b, c}} with the weights λB′ = (λ{a} = 1, λ{b,c} = 1) are examples of
+balanced sets of coalitions.
+Definition 3. Let B ⊆ N be a set of coalitions. We say that B is a balanced set if
+there exists a balancing vector λB = (λB
+S)S∈B such that, for each player i ∈ N, we have
+�
+S∈Bi λB
+S = 1, with Bi = {S ∈ B | i ∈ S}.
+The balanced sets are defined by Shapley (1965) but were studied a few years be-
+fore by Bondareva (1963) without defining them explicitly. They both independently
+gave a characterisation of the set of balanced games based on the balanced sets. The
+balanced games are exactly the games for which the weighted sum of the values of the
+coalitions of any balanced set does not exceed the value of the grand coalition, i.e., for
+any balanced set B with a system of weights λB, we have �
+S∈B λB
+Sv(S) ≤ v(N). In
+the same paper, Shapley identified a subset of the set of balanced sets, on which the
+characterisation of balanced games remains valid. This characterisation is now known
+as the Bondareva-Shapley Theorem and is maybe the fundamental theorem for this
+study of cooperative behaviour. The Bondareva-Shapley Theorem will be discussed in
+Section 4.
+Now, let us focus on a given coalition S ∈ N \ {N}. If the proposed preimputation
+x ∈ X(N, v) satisfies x(S) < v(S), the coalition S can get a better value by leaving
+the grand coalition. Let y ∈ X(N, v) be a second preimputation.
+Definition 4. We say that y dominates x via S, denoted y domS x, if
+• y is affordable, i.e., y(S) ≤ v(S),
+• y is improving, i.e., for all i ∈ S, we have yi > xi, .
+We say that y dominates x, noted y dom x, if a coalition S exists such that y domS x.
+For illustration, consider the game described in Table (1), and let x = (6, 2, 2),
+y = (2, 6, 2) and z = (2, 2, 6) be preimputations. We have that x dom{a,b} z, also
+z dom{a,c} y and y dom{b,c} x. We see in particular that the domination relation is not
+transitive, however, the relation of being dominated via a specific coalition is transitive.
+
+GEOMETRY OF SET FUNCTIONS IN COOPERATIVE GAME THEORY
+5
+The first condition expresses that the threat from S to leave the grand coalition only
+exists because they can get v(S) on their own. The second condition requires that
+everyone in the coalition S benefits from leaving the grand coalition. According to our
+interpretation of the characteristic function, this is what v(S) represents: the relative
+importance of their threat of leaving the grand coalition, and therefore the relative
+importance the coalition gets when we seek a preimputation that benefits everyone.
+In today’s literature, we call solution a map associating to each game a subset of
+RN. But von Neumann and Morgenstern (1944), when they made the first proposal
+for a solution concept in the broad sense, defined a solution as being a subset of RN
+that we today call stable set, or von Neumann-Morgenstern stable set. They are the
+set of “maximal” elements relative to the intransitive domination relation.
+Definition 5 (von Neumann and Morgenstern, 1944).
+A subset K of V ⊆ RN is a V -stable set if it is
+• internally stable, i.e., for all x ∈ K, there is no y ∈ K such that y dom x,
+• externally stable, i.e., for all x ∈ V \K, there exists y ∈ K such that y dom x.
+We simply call stable sets the {x ∈ RN | x(N) ≤ v(N)}-stable sets.
+Originally, von Neumann and Morgenstern defined their solutions of a game as
+I(N, v)-stable sets, ignoring any payoff vector that was not efficient nor individually
+rational. In this paper, we are interested in the most general stable sets. Shapley
+(1952) showed that a set is a stable set if and only if it is a X(N, v)-stable set. We
+see from the definition that any stable set included in I(N, v) is a I(N, v)-stable set.
+Moreover, Shapley showed that a I(N, v)-stable set K is a stable set, if and only if,
+for any player i ∈ N, there exists an element x ∈ K such that xi = v({i}).
+The definition of stable sets is very appealing, but suffers from several defaults:
+there may be no stable sets, several stable sets, or even a continuum of them, and it
+is difficult to identify them (Lucas, 1969, 1992).
+By contrast, when nonempty, the core is unique, contains all the coalitionally ra-
+tional preimputations and is a polytope. Also, notice that all the core elements are
+undominated: they cannot be affordable and improvable at the same time for any
+coalition by definition. Therefore, the core is always internally stable and is contained
+in all the stable sets if they exist. However, the core is not necessarily externally sta-
+ble: there can exist preimputations that are dominated solely by preimputations not
+belonging to the core. Nevertheless, if the core is externally stable, it is the unique
+stable set (Driessen, 2013).
+By definition, a preimputation x can be dominated via a coalition S only if the
+coalition can improve upon x by leaving the grand coalition, i.e., x(S) < v(S). In the
+following, we denote by e(S, x) the additional quantity of resources that coalition S
+can acquire by itself, i.e., e(S, x) := v(S) − x(S), that we call the excess of S at x. Let
+S ⊆ N be a set of coalitions. We denote by XS(N, v) the set of preimputations upon
+
+6
+DYLAN LAPLACE MERMOUD
+which a coalition can improve if and only if it belongs to S, i.e.,
+XS(N, v) := {x ∈ X(N, v) | x(S) < v(S) if and only if S ∈ S}.
+We say that the set S is feasible if its associated region XS(N, v) is non-empty. The
+regions correspond to the connected parts of the complement of the union of all the
+affine subspaces AS, that form a hyperplane arrangement in X(N, v). However, the
+regions are slightly bigger than the chambers of the hyperplane arrangement: the
+chambers are open sets, while the regions include a part of their frontier. If we consider
+that the core is the region associated with the empty set of coalitions, the regions form
+a partition of X(N, v).
+Definition 6. The resonance arrangement AR is defined by
+AR := {HR
+S | S ∈ N \ {N}}
+where
+HR
+S := {x ∈ RN | x(N) = 0 and x(S) = 0}.
+Let o be the set function assigning 0 to any coalition.
+Proposition 1. The set A(N, o) := {AS(N, o) | S ∈ N \ {N}} and AR define the
+same hyperplane arrangement.
+Proof. Let S be a coalition. We have
+AS(N, o) = {x ∈ X(N, o) | x(S) = o(S)}
+= {x ∈ RN | x(N) = 0 and x(S) = 0} = HR
+S .
+□
+From Prop. 1, we see that any game (N, v) induces a deformation of the resonance
+arrangement leading to the arrangement A(N, v) := {AS(N, v) | S ∈ N \ {N}}. Also,
+the regions XS(N, o) are in bijection with the chambers of AR, i.e., the connected part
+of RN \ AR. Another well-known hyperplane arrangement interests us in this paper.
+Definition 7. The (restricted) braid arrangement AB is defined by
+AB := {HB
+ij | 1 ≤ i < j ≤ n}
+where
+HB
+ij := {x ∈ RN | x(N) = 0 and xi = xj}.
+The permutohedron is the polytope the normal fan of which is the restricted braid
+arrangement. Also, the chambers of AB correspond to the polar regions defined in
+Section 3 the same way the chambers of AR correspond to the regions XS(N, v).
+Throughout this paper, we equip the Euclidean vector space RN, and therefore
+X(N, v), with the usual scalar product, denoted by ⟨·, ·⟩ and the associated norm,
+denoted by ∥·∥ defined, for all x, y ∈ RN, by
+⟨x, y⟩ =
+�
+i∈N
+xiyi,
+and
+∥x∥ =
+�
+⟨x, x⟩.
+Definition 8. Let K ⊆ RN be nonempty, closed and convex and let x ∈ RN. We
+denote by πK(x) the element of RN defined by πK(x) := argminp∈K∥x − p∥. The map
+πK : x �→ πK(x) is called the projector onto K.
+
+GEOMETRY OF SET FUNCTIONS IN COOPERATIVE GAME THEORY
+7
+In the definition above, the existence of the projection comes from the closedness
+and the nonemptiness of K, and the uniqueness comes from the convexity. The fol-
+lowing result, as formulated by Bauschke and Combettes (2011) is well-known and
+characterises the projectors we use in the space of preimputations.
+Theorem 1 (Projection Theorem).
+Let K be a nonempty closed convex subset of RN. For all payoff vectors x and y,
+y = πK(x)
+if and only if
+�
+y ∈ K
+and for all z ∈ K, ⟨z − y, x − y⟩ ≤ 0
+�
+.
+3. Affine geometry and hyperplane arrangements
+This section focuses on the hyperplane X(N, v). Recall that the affine subspace AS
+and the affine halfsubspace A≥
+S are defined by
+AS = {x ∈ X(N, v) | x(S) = v(S)},
+and
+A≥
+S = {x ∈ X(N, v) | x(S) ≥ v(S)}.
+Proposition 2. Let S be a coalition. The vector ηS, defined by
+ηS = 1S − |S|
+n 1N,
+with
+1S
+i =
+�1,
+if i ∈ S,
+0,
+otherwise.
+is a side payment and a normal of AS.
+Proof. For any coalition S, we have ηS(N) = 1S − |S|
+n 1N(S) = |S| − |S| = 0, then ηS
+is a side payment. Let x and y be two elements of AS. We have
+⟨ηS, x − y⟩ = ⟨ηS, x⟩ − ⟨ηS, y⟩ = x(S) − |S|
+n x(N) − y(S) + |S|
+n y(N).
+The coalitions S and N are effective for both x and y, then ⟨ηS, x − y⟩ = 0.
+□
+For convenience, we write ηi to denote η{i}. The set of vectors {ηS | S ∈ N} does not
+depend on the game we are considering, and the characteristic function only defines
+the position of AS along the line generated by ηS. We now define the projectors used
+in this paper.
+Proposition 3. Let S be a coalition and x be a preimputation. Then
+πAS(x) = x + γS(x)ηS,
+with γS(x) := e(S, x)/∥ηS∥2, is a projector onto AS. Moreover, if πA≥
+S (x) differs from
+x, then πAS(x) dominates x via S.
+Proof. First, notice that ∥ηS∥2 = ⟨ηS, ηS⟩ = ηS(S). Following the Projection Theorem,
+we first prove that πAS(x) belongs to AS.
+πAS(x)(S) = x(S) + γS(x)ηS(S) = x(S) + e(S, x)
+∥ηS∥ ∥ηS∥2 = v(S).
+
+8
+DYLAN LAPLACE MERMOUD
+Let z be an element of AS. We have
+⟨z − πAS(x), x − πAS(s)⟩ = ⟨z − x − γS(x)ηS, −γS(x)ηS⟩
+= −γS(x)
+�
+⟨z, ηS⟩ − ⟨x, ηS⟩ − γS(x)⟨ηS, ηS⟩
+�
+= −γS(x)
+�
+z(S) − x(S) − γS(x)∥ηS∥
+�
+= −γS(x) (e(S, x) − e(S, x)) = 0.
+We use the Projection Theorem to conclude the construction of the projector. To end
+this proof, notice that, for all i ∈ S, we have ηS
+i = 1 − |S|
+n > 0.
+□
+If a coalition S can improve upon a preimputation x ∈ X(N, v), the projection
+πAS(x) shares the excess of S at x equally among the players in S, and therefore the
+projection dominates x via S. This fact motivates the use of these projectors in this
+work to study the domination relations between payoff vectors and, more specifically,
+between preimputations and core elements. Moreover, γS(x)ηS is the shortest side
+payment able to map the preimputation x onto a preimputation that benefits the
+coalition S. It is the solution to the problem consisting of satisfying coalition S with
+the smallest value transfer between players.
+The idea of projections between preimputations was already existing in the transfer
+scheme defined by Cesco (1998). A transfer scheme, first defined and used by Stearns
+(1968), is a sequence of preimputations defined by a sequence of transfers, i.e., trans-
+lation by a side payment. The sequence of preimputations defined by Cesco is defined
+recursively by projecting onto AS the last preimputation produced. His projector is
+defined, for all preimputation x, by
+πAS(x) = x + e(S, x)
+� 1S
+|S| −
+1N\S
+|N \ S|
+�
+.
+Cesco’s projectors are identical to ours, but the choice of normals is different. He chose
+to have a normal of a different norm to have a formulation depending explicitly on the
+excess e(S, x). Our choice of normals is motivated by the following theorem.
+Theorem 2. The map η : S → ηS is modular, and for all S, T ∈ N, we have
+⟨ηS, ηT⟩ = ηS(T).
+Proof. We start by proving that η : S → ηS is modular. Notice that if S and T are
+disjoint, we have
+ηS + ηT = 1S − |S|
+n 1N + 1T − |T|
+n 1N = 1S∪T − |S| + |T|
+n
+1N = ηS∪T.
+Now we decompose ηS = ηS\T + ηS∩T, similarly for ηT, and we have
+ηS + ηT = ηS\T + ηS∩T + ηT\SηS∩T = ηS∪T + ηS∩T.
+For the second property, the fact that ηS is a side payment leads to
+⟨ηS, ηT⟩ = ⟨ηS, 1T⟩ − |S|
+n ⟨ηS, 1N⟩ = ηS(T) − |S|
+n ηS(N) = ηS(T).
+
+GEOMETRY OF SET FUNCTIONS IN COOPERATIVE GAME THEORY
+9
+□
+From this theorem, we can derive several properties on the set {ηS | S ∈ N}, in
+particular, the repartition of the ηS is balanced around the origin of the vector space
+parallel to X(N, v).
+Corollary 3. A set B of coalitions is balanced if and only if there exists a set of
+positive scalars {λS | S ∈ B} such that
+�
+S∈B
+λSηS = ⃗o.
+Proof. First, remark that
+�
+S∈B
+λS|S| =
+�
+S∈B
+λS1S(N) =
+��
+S∈B
+λS1S
+�
+(N) = 1N(N) = n.
+Assume that B is balanced with balancing weights {λS}S∈B. Then
+�
+S∈B
+λSηS =
+�
+S∈B
+λS1S − 1
+n
+�
+S∈B
+λS|S|1N = 1N − 1
+nn1N = ⃗o.
+Assume now that �
+S∈B λSηS = ⃗o.
+�
+S∈B
+λS1S =
+�
+S∈B
+λS
+�
+ηS + |S|
+n 1N
+�
+=
+�
+S∈B
+λSηS + 1
+n1N �
+S∈B
+λS|S| = 1
+nn1N = 1N.
+□
+Another surprising result following from the theorem is that, for any coalition S
+and T, if n is a prime number, ηS and ηT cannot be orthogonal, because
+⟨ηS, ηT⟩ = ηS(T) = 1S(T) − |S|
+n 1N(T) = |S ∩ T| − |S| · |T|
+n
+,
+and |S| · |T| cannot be a multiple of n.
+We have seen earlier that projecting onto A≥
+S generates a preimputation dominating
+the initial one if they differ. For any preimputation x, the relative position of πAS(x)
+with respect to AT depends on the excesses e(S, x) and e(T, x), but also on the scalar
+product ηS(T) = ⟨ηS, ηT⟩. The value and the sign of ηS(T) indicate how correlated the
+fluctuations of coalitions S and T are. To summarize all this information, we define
+χS(T, x) := e(S, x)ηS(T) − e(T, x)ηS(S).
+Proposition 4. Let S and T be two coalitions, and let x ∈ X(N, v). Then
+πAS(x) ∈ A≥
+T
+if and only if
+χS(T, x) ≥ 0.
+In other words, πAS(x) ∈ A≥
+S ∩ A≥
+T and πAS(x) dom x.
+
+10
+DYLAN LAPLACE MERMOUD
+Proof. First, study the excess of T at the projection onto AS:
+e(T, πAS(x)) = v(T) − x(T) − γS(x)ηS(T)
+= e(T, x) − e(S, x)
+ηS(S) ηS(T)
+=
+1
+ηS(S)
+�
+e(T, x)ηS(S) − e(S, x)ηS(T)
+�
+=
+−1
+ηS(S)χS(T, x).
+Then, the projection lies into A≥
+T if and only if e(T, πAS(x)) is nonpositive, by definition,
+and therefore if and only if χS(T, x) is nonnegative. For domination, we apply Prop.
+3.
+□
+There are intuitive implications for this result. If ⟨ηS, ηT⟩ ≥ 0, an increase in the
+value of S will also generate an increase for T. Therefore, assuming that e(S, x) ≥
+e(T, x), projecting onto AS will also provide, if needed, what coalition T requires to
+be satisfied with the preimputation. It is implied by the previous result, knowing that
+ηS(T) ≤ ηS(S) = ∥ηS∥2.
+For most of the preimputations, projecting onto an affine subspace AS is not suffi-
+cient to be projected on the core. To do so, we need to be able to define projectors
+on any intersection of affine subspaces, to simultaneously improve the value of all the
+coalitions that can improve upon the considered preimputation. In the following, we
+provide a closed formula that defines any needed projectors, without having to use
+algorithmic sequential methods such as Dykstra’s projection algorithm.
+Definition 9. Let S ⊆ N be a set of coalitions. We say that S is an independent set
+if {ηS}S∈S forms a linearly independent set of vectors.
+A set which is not independent is said to be dependent.
+Denote by AS the set
+AS := �
+S∈S AS and by LS the matrix with {ηS | S ∈ S} as rows. This unusual
+definition of a matrix by its rows is motivated by an easier proof for a forthcoming
+result. Define by ΓS := LSL⊤
+S the Gram matrix associated with the set S. If we write
+S = {S1, . . . , Sk}, the Gram matrix is a square matrix of size k whose coefficients are
+(ΓS)ij = ⟨ηSi, ηSj⟩. The symmetry of the scalar product implies the symmetry of ΓS.
+In addition, ΓS is positive semidefinite: for all k-dimensional vector x ∈ RS,
+x⊤ΓSx =
+�
+S∈S
+�
+T∈S
+⟨ηS, ηT⟩xSxT =
+��
+S∈S
+xSηS,
+�
+T∈S
+xTηT
+�
+=
+�����
+�
+S∈S
+xSηS
+�����
+2
+≥ 0.
+Proposition 5. A coalition set S is independent if and only if ΓS is nonsingular.
+Proof. Using the above formula, we have that for all x ∈ RS,
+x⊤ΓSx =
+�����
+�
+S∈S
+xSηS
+�����
+2
+≥ 0.
+It shows that ΓS is positive semidefinite, i.e., that all its eigenvalues are nonnegative.
+To have ΓS nonsingular, we need to have only positive eigenvalues, i.e., to have ΓS
+
+GEOMETRY OF SET FUNCTIONS IN COOPERATIVE GAME THEORY
+11
+positive definite. Then ΓS is nonsingular if and only if, for all x ∈ RS \ {⃗o},
+x⊤ΓSx =
+�����
+�
+S∈S
+xSηS
+�����
+2
+> 0,
+i.e., if and only if for all x ∈ RS \ {⃗o}, we have �
+S∈S xSηS ̸= 0; that is the definition
+of {ηS | S ∈ S} being a linearly independent set of vectors.
+□
+Another implication of the linear independence of the set {ηS | S ∈ S} is that AS
+is nonempty. Therefore, if a coalition set S is independent, it is possible to satisfy all
+the coalitions at once, thanks to a translation of the initial preimputation by a well-
+chosen side payment. For a coalition set S, a coalition S ∈ S, and a preimputation
+x ∈ X(N, v), denote by ΓS,x
+S
+the matrix formed from ΓS by replacing each coefficient
+⟨ηT, ηS⟩ of the column of scalar product with ηS by e(T, x).
+Theorem 4. Let S be an independent coalition set. Then AS is nonempty and for all
+preimputation x,
+πAS(x) = x +
+1
+det ΓS
+�
+S∈S
+det
+�
+ΓS,x
+S
+�
+ηS.
+Proof. The proof relies on the following result.
+Proposition 6 (Bauschke and Combettes, 2011, Example 29.17). Let L : RN → RS
+be linear and let y ∈ ranL. Then, for all x ∈ RN,
+πL−1({y})(x) = x − L⊤Γ−1
+S (Lx − y) .
+We apply this formula to the linear map LS that we defined earlier, with the vector
+y ∈ RS defined, for all S ∈ S, by
+yS = v(S) − |S|
+n v(N).
+Therefore the image of a preimputation x through LS is, for all S ∈ S,
+(LSx)S = ⟨ηS, x⟩ = ⟨1S, x⟩ − |S|
+n ⟨1N, x⟩ = x(S) − |S|
+n v(N).
+Because S is independent, AS is nonempty, therefore we have y ∈ ranLS and L−1
+S ({y}) =
+AS. We consider the singleton {y} to emphasise the set-theoretical notation for L−1
+S .
+Implementing these in the formula yields
+πAS(x) = x − L⊤
+S Γ−1
+S (LSx − y) = x + L⊤
+S Γ−1
+S e(S, x),
+with e(S, x) being the column vector composed of the excesses (e(S, x))S∈S. Using
+Cramer’s rule, the solution to the linear system ΓSβ = e(S, x) can be expressed as,
+for each S ∈ S,
+βS =
+det
+�
+ΓS,x
+S
+�
+det ΓS
+.
+
+12
+DYLAN LAPLACE MERMOUD
+Therefore the projection formula becomes
+πAS(x) = x + L⊤
+S β = x +
+1
+det ΓS
+�
+S∈S
+det
+�
+ΓS,x
+S
+�
+ηS.
+□
+In general, the set AS is empty if S is dependent, making this result sharp. Here we
+understand in general in the sense that an overdetermined system of linear equations
+has, in general, no solution. We say that a set function v acts generally on a coalition
+set S if for all dependent subsets T ⊆ S we have AT = ∅. There are two main impli-
+cations of this result: for a preimputation upon which a set of coalitions can improve,
+and a game acting generally on N, we have a necessary and sufficient condition to
+know whether we can find another preimputation putting every excess to 0, and the
+explicit formula of the side payment between the two preimputations that minimises
+the transfers of value between the players. This projector is especially useful for a
+feasible coalition set which is independent.
+Theorem 5. Let (N, v) be a balanced game and let S be an independent and feasible
+coalition set. Then cl(XS(N, v)) ∩ C(N, v) ̸= ∅ and, for all x ∈ XS(N, v),
+πAS(x) ∈ C(N, v).
+Proof. Let x ∈ XS(N, v). By Theorem 4, the independence of S implies the nonempti-
+ness of AS. Therefore set y = πAS(x). Define
+ct = tx + (1 − t)y,
+for t ∈ [0, 1].
+Because cl(XS(N, v)) is a closed polyhedron defined by inequalities, it is convex, and
+ct ∈ cl(XS(N, v)), with ct ∈ XS, if t ̸= 0. By contradiction, assume that y ̸∈ C(N, v).
+Therefore, there exists a coalition T ̸∈ S such that e(T, y) > 0. Because T ̸∈ S, we
+have e(T, x) ≤ 0, and then there exists t∗ ∈ ]0, 1] such that e(T, ct∗) = 0. Set τ = t∗/2.
+Because t∗ ̸= 0, we have τ ̸= 0 and
+cτ ∈ XS(N, v),
+e(T, cτ) > 0,
+which is impossible. Then y ∈ C(N, v) ∩ cl(XS(N, v)).
+□
+4. Polyhedral geometry
+In this section, we will discuss the core, its properties, what it represents in eco-
+nomics or any cooperative environment, and how this particular object can be found
+somewhere else in mathematics, in particular in the theory of set functions. First,
+recall the modern definition of the core from Gillies (1959). For a game (N, v), the
+core, denoted by C(N, v), is the intersection between the set of preimputations and
+the set of upper vectors, i.e., the set of preimputations benefitting every coalition:
+C(N, v) = X(N, v) ∩ U(N, v) =
+�
+S∈N
+A≥
+S .
+
+GEOMETRY OF SET FUNCTIONS IN COOPERATIVE GAME THEORY
+13
+Let v and w be two set functions on the same domain 2N. We write w ≥ v if, for every
+S ∈ 2N, we have w(S) ≥ v(S). Then, the core of v can be seen as the set of additive
+set functions w such that w ≥ v but still with w(N) = v(N).
+Figure 1. Graphical illustrations of cores from Shapley (1971).
+In this paper, we interpret the nonemptiness of the core as a necessary condition
+for the grand coalition N to form, but not as a sufficient condition. It is necessary
+because if the core is empty, there exists no preimputation that benefits every coali-
+tion, and then some players will prefer to form other coalitions than N. However,
+even if the core is nonempty, there may exist no mechanism for the players to reach
+these coalitionally rational preimputations, nothing indicates that all the players and
+coalitions will succeed to defend their interests. This mechanism can be the result
+of external intervention, for instance, a planner that has the power to choose which
+preimputations will be used to share the value v(N) among the players, in which case
+the core’s nonemptiness is a sufficient condition.
+Nevertheless, if there is no external mediator between the players, a coalition able to
+improve upon a current preimputation can still threaten to leave the grand coalition
+N, and then require a preimputation that dominates this current preimputation. To
+end this interlude on the game-theoretical interpretation, we can conclude that, if
+there is no external mediator, the core’s stability, especially its external stability, is a
+sufficient condition for the grand coalition to form and cooperate behaviour to emerge.
+4.1. The balanced sets. Let (N, v) be a game. Recall that a balanced set on N is
+a set B of subsets of N together with a set of positive weights {λS}S∈B, called the
+balancing weights, such that, for every player i ∈ N, the sum �
+S∈Bi λS is equal to
+one, with Bi being the set of coalitions of B containing i. In other words, we have
+�
+S∈B
+λS1S = 1N,
+
+Characteristic function:
+v(s)
+v(S)=/2i 12
+0
+0
+jes
+23
+H23414
+DYLAN LAPLACE MERMOUD
+with 1S being the n-dimensional (0, 1)-vector such that 1S
+i = 1 if and only if i ∈ S.
+From a geometrical point of view, the set B is balanced if
+1N ∈ relint
+�
+cone
+�
+{1S}S∈B
+��
+,
+when relint(X) denotes the relative interior of a set X. Notice that B is minimal if
+the cone is simplicial, i.e., if its number of rays exceeds by 1 the dimension of its affine
+span. Using the balanced sets, Bondareva (1963), and then Shapley (1965) have found
+a characterisation of games with nonempty cores.
+Theorem 6 (Bondareva-Shapley Theorem, first version).
+A game (N, v) has a nonempty core if and only if for every balanced set B on N
+together with its balancing weights {λS}S∈B, we have
+�
+S∈B
+λSv(S) ≤ v(N).
+This well-known theorem provides the adjective balanced to name games having a
+nonempty core. The interpretation of the result is very natural: a balanced set is a
+possible organisation for the players, possibly spending fractions of their time among
+different coalitions. The weighted sum is the total amount of resources gathered by
+the players in N at the end of one unit of time. Then, if the biggest weighted sum
+is the one corresponding to the balanced set {N}, the players should be organised as
+one unique grand coalition, and therefore there should exist preimputations allocat-
+ing v(N) among the players, with each subcoalition of N acquiring at least the same
+amount of resources as if it was working on its own.
+One way to study the existence of a preimputation which is also an upper vector
+is to study the linear program minx∈U(N,v) x(N) and to use the duality theory. The
+program can be rewritten as
+(P)
+min x(N)
+s.t. x(S) ≥ v(S),
+for all S ∈ N.
+Remark that this program is always feasible and that the game’s core is nonempty if
+and only if the program’s value is v(N). The dual program is:
+(D)
+max
+�
+S∈N
+λSv(S)
+s.t.
+�
+�
+�
+�
+S∈N
+λS1S = 1N, and
+λS ≥ 0, for all S ∈ N.
+The dual program’s constraints define the aforementioned balanced sets. Because the
+vector λ∗ defined by λ∗
+S = 0 for all S ∈ N and λ∗
+N = 1 satisfies the constraints, the
+program has a solution. Furthermore, the value of the objective function for λ∗ is
+v(N). It follows from the duality theorem that the game’s core is nonempty if and
+only if the value of (D) is v(N), i.e., for all balanced sets B on N with balancing
+weights {λS}S∈B, we have �
+S∈B λSv(S) ≤ v(N).
+
+GEOMETRY OF SET FUNCTIONS IN COOPERATIVE GAME THEORY
+15
+In practice, this characterisation cannot be used for algorithmic purposes because
+most of the balanced sets can have an infinity of balancing weights. Notice that the
+balancing weights form a polytope in RN, described by
+F =
+�
+λ ∈ RN
++
+�����
+�
+S∈N
+λS1S = 1N
+�
+.
+Each point λ ∈ F represents a balanced set B corresponding to the support of λ, and
+the balancing weights of B are the corresponding positive coefficients of λ. Similarly,
+we can identify each characteristic function v with an (2n − 1)-dimensional vector;
+therefore, the vector space RN can be seen as the space of games. Then, the ambient
+space of the polytope of balancing weights and the vector space of games, together
+with the usual scalar product is a dual system, or a dual pair, studied in functional
+analysis. The set of balanced normalised (i.e. v(N) = 1) games is the polar set of
+F because the weighted sum in Bondareva-Shapley Theorem is the aforementioned
+scalar product.
+Because the balanced sets and their weights are associated with a polytope, they are
+determined by convex combinations of the balanced sets represented by the vertices.
+Definition 10. A balanced set B on N is minimal if there is no balanced subset of B.
+Shapley proved that the set of minimal balanced sets corresponds to the set of
+extremal points of F, therefore a minimal balanced set B has a unique set of balancing
+weights λB. From this, he stated this sharp version of the Bondareva-Shapley Theorem.
+For more details about this, see the monograph of Grabisch (2016).
+Theorem 7 (Bondareva-Shapley, sharp version).
+A game (N, v) has a nonempty core if and only if for every minimal balanced set B on
+N, we have
+�
+S∈B
+λB
+Sv(S) ≤ v(N).
+Furthermore, none of these equalities is redundant except for B = {N}.
+The sharpness of the result comes from the fact that, if we choose a minimal bal-
+anced set B, we can find a game that satisfies all the inequalities for minimal balanced
+sets different from B, but not the one corresponding to B. This new characterisation
+only requires a finite number of inequalities to check, provided that we know the min-
+imal balanced sets on N.
+As a consequence, it is possible to design an algorithm checking the nonemptiness
+of the core, significantly faster than the usual algorithms used in linear programming.
+The only requirement to use a Bondareva-Shapley-like algorithm is to know the set of
+minimal balanced sets, which is very complex (see Laplace Mermoud et al., 2022).
+The balanced collections and the Bondareva-Shapley theorem are thus defining
+which deformations of the permutohedron are acceptable to keep it non-empty, even
+
+16
+DYLAN LAPLACE MERMOUD
+if, in general, the deformation induced by a game of the permutohedron does not give
+a generalised permutohedron. Only the deformations induced by convex games satisfy
+this property.
+4.2. Cores of convex games. So far, we have studied games in a very general frame-
+work, coming from arbitrary grounded set functions. In this paper, we are mainly fo-
+cused on the core and the domination relation between the preimputations. Moreover,
+a lot of economic problems modelled with a game-theoretic framework as described
+here have some common and handy properties.
+Definition 11. Let ξ : 2N → R be a set function. We say that ξ is
+• supermodular if, for all S, T ∈ N, we have ξ(S)+ξ(T) ≤ ξ(S ∪T)+ξ(S ∩T),
+• submodular if, for all S, T ∈ N, we have ξ(S) + ξ(T) ≥ ξ(S ∪ T) + ξ(S ∩ T).
+Games with supermodular characteristic functions are called convex (Shapley, 1971).
+Many interaction situations can be modelled by these, for instance: production econ-
+omy with landowners (Shapley & Shubik, 1967; Driessen, 2013), bankruptcy games
+(Aumann & Maschler, 1985; Driessen, 2013), common pool games with linear cost
+functions (O’Neill, 1982; Driessen, 2013), etc. Shapley (1971) studied in great depth
+the properties of convex games, thanks to the extensive study of submodular set func-
+tions, and more specifically polymatroids, by Edmonds (1970). In particular, convex
+games have nonempty cores, and their cores are (externally) stable. The result of the
+nonemptiness of the core comes from the theory of submodular set functions. For an
+exposition of this theory, see the monograph of Fujishige (2005).
+Let ξ : 2N → R be a grounded submodular set function. The submodular polyhedron
+P(ξ) of ξ is defined by P(ξ) := {x ∈ RN | x(A) ≤ ξ(A), for all A ∈ N}, and the base
+polyhedron of ξ by B(ξ) := {x ∈ P(ξ) | x(N) = v(N)}.
+Figure 2. Example of a submodular/base polyhedron from Fujishige
+(2005).
+Denote by ξ# the conjugate set function of ξ, defined, for all S ∈ N, by
+ξ#(N \ S) = ξ(N) − ξ(S).
+Lemma 8. Let ξ : 2N → R be a grounded submodular set function and let (N, v) be a
+game such that v ≡ ξ#. Then, (N, v) is convex and B(ξ) = C(N, v).
+
+r(2)
+B(f)
+P(f)GEOMETRY OF SET FUNCTIONS IN COOPERATIVE GAME THEORY
+17
+Proof. This result is based on Lemma 2.4 from Fujishige (2005). Let x ∈ B(ξ). Then,
+for all S ∈ N, we have x(S) ≤ ξ(S). Taking the opposite value and adding ξ(N) =
+x(N) of each side yields x(N) − x(S) ≥ ξ(N) − ξ(S), then we have, for all S ∈ N,
+x(N \ S) ≥ ξ#(N \ S) = v(N \ S).
+Therefore, B(ξ) and C(N, v) are the same set. Let us now prove that (N, v) is convex.
+For all S and T ∈ N, we have
+v(S) + v(T) = ξ#(S) + ξ#(T) = 2ξ(N) − (ξ(N \ S) + ξ(N \ T)) .
+Notice that (N \ S) ∪ (N \ T) = N \ (S ∩ T) and (N \ S) ∩ (N \ T) = N \ (S ∪ T).
+Because ξ is submodular, we have that
+ξ(N \ S) + ξ(N \ T) ≥ ξ(N \ (S ∩ T)) + ξ(N \ (S ∪ T)).
+Then, we finish the calculations with
+v(S) + v(T) ≤ 2ξ(N) − (ξ(N \ (S ∩ T)) + ξ(N \ (S ∪ T)))
+= ξ#(S ∩ T) + ξ#(S ∪ T)
+= v(S ∩ T) + v(S ∪ T).
+□
+Figure 3. Example of submodular and base polyhedra for a set func-
+tion f and its conjugate f # from Fujishige (2005).
+Then the theory of submodular functions (Edmonds, 1970; Fujishige, 2005, etc) is,
+to some extent, dual to Shapley’s theory of convex games. There remains an impor-
+tant difference between these two theories: the concept of external stability of the
+core/base polyhedron. There is no analogue in the set function theory.
+Originally, Edmonds studied what he called a polymatroid, which is a polytope.
+His definition differs from the one of Fujishige, who defined a submodular set func-
+tion under the same name of polymatroid. These two definitions are however deeply
+connected. For two vectors x, y ∈ RN, we write x ≤ y if, for all i ∈ N, we have xi ≤ yi.
+
+T(2)
+P(f)
+0
+B(f)18
+DYLAN LAPLACE MERMOUD
+Definition 12 (Edmonds, 1970; Schrijver, 2003).
+A polymatroid P in RN
++ is a compact nonempty subset of RN
++ such that
+(1) if ⃗o ≤ y ≤ x ∈ P, then y ∈ P,
+(2) for each z ∈ RN
++ there exists a number ρ(z) such that each maximal vector x
+of P ∩ {x | x ≤ z} satisfies x(N) = ρ(z).
+This definition resembles a polytopal equivalent of the following definition of a ma-
+troid: a matroid is a pair M = (N, B) with N a finite set and B a nonempty set of
+so-called independent subsets of N such that
+(1) every subset of an independent set is an independent set,
+(2) for every A ⊆ B, every maximal independent subset of A has the same cardi-
+nality, called the rank, r(A), of A (w.r.t. M).
+Definition 13 (Fujishige, 2005).
+Let ρ : 2N → R be a grounded set function satisfying
+(1) A ⊆ B ⊆ N implies ρ(A) ≤ ρ(B),
+(2) for all A, B ⊆ N, we have ρ(A) + ρ(B) ≥ ρ(A ∪ B) + ρ(A ∩ B).
+The pair (N, ρ) is called a polymatroid and ρ is called rank function of the polymatroid.
+The two definitions are connected thanks to the forthcoming theorem.
+Theorem 9 (Edmonds, 1970).
+Let ξ : 2N → R be a grounded, non-decreasing, submodular set function. Then the
+following polyhedron is a polymatroid:
+P(ξ) = {x ∈ RN
++ | x(A) ≤ ξ(A), for all A ∈ N}.
+Another interesting polytope in the polyhedral combinatorics literature linked to the
+core is the (generalised) permutohedron. It is generated by an ‘admissible’ deformation
+of a permutohedron, a polytope extensively studied in polyhedral combinatorics.
+Definition 14 (Postnikov, 2009).
+Let x ∈ RN such that, for all i, j ∈ N, xi ̸= xj. The permutohedron ΠN(x) is the
+convex polytope in RN defined as the convex hull of all vectors obtained from x by
+permutation of the coordinates:
+ΠN(x) := conv {xσ | σ ∈ SN} ,
+where SN is the group of permutations of N and xσ = (xσ(1), . . . , xσ(n)).
+We see that cores of convex games, polymatroids, base polyhedra of submodular
+set functions and generalised permutahedra are the same objects, seen from differ-
+ent perspectives. However, definitions of permutohedra are not yet consistent in the
+literature. For some authors, the standard permutahedron is defined as
+ΠN
+�
+(n, n − 1, . . . , 2, 1)
+�
+.
+It is the core of the strictly convex game (N, v) defined, for all S ∈ N, by
+v(S) = |S| (|S| − 1)
+2
+.
+
+GEOMETRY OF SET FUNCTIONS IN COOPERATIVE GAME THEORY
+19
+In polyhedral combinatorics, since Postnikov (2009), people are studying deformations
+of permutohedra to deal with a more general type of polytopes. A generalised per-
+mutohedron is a polytope obtained from a permutohedron by moving vertices so that
+the directions of all edges are preserved, while some of the edges may accidentally
+degenerate into a single point. More formally, we have the following characterisation.
+Lemma 10 (Castillo and Liu, 2022).
+A polytope P ⊆ RN is a generalised permutohedron if and only if, all of its edge
+directions are in the form of
+1{i} − 1{j},
+for some i, j ∈ N such that i ̸= j.
+Figure 4. A permutohedron and two of its generalised permutohedra
+from Postnikov (2009).
+According to Postnikov, generalised permutohedra are parametrised by sets of 2n−1
+coordinates {zS}S∈N indexed by the nonempty subsets S ⊆ N.
+Each generalised
+permutohedron is of the form
+�
+x ∈ RN | x(N) = zN, and x(S) ≥ zS, for all S ∈ N
+�
+.
+Here, we can recognise the core’s definition, where the “parameters” {zS | S ∈ N}
+are the value v(S). Castillo and Liu (2022) called these sets of parameters deforming
+vectors. A game can then be viewed as a deformation of the standard permutohedron.
+This point of view joins the one we had earlier with the hyperplane arrangement
+{AS | S ∈ N} in X(N, v). Moreover, generalised permutohedra are deeply connected
+to submodular functions by the Subdmodulartiy Theorem (Castillo & Liu, 2022).
+Theorem 11 (Submodularity Theorem).
+There exists a bijection between generalised permutohedra P with dim(P) ≤ n − 1 and
+grounded submodular set functions on 2N.
+Another proof of the Submodularity Theorem can be found in Rehberg (2021). To
+conclude this part on convex games, let us introduce the convex cover of a balanced
+game.
+Definition 15. Let (N, v) be a balanced game. A convex cover of (N, v) is a convex
+game (N, v∗) such that C(N, v) = C(N, v∗).
+
+1,2,3)20
+DYLAN LAPLACE MERMOUD
+Unlike the exact cover and the totally balanced cover, which will be defined later,
+not all games have a convex cover.
+Proposition 7. A balanced game (N, v) has a convex cover if and only if its core is
+a generalised permutohedron. Moreover, its convex cover is unique.
+Proof. It is a corollary of the Submodularity Theorem. For each generalised permuto-
+hedron P, a unique submodular function ξ exists for which B(ξ) = P. Then, we use
+Lemma 8 to go from the submodular function to the convex game.
+□
+The relations between the polytopes presented in this subsection are summarised in
+the following table.
+Set functions
+Polymatroids
+Permutohedra
+Convex games
+Submodular
+polyhedron
+Extended
+polymatroid
+-
+Upper vectors
+-
+Polymatroid
+-
+-
+Base polyhedron
+Base polytope
+Generalised
+permutohedron
+Core
+-
+-
+Permutohedron
+C(N, v) with
+v : S �→ |S|(|S|−1)
+2
+Schrijver (2003),
+Fujishige (2005)
+Edmonds (1970),
+Schrijver (2003)
+Postnikov (2009),
+Castillo and Liu (2022)
+Shapley (1971),
+Grabisch (2016)
+Figure 5. Similarly defined polyhedra in different theories.
+As we have seen, convex games’ cores are very well-known, from at least four different
+perspectives. But a supermodular characteristic function is a restrictive condition, and
+many social interactions or economic situations can not be modelled by convex games.
+To model these problems, we need a more general theory.
+4.3. Totally balanced games.
+Definition 16. Let (N, v) be a game, and let S ∈ N be a coalition. The subgame (S, v)
+is the game on S where its characteristic function is v restricted to the subcoalitions
+of S. A game is totally balanced if all of its subgames are balanced.
+These totally balanced games appear from various and common situations.
+Market games. The definition of a mathematical market comes from Shapley and
+Shubik (1969).
+Let N be a finite set of players, or traders, let G be the nonneg-
+ative orthant of a finite-dimensional vector space, called the commodity space, let
+A = {ai | i ∈ N} ∈ GN be an indexed set of elements in G, called the initial endow-
+ments, and let U = {ui | i ∈ N} is an indexed set of continuous, concave functions
+ui : G → R called the utility functions. For any coalition S ∈ N, a set of endowments
+
+GEOMETRY OF SET FUNCTIONS IN COOPERATIVE GAME THEORY
+21
+{xi}i∈S ⊆ G such that �
+i∈S xi = �
+i∈S ai is called a feasible S-allocation of the mar-
+ket (N, G, A, U), and we denote their set by XS. The market game generated by this
+market is a game (N, v) such that, for all S ∈ N,
+v(S) = max
+x∈XS
+�
+i∈S
+ui(xi).
+Theorem 12 (Shapley and Shubik, 1969).
+A game is a market game if and only if it is totally balanced.
+For a specific subclass of market games, called assignment games, Solymosi and
+Raghavan (2001) found the set of games with a stable core.
+Flow games. Another example of a class of games with a wide range of applications
+is the class of flow games. These games are described by Kalai and Zemel (1982)
+as being useful for modelling problems of profit sharing in an integrated production
+system with alternative production routes. Let G = (V, E) be a directed graph, with
+V being the set of vertices and E being the set of edges, and let N be a finite set of
+players. Define two functions u : e ∈ E �→ u(e) ∈ R and p : e ∈ E �→ p(n) ∈ N, with
+u describing the “capacity” of an edge and p pointing to the owner of the edge. For a
+coalition S ∈ N, let GS be the subgraph restricted to the edges owned by a player in
+S. Then the characteristic function v of the flow game (N, v) associated with G maps
+the coalitions S to the maximal amount of flow carried throughout GS.
+Theorem 13 (Kalai and Zemel, 1982).
+A game is a flow game if and only if it is totally balanced.
+These games represent a much wider class of economic situations and are solely
+defined as games having nonempty cores, no matter at which level we study the ap-
+pearance of cooperation. Any totally balanced game, i.e., any market game or any
+flow game, is balanced, and this does not depend on a choice of a grand coalition. We
+can say that they are games which lead naturally the players to cooperate, at least
+the necessary condition of having a nonempty core is always satisfied, by definition.
+There are at least two other strong assets for totally balanced games.
+Definition 17. Two games are called d-equivalent (domination-equivalent) if they
+have the same imputation sets and the same domination relations on them.
+Then two d-equivalent games have precisely the same stable sets, or both have none.
+Also, if they have cores, they have the same cores. The notion of d-equivalence comes
+from Gillies (1959), and the definition above is the reformulation by Shapley and
+Shubik (1969). From the last authors, we also have the concept of a (totally balanced)
+cover of a game.
+Lemma 14. Let (N, v) be a game. The game (N, v) defined, for all S ∈ N, by
+v(S) = max
+B
+�
+T∈B
+λB
+T v(T),
+where the maximum is taken over the minimal balanced sets on S, is the cover of
+(N, v). If (N, v) is balanced, then it is d-equivalent to its cover (N, v).
+
+22
+DYLAN LAPLACE MERMOUD
+This result reduces the study of the external stability of cores of balanced games
+to the study of the external stability of cores of totally balanced games. Therefore,
+studying totally balanced games allows us to study a very broad class of games, but
+also through the d-equivalence, all the balanced games. This is why in this paper
+we always consider if needed, totally balanced games. But, unlike the convex games,
+it is not clear which type of set functions characterises the totally balanced games,
+but Kalai and Zemel (1982) found a very interesting and useful characterisation while
+studying flow games. The games which have an additive characteristic function are
+called inessential.
+Theorem 15 (Kalai and Zemel, 1982).
+A game (N, v) is totally balanced if and only if it is the minimum of a finite set of
+inessential games.
+Let {ai | i ∈ I} be a set of additive set functions indexed by a finite set I. When
+we say that a game (N, v) is the minimum of inessential games {(N, ai)}i∈I, we mean
+that, for all S ∈ N,
+v(S) = min
+i∈I ai(S).
+The set {ai | i ∈ I} is called a representation of the game (see Rosenm¨uller (2013)).
+Definition 18. We say that a set function ξ : 2N → R is min-additive is there exists
+a finite set of additive set functions {ai : 2N → R}i∈I such that, for all S ∈ 2N,
+ξ(S) = min
+i∈I ai(S).
+Then the min-additive set functions are to the totally balanced games what the
+super/submodular were to the convex games. Let a be an additive set function. Then,
+it is completely determined by its values on singletons, and we can therefore identify
+a to a n-dimensional vector a = (a({i}))i∈N. From this vector, we can associate a
+unique linear form ϕ on RN, defined, for all x ∈ RN, by
+ϕ(x) := ⟨a, x⟩.
+Definition 19. A tropical polynomial is a linear form defined, for all x ∈ RN, by
+φ(x) = min
+i∈I
+�
+ci + ⟨zi, x⟩
+�
+,
+with I a finite set of indices, and for all i ∈ I, ci ∈ R ∪ {−∞} and zi ∈ RN.
+Theorem 16. Let (N, v) be a totally balanced game. There exists a tropical polynomial
+φ : RN → R such that, for all S ∈ N, we have φ
+�
+1S�
+= v(S). If φ is a monic tropical
+polynomial, there exists a totally balanced game that can be extended to φ.
+Proof. Let (N, v) be a totally balanced game. Then, there exists a finite set of additive
+set functions {ai}i∈I such that, for all S ∈ N, we have v(S) = mini∈I{ai(S)}. Denote
+by ai the n-dimensional vector corresponding to the additive set function ai. Now,
+denote by φ the map φ : RN → R defined, for all x ∈ RN, by
+φ(x) = min
+i∈I ⟨ai, x⟩.
+
+REFERENCES
+23
+Therefore, for every coalition S ∈ N, we have
+φ
+�
+1S�
+= min
+i∈I ⟨ai, 1S⟩ = min
+i∈I ai(S) = v(S).
+Now, let φ be a tropical polynomial defined as the minimum of a finite set of linear
+forms ϕi, i.e., for all x ∈ RN,
+φ(x) = min
+i∈I ϕi(x).
+For all linear forms ϕi, there exists a unique vector ai such that, for all x ∈ RN,
+ϕi(x) = ⟨ai, x⟩.
+For each vector ai, we define an additive function ai by ai(S) = ai(S). The game
+(N, v) defined by v(S) = mini∈I ai(S) is totally balanced.
+□
+In the game-theoretical literature, there already exist different concepts of exten-
+sions of games. Back in 1972, Owen (1972) defined a multilinear extension for any
+game (N, v). Algaba et al. (2004) studied the Choquet integral (also called the Lov´asz
+extension, see Grabisch (2016)) in the context of cooperative game theory. The Cho-
+quet and the Sugeno integrals are also used as extensions of capacities in decision
+theory (Grabisch, 2016). But our extension aims to build deeper connections with
+combinatorial optimisation and discrete geometry.
+5. Concluding remarks
+We investigated many links between cooperative game theory and various parts
+of discrete mathematics. Seeing cooperative games as deformations of the resonance
+hyperplane arrangement, and cores as deformations of generalised permutohedra may
+help to import new results and tools from other fields into the study of cooperative
+games, and the other way around. Furthermore, the study of totally balanced games
+with the help of tropical polynomial and tropical hypersurfaces has to be continued,
+in the same way, that set functions, polymatroids and generalised permutohedra have
+contributed to the study of convex games.
+References
+Aguiar, M., & Mahajan, S. (2017). Topics in hyperplane arrangements (Vol. 226).
+American Mathematical Society.
+Algaba, E., Bilbao, J. M., Fern´andez, J. R., & Jimenez, A. (2004). The Lov´asz exten-
+sion of market games. Essays in cooperative games (pp. 229–238). Springer.
+Aumann, R. J., & Maschler, M. (1985). Game theoretic analysis of a bankruptcy
+problem from the talmud. Journal of economic theory, 36(2), 195–213.
+Barvinok, A. (2002). A course in convexity (Vol. 54). American Mathematical Society.
+Bauschke, H. H., & Combettes, P. L. (2011). Convex analysis and monotone operator
+theory in hilbert spaces (Vol. 408). Springer.
+Billera, L. J., Moore, J. T., Moraites, C. D., Wang, Y., & Williams, K. (2012). Maximal
+unbalanced families. arXiv preprint arXiv:1209.2309.
+Bondareva, O. N. (1963). Some applications of linear programming methods to the
+theory of cooperative games. Problemy kibernetiki, 10(119), 139.
+
+24
+REFERENCES
+Castillo, F., & Liu, F. (2022). Deformation cones of nested braid fans. International
+Mathematics Research Notices, 2022(3), 1973–2026.
+Cesco, J. C. (1998). A convergent transfer scheme to the core of a tu-game. Revista
+de Matem´aticas Aplicadas, 19(1-2), 23–35.
+Driessen, T. S. (2013). Cooperative games, solutions and applications (Vol. 3). Springer
+Science & Business Media.
+Edmonds, J. (1970). Submodular functions, matroids and certain polyhedra. Combi-
+natorial Structures and Their Applications.
+Fujishige, S. (2005). Submodular functions and optimization. Elsevier.
+Gillies, D. B. (1959). Solutions to general non-zero-sum games. Contributions to the
+Theory of Games, 4, 47–85.
+Grabisch, M. (2016). Set functions, games and capacities in decision making (Vol. 46).
+Springer.
+Grabisch, M., & Sudh¨olter, P. (2021). Characterization of tu games with stable cores
+by nested balancedness. Mathematical Programming, 1–26.
+Kalai, E., & Zemel, E. (1982). Totally balanced games and games of flow. Mathematics
+of Operations Research, 7(3), 476–478.
+Kamiya, H., Takemura, A., & Terao, H. (2012). Arrangements stable under the coxeter
+groups. Configuration spaces (pp. 327–354). Springer.
+K¨uhne, L. (2020). The universality of the resonance arrangement and its betti numbers.
+arXiv preprint arXiv:2008.10553.
+Laplace Mermoud, D., Grabisch, M., & Sudh¨olter, P. (2022). Core stability and other
+applications of minimal balanced collections. (No. 4/2022). University of South-
+ern Denmark, Department of Economics.
+Lucas, W. F. (1969). The proof that a game may not have a solution. Transactions of
+the American Mathematical Society, 137, 219–229.
+Lucas, W. F. (1992). Von neumann-morgenstern stable sets. Handbook of game theory
+with economic applications, 1, 543–590.
+O’Neill, B. (1982). A problem of rights arbitration from the talmud. Mathematical
+social sciences, 2(4), 345–371.
+Owen, G. (1972). Multilinear extensions of games. Management Science, 18(5-part-2),
+64–79.
+Postnikov, A. (2009). Permutohedra, associahedra, and beyond. International Mathe-
+matics Research Notices, 2009(6), 1026–1106.
+Rehberg, S. (2021). Combinatorial reciprocity theorems for generalized permutahedra,
+hypergraphs, and pruned inside-out polytopes. arXiv preprint arXiv:2103.09073.
+Rosenm¨uller, J. (2013). Game theory: Stochastics, information, strategies and cooper-
+ation (Vol. 25). Springer Science & Business Media.
+Schrijver, A. (2003). Combinatorial optimization: Polyhedra and efficiency (Vol. 24).
+Springer.
+Shapley, L. S. (1952). Notes on the n-person game, III: Some variants of the von
+Neumann-Morgenstern definition of solution. RAND Corporation. https://doi.
+org/10.7249/RM817
+
+REFERENCES
+25
+Shapley, L. S. (1965). On balanced sets and cores (tech. rep.). Santa Monica, CA,
+RAND Corporation.
+Shapley, L. S. (1971). Cores of convex games. International journal of game theory,
+1(1), 11–26.
+Shapley, L. S., & Shubik, M. (1967). Ownership and the production function. The
+Quarterly Journal of Economics, 81(1), 88–111.
+Shapley, L. S., & Shubik, M. (1969). On market games. Journal of Economic Theory,
+1(1), 9–25.
+Solymosi, T., & Raghavan, T. (2001). Assignment games with stable core. Interna-
+tional Journal of Game Theory, 30(2), 177–185.
+Stearns, R. E. (1968). Convergent transfer schemes for n-person games. Transactions
+of the American Mathematical Society, 134(3), 449–459.
+van Gellekom, J., Potters, J. A., & Reijnierse, J. (1999). Prosperity properties of tu-
+games. International Journal of Game Theory, 28(2), 211���227.
+von Neumann, J. (1928). On the theory of games of strategy. Zur theorie der gesellschaftsspiele
+(S. Bargmann, Trans.). Mathematische annalen, 100(1), 295–320.
+von Neumann, J., & Morgenstern, O. (1944). Theory of games and economic behavior.
+Princeton University Press.
+Centre d’´Economie de la Sorbonne, Universit´e Paris I Panth´eon-Sorbonne, 106-112
+boulevard de l’Hˆopital, 75013, Paris, France
+Email address: dylan.laplace.mermoud@gmail.com
+

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+arXiv:2301.00176v1  [math.NA]  31 Dec 2022
+RANDOMIZED KACZMARZ METHOD WITH ADAPTIVE STEPSIZES
+FOR INCONSISTENT LINEAR SYSTEMS
+YUN ZENG, DEREN HAN, YANSHENG SU, AND JIAXIN XIE
+Abstract. We investigate the randomized Kaczmarz method that adaptively updates the
+stepsize using readily available information for solving inconsistent linear systems. A novel
+geometric interpretation is provided which shows that the proposed method can be viewed
+as an orthogonal projection method in some sense. We prove that this method converges
+linearly in expectation to the unique minimum Euclidean norm least-squares solution of
+the linear system. Numerical experiments are given to illustrate the theoretical results.
+1. Introduction
+Solving systems of linear equations is a fundamental problem in scientific computing
+and engineering. It comes up in many real-world applications such as signal processing
+[6], optimal control [41], machine learning [8], and partial differential equations [40]. The
+Kaczmarz method [24], also known as algebraic reconstruction technique (ART) [15,22], is
+a classic while effective row-action iteration solver for solving the large-scale linear system
+of equations
+(1)
+Ax = b, A ∈ Rm×n, b ∈ Rm.
+At each step of the original Kaczmarz method, a row of the system is sampled and the
+previous iterate is orthogonally projected onto the hyperplane defined by that row.
+In the literature, there was empirical evidence that using the rows of the matrix A in
+random order rather than in their given order can often accelerate the convergence of the
+Kaczmarz method [13, 22, 32]. In the seminal paper [47], Strohmer and Vershynin studied
+the randomized Kaczmarz (RK) method and proved its linear convergence in expectation
+provided that the linear system (1) is consistent. Subsequently, there is a large amount of
+work on the development of the Kaczmarz-type method including accelerated randomized
+Key words: system of linear equations, inconsistency, Kaczmarz, adaptive stepsize, minimum Euclidean
+norm least-squares solution
+Mathematics subject classification (2020): 65F10, 65F20, 90C25, 15A06, 68W20
+1
+
+2
+YUN ZENG, DEREN HAN, YANSHENG SU, AND JIAXIN XIE
+Kaczmarz methods [19,27,28], block Kaczmarz methods [17,31,33,36], greedy randomized
+Kaczmarz methods [2,16], randomized sparse Kaczmarz methods [9,45], etc. Nevertheless,
+all of these methods will not converge if the linear system (1) is not consistent. Indeed,
+Needell [34] showed that RK applied to inconsistent linear systems converges only to within
+a radius (convergence horizon) of the least-squares solution (see Theorem 2.1); see also [4]
+for some further comments.
+It is well-known that the so-called relaxation parameters or stepsizes λ1, . . . , λk are impor-
+tant for the Kaczmarz method in practice. The original Kaczmarz method with decreasing
+stepsizes for solving inconsistent systems has been investigated in [7,20]. It has shown that
+with the stepsizes being nearly zero and appropriate initial point, the Kaczmarz method
+converges inside the convergence horizon to the minimum Euclidean norm least-squares
+solution. However, its convergence rate is difficult to obtain. Hence, for the RK method,
+a natural and interesting question is that Is it possible that with carefully designed step-
+sizes the RK method is convergent for solving inconsistent systems? Furthermore, can the
+convergence rate of the proposed method be obtained easily?
+In this paper, we answer positively to the above questions. We investigate the RK method
+with adaptive stepsizes for solving inconsistent systems.
+To the best of our knowledge,
+this is the first time that dynamic stepsizes have been identified as a key property for
+proving the convergence of the RK method for inconsistent systems. We show that our
+strategy is effective in simplifying the analysis and endows the proposed method with a
+linear convergence rate.
+Finally, we mention that in a fruitful line of research, the extended randomized methods
+have been proposed for solving inconsistent systems, which include randomized extended
+Kaczmarz (REK) method [10, 51], its block or deterministic variants [3, 11, 12, 37, 42, 43,
+48–50], greedy randomized augmented Kaczmarz (GRAK) method [4], randomized extended
+Gauss-Seidel (REGS) method [10, 30], etc.
+These methods make use of both rows and
+columns of A in each step (see (8)) and work for general linear systems (consistent or
+inconsistent, full-rank or rank-deficient). Another randomized method that can be used
+to solve inconsistent systems with full column-rank coefficient matrix is the randomized
+coordinate descent (RCD) method [26], we refer to [1] for more discussions about the RCD
+method.
+
+RKAS FOR INCONSISTENT SYSTEMS
+3
+Table 1. The pseudoinverse solution A†b of Ax = b.
+Ax = b
+rank(A)
+A†b
+consistent
+= n
+unique solution
+consistent
+< n
+unique minimum Euclidean norm solution
+inconsistent
+= n
+unique least-squares (LS) solution
+inconsistent
+< n
+unique minimum Euclidean norm LS solution
+The remainder of the paper is organized as follows. After introducing some preliminaries
+in Section 2, we present and analyze the RK method with adaptive stepsizes in Section
+3. In Section 4, we perform some numerical experiments to show the effectiveness of the
+proposed method. Finally, we conclude the paper in Section 5.
+2. Preliminaries
+2.1. Notations. Throughout the paper, for any random variables ξ and ζ, we use E[ξ] and
+E[ξ|ζ] to denote the expectation of ξ and the conditional expectation of ξ given ζ. For an
+integer m ≥ 1, let [m] := {1, . . . , m}. For any vector x ∈ Rn, we use xi, x⊤, and ∥x∥2 to
+denote the i-th entry, the transpose and the Euclidean norm of x, respectively. For any
+matrix A ∈ Rm×n, we use Ai,:, A:,j, A⊤, A†, ∥A∥2, ∥A∥F , Range(A), and Null(A) to denote
+the i-th row, the j-th column, the transpose, the Moore-Penrose pseudoinverse, the spectral
+norm, the Frobenius norm, the column space, and the null space of A, respectively. The
+nonzero singular values of a matrix A are σ1(A) ≥ σ2(A) ≥ . . . ≥ σr(A) := σmin(A) > 0,
+where r is the rank of A and σmin(A) denotes the smallest nonzero singular values of A. We
+see that ∥A∥2 = σ1(A) and ∥A∥F =
+�
+r�
+i=1
+σi(A)2.
+2.2. The pseudoinverse solution. In this paper, we are interested in the pseudoinverse
+solution A†b of the linear system (1). Here we would like to make clear what A†b represents
+in different cases of linear systems [5,11,14]. Table 1 summarizes the results.
+2.3. The RK method. The RK method for solving the linear system (1) begins with an
+arbitrary vector x0, and in the k-th iteration iterates by
+(2)
+xk+1 = xk − λAik,:xk − bik
+∥Aik,:∥2
+2
+A⊤
+ik,:,
+
+4
+YUN ZENG, DEREN HAN, YANSHENG SU, AND JIAXIN XIE
+where the index ik is i.i.d. selected from [m] and λ ∈ (0, 2) is the stepsize. When λ = 1,
+it reduces to the classical RK method. In the seminal paper [47], Strohmer and Vershynin
+proved the first linear convergence rate of the RK method for consistent systems. Later,
+Needell [34,35] studied the RK method for inconsistent cases. The result is precisely restated
+below.
+Theorem 2.1 ( [35], Corollary 5.1). Starting from any initial vector x0 ∈ Range(A⊤), the
+expected error of the RK method (2) in the k-th iteration satisfies
+(3)
+E
+����xk − A†b
+���
+2
+2
+�
+≤
+�
+1 − 2λ(1 − λ)σ2
+min(A)
+∥A∥2
+F
+�k ���x0 − A†b
+���
+2
+2 +
+λa2
+max∥e∥2
+2
+(1 − λ)a2
+minσ2
+min(A),
+where the index i is selected with probability
+∥Ai,:∥2
+2
+∥A∥2
+F , λ ∈ (0, 1), e = AA†b − b, a2
+min =
+min
+i∈[m] ∥Ai,:∥2
+2, and a2
+max = max
+i∈[m] ∥Ai,:∥2
+2.
+It can be seen that for arbitrary λ, the above result implies a tradeoff between a smaller
+convergence horizon and a slower convergence. In this paper, we study the RK method
+with adaptive stepsizes λk to eliminate the last term in (3) (see Corollary 3.2).
+3. Adaptive stepsizes for RK
+In this section, we introduce RK with adaptive stepsizes (RKAS) for solving the linear
+system (1). The method is formally described in Algorithm 1.
+Algorithm 1 RK with adaptive stepsizes (RKAS)
+Input: A ∈ Rm×n, b ∈ Rm, k = 0 and initial points x0 = 0, r0 = Ax0 − b = −b.
+1: Select ik ∈ [m] with probability Pr(ik = i) = ∥Ai,:∥2
+2
+∥A∥2
+F .
+2: Compute
+αk =
+⟨AA⊤
+ik,:, rk⟩
+∥AA⊤
+ik,:∥2
+2
+.
+3: Update
+xk+1 = xk − αkA⊤
+ik,:,
+rk+1 = rk − αkAA⊤
+ik,:.
+4: If the stopping rule is satisfied, stop and go to output. Otherwise, set k = k + 1 and
+return to Step 1.
+Output: The approximate solution.
+
+RKAS FOR INCONSISTENT SYSTEMS
+5
+Note that the most expensive computational cost in the k-th iteration of Algorithm 1
+is to compute AA⊤
+ik,:.
+Let B := AA⊤, then B:,ik = AA⊤
+ik,:.
+Thus, if it is possible to
+store B = AA⊤ at the initialization, Algorithm 1 could be faster in practice. In fact, this
+strategy is also adopted by the greedy randomized Kaczmarz method [2,4] and the weighted
+randomized Kaczmarz method [46].
+3.1. A geometric interpretation. We present an intuitive geometric explanation of Al-
+gorithm 1 in this subsection. Consider the following least-squares problem
+min
+x∈Rn f(x) := 1
+2∥Ax − b∥2
+2.
+Since
+f(x) = 1
+2x⊤A⊤Ax − x⊤A⊤b + 1
+2∥b∥2
+2
+= 1
+2x⊤A⊤Ax − x⊤A⊤AA†b + 1
+2∥b∥2
+2
+= 1
+2∥Ax − AA†b∥2
+2 − 1
+2∥AA†b∥2
+2 + 1
+2∥b∥2
+2,
+where the second equality follows from the fact that A⊤AA†b = A⊤b. This implies that the
+least-squares problem can be equivalently reformulated as
+(4)
+min
+x∈Rn
+1
+2∥Ax − AA†b∥2
+2.
+When using RK (2) to solve the least-squares problem (4), in the k-th iteration, we may
+expect the distance between Axk+1 and AA†b to be as small as possible. This leads to the
+following optimization problem:
+(5)
+xk+1 = arg min
+x∈Rn ∥Ax − AA†b∥2
+2 subject to x = xk − λAik,:xk − bik
+∥Aik,:∥2
+2
+A⊤
+ik,:, λ ∈ R.
+Note that rk+1 in Step 3 is actually obtained by an incremental method
+rk+1 = rk − αkAA⊤
+ik,: = Axk − b − αkAA⊤
+ik,: = Axk+1 − b.
+Using the fact that A⊤AA†b = A⊤b and letting γk =
+Aik,:xk−bik
+∥Aik,:∥2
+2
+, then the minimizer of (5)
+is achieved when
+λ∗
+k =
+⟨AA⊤
+ik,:, Axk − AA†b⟩
+∥AA⊤
+ik,:∥2
+2γk
+=
+⟨AA⊤
+ik,:, Axk − b⟩
+∥AA⊤
+ik,:∥2
+2γk
+=
+⟨AA⊤
+ik,:, rk⟩
+∥AA⊤
+ik,:∥2
+2γk
+.
+Hence
+xk+1 = xk − λ∗
+kγkAik,: = xk −
+⟨AA⊤
+ik,:, rk⟩
+∥AA⊤
+ik,:∥2
+2
+A⊤
+ik,: = xk − αkA⊤
+ik,:,
+
+6
+YUN ZENG, DEREN HAN, YANSHENG SU, AND JIAXIN XIE
+which is exactly the iteration in Step 3 of Algorithm 1.
+It follows from (5) that xk+1
+obtained by Algorithm 1 satisfies that Axk+1 is the orthogonal projection of AA†b onto
+Axk + Span{AA⊤
+ik,:}. The geometric interpretation of Algorithm 1 is presented in Figure 1.
+Axk + Span{AA⊤
+ik,:}
+Axk+1 + Span{AA⊤
+ik+1,:}
+AA†b
+Axk+1
+Axk
+Axk+2
+Figure 1. A geometric interpretation of Algorithm 1. The next iterate xk+1
+arises such that Axk+1 is the projection of AA†b onto Axk + Span{AA⊤
+ik,:}.
+3.2. Convergence analysis. We now state our convergence results. The following result
+is about the convergence for E[∥Axk − AA†b∥2
+2].
+Theorem 3.1. For any given linear system Ax = b, let {xk}∞
+k=0 be the iteration sequence
+generated by Algorithm 1 with x0 = 0. Then
+E
+����Axk − AA†b
+���
+2
+2
+�
+≤
+�
+1 −
+σ4
+min(A)
+∥A∥2
+2∥A∥2
+F
+�k ���Ax0 − AA†b
+���
+2
+2 .
+Proof. Since Axk+1 − AA†b and AA⊤
+ik,: are orthogonal, we have
+���Axk+1 − AA†b
+���
+2
+2 =
+���Axk − AA†b
+���
+2
+2 −
+���Axk+1 − Axk���
+2
+2
+=
+���Axk − AA†b
+���
+2
+2 −
+⟨AA⊤
+ik,:, rk⟩2
+∥AA⊤
+ik,:∥2
+2
+=
+���Axk − AA†b
+���
+2
+2 −
+⟨AA⊤
+ik,:, Axk − b⟩2
+∥AA⊤
+ik,:∥2
+2
+=
+���Axk − AA†b
+���
+2
+2 −
+⟨AA⊤
+ik,:, Axk − AA†b⟩2
+∥AA⊤
+ik,:∥2
+2
+,
+
+RKAS FOR INCONSISTENT SYSTEMS
+7
+where the last equality follows from A⊤b = A⊤AA†b. Hence
+E
+����Axk+1 − AA†b
+���
+2
+2
+����xk
+�
+=
+���Axk − AA†b
+���
+2
+2 −
+m
+�
+i=1
+∥Ai,:∥2
+2
+∥A∥2
+F
+⟨AA⊤
+i,:, Axk − AA†b⟩2
+∥AA⊤
+i,:∥2
+2
+≤
+���Axk − AA†b
+���
+2
+2 −
+m
+�
+i=1
+∥Ai,:∥2
+2
+∥A∥2
+F
+⟨AA⊤
+i,:, Axk − AA†b⟩2
+∥A∥2
+2∥Ai,:∥2
+2
+=
+���Axk − AA†b
+���
+2
+2 −
+m
+�
+i=1
+⟨AA⊤
+i,:, Axk − AA†b⟩2
+∥A∥2
+2∥A∥2
+F
+=
+���Axk − AA†b
+���
+2
+2 − ∥AA⊤(Axk − AA†b)∥2
+2
+∥A∥2
+2∥A∥2
+F
+≤
+�
+1 −
+σ4
+min(A)
+∥A∥2
+2∥A∥2
+F
+� ���Axk − AA†b
+���
+2
+2 ,
+where the first inequality follows from ∥AA⊤
+i,:∥2 ≤ ∥A∥2∥Ai,:∥2 and the last inequality follows
+from the fact that A(xk − A†b) ∈ Range(AA⊤). Taking expectation over the entire history
+we have
+E
+����Axk+1 − AA†b
+���
+2
+2
+�
+≤
+�
+1 −
+σ4
+min(A)
+∥A∥2
+2∥A∥2
+F
+�
+E
+����Axk − AA†b
+���
+2
+2
+�
+.
+By induction on the iteration index k, we can obtain the desired result.
+□
+By Theorem 3.1, we can obtain the following linear convergence for the expected norm
+of the error.
+Corollary 3.2. For any given linear system Ax = b, let {xk}∞
+k=0 be the iteration sequence
+generated by Algorithm 1 with x0 = 0. Then
+E
+����xk − A†b
+���
+2
+2
+�
+≤ σ−2
+min(A)
+�
+1 −
+σ4
+min(A)
+∥A∥2
+2∥A∥2
+F
+�k ���Ax0 − AA†b
+���
+2
+2 .
+Proof. According to the iteration of Algorithm 1 and x0 = 0, we know that xk ∈ Range(A⊤).
+As A†b ∈ Range(A⊤), we have xk − A†b ∈ Range(A⊤). This implies
+���Axk − AA†b
+���
+2
+2 ≥ σ2
+min(A)
+���xk − A†b
+���
+2
+2 .
+Then by Theorem 3.1, we arrive at this corollary.
+□
+Remark 3.3. We note that one can choose any x0 ∈ Rn as the initial vector. In this case,
+we can prove that
+E
+����xk − x0
+∗
+���
+2
+2
+�
+≤ σ−2
+min(A)
+�
+1 −
+σ4
+min(A)
+∥A∥2
+2∥A∥2
+F
+�k ��Ax0 − Ax0
+∗
+��2
+2 ,
+
+8
+YUN ZENG, DEREN HAN, YANSHENG SU, AND JIAXIN XIE
+where x0
+∗ := A†b + (I − A†A)x0. We refer the reader to [12,18,19] for more details.
+Remark 3.4. For the RK method (2), from (3) and with an analysis analogous to [35,
+Corollary 2.2], for any desired ε, using a stepsize
+λ =
+εσ2
+min(A)a2
+min
+2εσ2
+min(A)α2
+min + 2∥e∥2
+2a2max
+,
+one has that after
+(6)
+k = 2 log (2ε0/ε)
+� ∥A∥2
+F
+σ2
+min(A) + ∥A∥2
+F ∥e∥2
+2a2
+max
+εσ4
+min(A)a2
+min
+�
+iterations, E[∥xk − A†b∥2
+2] ≤ ε, where ε0 = ∥x0 − A†b∥2
+2. For Algorithm 1, according to
+corollary 3.2, we have that after
+(7)
+k = log
+�
+ε1
+εσ2
+min(A)
+� ∥A∥2
+F ∥A∥2
+2
+σ4
+min(A)
+iterations, E[∥xk −A†b∥2
+2] ≤ ε, where ε1 = ∥Ax0 −AA†b∥2
+2. From (6) and (7), we know that
+the RKAS method shall use less number of iterations than that of the RK method to obtain
+an iterative solution with the accuracy ε < O(1/∥A∥2
+2).
+Remark 3.5. Recently, the randomized extended Kaczmarz (REK) method [10,51] has at-
+tracted much attention for solving inconsistent systems. As shown by Du [10, Theorem 2],
+the convergence factor for REK is 1 − σ2
+min(A)
+∥A∥2
+F , which is better than 1 −
+σ4
+min(A)
+∥A∥2
+2∥A∥2
+F established
+in Corollary 3.2. However, we note that RKAS is a row-action method, while REK uses
+both a row and a column of the matrix A in each step. Therefore, RKAS would be partic-
+ularly suitable for handling large-scale and sparse systems, which is also confirmed by our
+numerical experiments.
+4. Numerical experiments
+In this section, we describe some numerical results for the RKAS method for inconsistent
+systems. We also compare RKAS with REK [10,51] on a variety of test problems. Recall
+that REK generates two sequences {zk}∞
+k=0 and {xk}∞
+k=0 via
+(8)
+zk+1
+= zk −
+A⊤
+:,jkzk
+∥A:,jk∥2
+2 A:,jk,
+xk+1
+= xk −
+Aik,:xk−bik+(zk+1)ik
+∥Aik,:∥2
+2
+A⊤
+ik,:,
+
+RKAS FOR INCONSISTENT SYSTEMS
+9
+where the column A:,jk is chosen with probability
+∥A:,jk∥2
+2
+∥A∥2
+F
+and the row Aik,: is chosen with
+probability
+∥Aik,:∥2
+2
+∥A∥2
+F , see [10]. All methods are implemented in Matlab R2022a for Windows
+10 on a desktop PC with the Intel(R) Core(TM) i7-10710U CPU @ 1.10GHz and 16 GB
+memory.
+As in Du et al [11], to construct an inconsistent linear system, we set b = Ax + r, where
+x is a vector with entries generated from a standard normal distribution and the residual
+r ∈ Null
+�
+A⊤�
+. Note that one can obtain such a vector r by the Matlab function null.
+For RKAS, we set x0 = 0, and for REK, we set z0 = b and x0 = 0. We stop the algorithms
+if the relative solution error (RSE) ∥xk−A†b∥2
+2
+∥A†b∥2
+2
+≤ 10−12. We report the average number of
+iterations (denoted as Iter) and the average computing time in seconds (denoted as CPU)
+of RKAS and REK.
+4.1. Synthetic data. We use the following two types of coefficient matrices.
+• For given m, n, r, and κ > 1, we construct a dense matrix A by A = UDV T , where
+U ∈ Rm×r, D ∈ Rr×r, and V ∈ Rn×r. Using Matlab colon notation, these matri-
+ces are generated by [U,∼]=qr(randn(m,r),0), [V,∼]=qr(randn(n,r),0), and
+D=diag(1+(κ-1).*rand(r,1)). So the condition number of A is upper bounded
+by κ.
+• We construct a random sparse matrix by using the Matlab sparse random matrix
+function sprandn(m,n,density,rc), where density is the percentage of nonzero
+entries and rc is the reciprocal of the condition number.
+Figures 2 and 3 illustrate our experimental results with a fixed n. In Figure 2, we plot
+the computing time of the REK and RKAS for inconsistent linear systems with coefficient
+matrices A = UDV ⊤, where m = 1000, 2000, . . . , 10000, n = 100, r = 80, κ(A) = 2 (left) or
+κ(A) = 10 (right). It can be observed from Figure 2 that REK is more efficient than RKAS
+for solving the dense problem, and the changing of parameter κ(A) affects the performance
+of RKAS greatly than that of REK.
+In Figure 3, we plot the computing time of the REK and RKAS with random sparse
+matrices A, where m = 1000, 2000, . . . , 10000, n = 100,κ(A) = 2 (left) or κ(A) = 10 (right),
+and the sparsity of the coefficient matrices A is 0.1. Noting that now κ(A) is exactly the
+condition number of A. It can be seen that RKAS performs better than REK. Indeed, this
+
+10
+YUN ZENG, DEREN HAN, YANSHENG SU, AND JIAXIN XIE
+is because the computational cost of RKAS has reduced a lot at each step. It can be also
+found that the more rows there are than columns, the better RKAS performs than REK.
+This is due to REK adopting both a row and a column at each step, while RKAS is a row-
+action method where only a single row is used at each step. To illustrate this observation
+more clearly, in Figure 4, we plot the computing times of REK and RKAS with a fixed
+m = 10000 and n = 100, 200, . . . , 1000. It is clear that the performance of RKAS is better
+than REK when n is small, and REK is better than RKAS when n is large.
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+8000
+9000
+10000
+Number of rows (m)
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+CPU time
+REK
+RKAS
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+8000
+9000
+10000
+Number of rows (m)
+0
+20
+40
+60
+80
+100
+120
+CPU time
+REK
+RKAS
+Figure 2. Figures depict the CPU time (in seconds) vs increasing number
+of rows for the case of the random dense matrix.
+The title of each plot
+indicates the values of n, r, and κ. All plots are averaged over 50 trials.
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+8000
+9000
+10000
+Number of rows (m)
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04
+0.045
+CPU time
+REK
+RKAS
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+8000
+9000
+10000
+Number of rows (m)
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+0.18
+0.2
+CPU time
+REK
+RKAS
+Figure 3. Figures depict the CPU time (in seconds) vs increasing number
+of rows for the case of the random sparse matrix. The title of each plot
+indicates the values of n, κ, and sparsity. All plots are averaged over 50
+trials.
+4.2. Real-world data. The real-world data are available via the SuiteSparse Matrix Col-
+lection [25]. The five matrices are nemsafm, df2177, ch8 8 b1, bibd 16 8, and ash958.
+Each dataset consists of a matrix A ∈ Rm×n and a vector b ∈ Rm. In our experiments, we
+only use the matrices A of the datasets and ignore the vector b. In Table 2, we report the
+
+RKAS FOR INCONSISTENT SYSTEMS
+11
+100
+200
+300
+400
+500
+600
+700
+800
+900
+1000
+Number of columns (n)
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+CPU time
+REK
+RKAS
+100
+200
+300
+400
+500
+600
+700
+800
+900
+1000
+Number of columns (n)
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+CPU time
+REK
+RKAS
+Figure 4. Figures depict the CPU time (in seconds) vs increasing number
+of columns for the case of the random sparse matrix. The title of each plot
+indicates the values of m, κ, and sparsity. All plots are averaged over 50
+trials.
+Table 2. The average (50 trials of each algorithm) Iter and CPU of REK
+and RKAS for inconsistent linear systems with coefficient matrices from [25].
+Matrix
+m × n
+rank
+σmax(A)
+σmin(A)
+REK
+RKAS
+Iter
+CPU
+Iter
+CPU
+nemsafm
+334 × 2348
+334
+4.77
+41308.70 1.3104 120565.48 2.3087
+df2177
+630 × 10358 630
+2.01
+20192.62 5.1010
+21480.34
+2.9148
+ch8 8 b1
+1568 × 64
+63
+3.48e+14 1800.96
+0.0186
+1686.84
+0.0136
+bibd 16 8 120 × 12870 120
+9.54
+7859.60 3.3143 151632.30 32.4403
+ash958
+958 × 292
+292
+3.20
+15711.02 0.1037 42197.00
+0.1924
+number of iterations and the computing times for REK and RKAS. It can be observed that
+RKAS is comparable with REK for solving inconsistent linear systems.
+5. Concluding remarks
+Consider the following the least-squares problem
+(9)
+min
+x∈Rn f(x) :=
+1
+2m∥Ax − b∥2
+2 = 1
+m
+m
+�
+i=1
+fi(x),
+where fi(x) = 1
+2 (Ai,:x − bi)2. To state conveniently, we assume that A ∈ Rm×n is normal-
+ized to ∥Ai,:∥2
+2 = 1 for each row of A. The RK method (2) can be seen as stochastic gradient
+descent (SGD) [21,29,44] applied to the least-squares problem (9). Indeed, SGD solves (9)
+using unbiased estimates for the gradient of the objective function, i.e. ∇fi(x) such that
+E [∇fi(x)] = ∇f(x). At each iteration, a random unbiased estimate ∇fi(x) is drawn and
+
+12
+YUN ZENG, DEREN HAN, YANSHENG SU, AND JIAXIN XIE
+SGD uses the following update formula
+(10)
+xk+1 = xk − λk∇fi(xk),
+where λk is an appropriately chosen stepsize. Noting that if a random row of the matrix A
+is selected and (10) is computed with ∇fi(xk) =
+�
+Ai,:xk − bi
+�
+A⊤
+i,:, then one can recover the
+RK method.
+It is well-known that SGD suffers from slow convergence as the variance of the gradient
+estimate ∇fi(x) does not naturally diminish, i.e.
+lim
+k→∞ E[∥∇fik(xk) − ∇f(xk)∥2
+2] ̸= 0. Let
+x∗ be an optimal point of (9) and consider the variance of its gradient estimate
+σ2 = E[∥∇fi(x∗) − ∇f(x∗)∥2
+2] = E[∥∇fi(x∗)∥2
+2] =
+m
+�
+i=1
+(Ai,:x∗ − bi)2
+m
+= ∥e∥2
+2
+m ,
+where e = Ax∗ − b is the residual at x∗. When the system is consistent, as the iterate
+approaches x∗, the residual gradually drops to zero and thus so does the variance, which
+ensures the convergence of SGD with a constant stepsize. When the system is inconsistent,
+however, e ̸= 0 and the variance does not decrease to zero. In this case, variance reduction
+techniques are introduced [23,39], otherwise a decreasing stepsize is required. Nevertheless,
+the decreasing stepsize brings about adverse effect on the convergence of SGD, which is
+sublinear even if the objective function is strongly convex [38].
+In this paper, we have
+shown that RK, i.e. SGD for (9), with our adaptive stepsize strategy enjoys a linear rate
+without any variance reduction procedure. A natural extension of our results is the design
+and analysis of adaptive stepsizes for SGD in the case of general convex or strongly convex
+functions. This should be an interesting and valuable topic that deserves in-depth study in
+the future.
+Finally, we note that a bunch of advanced probability criteria have been investigated in
+the literature for the RK method, such as the greedy selection rule [2] and the weighted
+version [46]. These criteria are convenient to be adapted to the RKAS context for further
+improvement in efficiency.
+References
+[1] Zhong-Zhi Bai, Lu Wang, and Wen-Ting Wu. On convergence rate of the randomized Gauss–Seidel
+method. Linear Algebra Appl., 611:237–252, 2021.
+[2] Zhong-Zhi Bai and Wen-Ting Wu. On greedy randomized Kaczmarz method for solving large sparse
+linear systems. SIAM J. Sci. Comput., 40(1):A592–A606, 2018.
+
+RKAS FOR INCONSISTENT SYSTEMS
+13
+[3] Zhong-Zhi Bai and Wen-Ting Wu. On partially randomized extended Kaczmarz method for solving
+large sparse overdetermined inconsistent linear systems. Linear Algebra Appl., 578:225–250, 2019.
+[4] Zhong-Zhi Bai and Wen-Ting Wu. On greedy randomized augmented Kaczmarz method for solving
+large sparse inconsistent linear systems. SIAM J. Sci. Comput., 43(6):A3892–A3911, 2021.
+[5] Adi Ben-Israel and Thomas NE Greville. Generalized inverses: theory and applications, volume 15.
+Springer Science & Business Media, 2003.
+[6] Charles Byrne. A unified treatment of some iterative algorithms in signal processing and image recon-
+struction. Inverse Problems, 20(1):103–120, 2003.
+[7] Yair Censor, Paul PB Eggermont, and Dan Gordon. Strong underrelaxation in Kaczmarz’s method for
+inconsistent systems. Numer. Math., 41(1):83–92, 1983.
+[8] Kai-Wei Chang, Cho-Jui Hsieh, and Chih-Jen Lin. Coordinate descent method for large-scale L2-loss
+linear support vector machines. J. Mach. Learn. Res., 9(7):1369—-1398, 2008.
+[9] Xuemei Chen and Jing Qin. Regularized Kaczmarz algorithms for tensor recovery. SIAM J. Imaging
+Sci., 14(4):1439–1471, 2021.
+[10] Kui Du. Tight upper bounds for the convergence of the randomized extended Kaczmarz and Gauss-
+Seidel algorithms. Numer. Linear Algebra Appl., 26(3):e2233, 2019.
+[11] Kui Du, Wu-Tao Si, and Xiao-Hui Sun. Randomized extended average block Kaczmarz for solving least
+squares. SIAM J. Sci. Comput., 42(6):A3541–A3559, 2020.
+[12] Kui Du and Xiao-Hui Sun. Pseudoinverse-free randomized block iterative algorithms for consistent and
+inconsistent linear systems. arXiv preprint arXiv:2011.10353, 2020.
+[13] Hans Georg Feichtinger, C Cenker, M Mayer, H Steier, and Thomas Strohmer. New variants of the
+POCS method using affine subspaces of finite codimension with applications to irregular sampling. In
+Visual Communications and Image Processing’92, volume 1818, pages 299–310. SPIE, 1992.
+[14] Gene H Golub and Charles F Van Loan. Matrix computations. JHU press, 2013.
+[15] Richard Gordon, Robert Bender, and Gabor T Herman. Algebraic reconstruction techniques (ART) for
+three-dimensional electron microscopy and X-ray photography. J. Theor. Biol., 29(3):471–481, 1970.
+[16] Robert M Gower, Denali Molitor, Jacob Moorman, and Deanna Needell. On adaptive sketch-and-project
+for solving linear systems. SIAM J. Matrix Anal. Appl., 42(2):954–989, 2021.
+[17] Robert M. Gower and Peter Richt´arik. Randomized iterative methods for linear systems. SIAM J.
+Matrix Anal. Appl., 36(4):1660–1690, 2015.
+[18] Deren Han, Yansheng Su, and Jiaxin Xie. Randomized Douglas-Rachford method for linear systems:
+Improved accuracy and efficiency. arXiv preprint arXiv:2207.04291, 2022.
+[19] Deren Han and Jiaxin Xie. On pseudoinverse-free randomized methods for linear systems: Unified
+framework and acceleration. arXiv preprint arXiv:2208.05437, 2022.
+[20] Martin Hanke and Wilhelm Niethammer. On the acceleration of Kaczmarz’s method for inconsistent
+linear systems. Linear Algebra Appl., 130:83–98, 1990.
+[21] Moritz Hardt, Ben Recht, and Yoram Singer. Train faster, generalize better: Stability of stochastic
+gradient descent. In Proc. 33th Int. Conf. Machine Learning, pages 1225–1234. PMLR, 2016.
+[22] Gabor T Herman and Lorraine B Meyer. Algebraic reconstruction techniques can be made computation-
+ally efficient (positron emission tomography application). IEEE Trans. Medical Imaging, 12(3):600–609,
+1993.
+[23] Rie Johnson and Tong Zhang. Accelerating stochastic gradient descent using predictive variance reduc-
+tion. In Proc. Adv. Neural Inf. Process. Syst., pages 315–323, 2013.
+[24] S Karczmarz. Angen¨aherte aufl¨osung von systemen linearer glei-chungen. Bull. Int. Acad. Pol. Sic. Let.,
+Cl. Sci. Math. Nat., pages 355–357, 1937.
+[25] Scott P Kolodziej, Mohsen Aznaveh, Matthew Bullock, Jarrett David, Timothy A Davis, Matthew
+Henderson, Yifan Hu, and Read Sandstrom. The suitesparse matrix collection website interface. J.
+Open Source Softw., 4(35):1244, 2019.
+[26] Dennis Leventhal and Adrian S Lewis. Randomized methods for linear constraints: convergence rates
+and conditioning. Math. Oper. Res., 35(3):641–654, 2010.
+[27] Ji Liu and Stephen Wright. An accelerated randomized Kaczmarz algorithm. Math. Comp., 85(297):153–
+178, 2016.
+
+14
+YUN ZENG, DEREN HAN, YANSHENG SU, AND JIAXIN XIE
+[28] Nicolas Loizou and Peter Richt´arik. Momentum and stochastic momentum for stochastic gradient,
+newton, proximal point and subspace descent methods. Comput. Optim. Appl., 77(3):653–710, 2020.
+[29] Anna Ma and Deanna Needell. Stochastic gradient descent for linear systems with missing data. Numer.
+Math. Theory Methods Appl., 12(1):1–20, 2019.
+[30] Anna Ma, Deanna Needell, and Aaditya Ramdas. Convergence properties of the randomized extended
+Gauss–Seidel and Kaczmarz methods. SIAM J. Matrix Anal. Appl., 36(4):1590–1604, 2015.
+[31] Jacob D Moorman, Thomas K Tu, Denali Molitor, and Deanna Needell. Randomized Kaczmarz with
+averaging. BIT., 61(1):337–359, 2021.
+[32] Frank Natterer. The mathematics of computerized tomography. SIAM, 2001.
+[33] Ion Necoara. Faster randomized block Kaczmarz algorithms. SIAM J. Matrix Anal. Appl., 40(4):1425–
+1452, 2019.
+[34] Deanna Needell. Randomized Kaczmarz solver for noisy linear systems. BIT., 50(2):395–403, 2010.
+[35] Deanna Needell, Nathan Srebro, and Rachel Ward. Stochastic gradient descent, weighted sampling, and
+the randomized Kaczmarz algorithm. Math. Program., 155:549–573, 2016.
+[36] Deanna Needell and Joel A Tropp. Paved with good intentions: analysis of a randomized block kaczmarz
+method. Linear Algebra Appl., 441:199–221, 2014.
+[37] Deanna Needell and Rachel Ward. Two-subspace projection method for coherent overdetermined sys-
+tems. J. Fourier Anal. Appl., 19(2):256–269, 2013.
+[38] Arkadi Nemirovski, Anatoli Juditsky, Guanghui Lan, and Alexander Shapiro. Robust stochastic ap-
+proximation approach to stochastic programming. SIAM J. Optim., 19(4):1574–1609, 2009.
+[39] Lam M Nguyen, Jie Liu, Katya Scheinberg, and Martin Tak´aˇc. Sarah: A novel method for machine
+learning problems using stochastic recursive gradient. In Proc. 34th Int. Conf. Machine Learning, pages
+2613–2621. PMLR, 2017.
+[40] Maxim A Olshanskii and Eugene E Tyrtyshnikov. Iterative methods for linear systems: theory and
+applications. SIAM, 2014.
+[41] Andrei Patrascu and Ion Necoara. Nonasymptotic convergence of stochastic proximal point methods
+for constrained convex optimization. J. Mach. Learn. Res., 18(1):7204–7245, 2017.
+[42] Constantin Popa. Extensions of block-projections methods with relaxation parameters to inconsistent
+and rank-deficient least-squares problems. BIT., 38(1):151–176, 1998.
+[43] Constantin Popa. Characterization of the solutions set of inconsistent least-squares problems by an
+extended Kaczmarz algorithm. Korean J. Comput. Appl. Math., 6(1):51–64, 1999.
+[44] Herbert Robbins and Sutton Monro. A stochastic approximation method. Ann. Math. Statistics, pages
+400–407, 1951.
+[45] Frank Sch¨opfer and Dirk A Lorenz. Linear convergence of the randomized sparse Kaczmarz method.
+Math. Program., 173(1):509–536, 2019.
+[46] Stefan Steinerberger. A weighted randomized Kaczmarz method for solving linear systems. Math.
+Comp., 90:2815–2826, 2021.
+[47] Thomas Strohmer and Roman Vershynin. A randomized Kaczmarz algorithm with exponential conver-
+gence. J. Fourier Anal. Appl., 15(2):262–278, 2009.
+[48] Nian-Ci Wu, Chengzhi Liu, Yatian Wang, and Qian Zuo. On the extended randomized multiple row
+method for solving linear least-squares problems. arXiv preprint arXiv:2210.03478, 2022.
+[49] Nian-Ci Wu and Hua Xiang. Semiconvergence analysis of the randomized row iterative method and its
+extended variants. Numer. Linear Algebra Appl., 28(1):e2334, 2021.
+[50] Wen-Ting Wu. On two-subspace randomized extended Kaczmarz method for solving large linear least-
+squares problems. Numer. Algorithms, 89(1):1–31, 2022.
+[51] Anastasios Zouzias and Nikolaos M. Freris. Randomized extended Kaczmarz for solving least squares.
+SIAM J. Matrix Anal. Appl., 34(2):773–793, 2013.
+
+RKAS FOR INCONSISTENT SYSTEMS
+15
+School of Mathematical Sciences, Beihang University, Beijing, 100191, China.
+Email address: zengyun@buaa.edu.cn
+LMIB of the Ministry of Education, School of Mathematical Sciences, Beihang University,
+Beijing, 100191, China.
+Email address: handr@buaa.edu.cn
+School of Mathematical Sciences, Beihang University, Beijing, 100191, China.
+Email address: suyansheng@buaa.edu.cn
+LMIB of the Ministry of Education, School of Mathematical Sciences, Beihang University,
+Beijing, 100191, China.
+Email address: xiejx@buaa.edu.cn
+

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+LOW COMPLEXITY ADAPTIVE MACHINE LEARNING
+APPROACHES FOR END-TO-END LATENCY PREDICTION ∗
+Pierre Larrenie
+Thales SIX & LIGM
+Université Gustave Eiffel, CNRS
+Marne-la-Vallée, France
+pierre.larrenie@esiee.fr
+Jean-François Bercher
+LIGM
+Université Gustave Eiffel, CNRS
+Marne-la-Vallée, France
+jean-francois.bercher@esiee.fr
+Olivier Venard
+ESYCOM
+Université Gustave Eiffel, CNRS
+Marne-la-Vallée, France
+olivier.venard@esiee.fr
+Iyad Lahsen-Cherif
+Institut National des Postes et Télécommunications (INPT)
+Rabat, Morocco
+lahsencherif@inpt.ac.ma
+ABSTRACT
+Software Defined Networks have opened the door to statistical and AI-based techniques
+to improve efficiency of networking. Especially to ensure a certain Quality of Service
+(QoS) for specific applications by routing packets with awareness on content nature
+(VoIP, video, files, etc.) and its needs (latency, bandwidth, etc.) to use efficiently
+resources of a network.
+Monitoring and predicting various Key Performance Indicators (KPIs) at any level may
+handle such problems while preserving network bandwidth.
+The question addressed in this work is the design of efficient, low-cost adaptive algo-
+rithms for KPI estimation, monitoring and prediction. We focus on end-to-end latency
+prediction, for which we illustrate our approaches and results on data obtained from a
+public generator provided after the recent international challenge on GNN [12].
+In this paper, we improve our previously proposed low-cost estimators [6] by adding
+the adaptive dimension, and show that the performances are minimally modified while
+gaining the ability to track varying networks.
+Keywords KPI Prediction · Machine Learning · Adaptivity · General Regression ·
+SDN · Networking
+1
+Introduction
+Routing while ensuring quality of service (QoS) remains a significant challenge in all networks. Whatever
+the resources, their use must be optimized to satisfy both throughput and QoS to users. This is true for
+static wide area networks, but even more so for mobile networks with dynamic topology.
+The emergence of software-defined networks (SDNs) [11, 1] has enabled data to be shared more ef-
+ficiently across communication layers. Services can provide network requirements to routers; routers
+acquire data about network performance and allocate resources to meet those requirements as best as pos-
+sible. However, acquiring overall network performance can result in high network bandwidth consump-
+tion for signaling, degrading the available resources, and is particularly limiting for resource-constrained
+networks such as mobile networks (MANETs).
+We consider a network for which we wish to reduce signaling and perform intelligent routing. In order to
+limit the amount of signaling, the first axis is to estimate some key performance indicators (KPIs) from
+other KPIs. A second axis would be to perform this prediction locally, at the node level, rather than a
+global estimation in the network. Finally, if predictions are to be performed locally, the complexity of
+the algorithms will need to be low while preserving good prediction quality. The last point is to be able
+to detect and track changes in the state of the network, which implies that the predictors will have to use
+only a small number of the previous states of the network and be able to readapt continuously.
+∗Note: Paper accepted at the 5th International Conference on Machine Learning for Networking (MLN’2022)
+and will be published as a post-proceedings in Springer’s LNCS.
+arXiv:2301.13536v1  [cs.NI]  31 Jan 2023
+
+The question addressed in this work is the design of efficient, low-cost adaptive algorithms for KPI
+estimation, monitoring and prediction. In the present paper, we improve our previously proposed low-
+cost estimators [6] by adding the adaptive dimension and show that the performances are minimally
+modified while gaining the ability to track varying networks. We focus on end-to-end latency prediction,
+for which we illustrate our approaches and results on data we generated using a public generator made
+available after the recent international challenge [12]. The best performances of the state-of-the-art are
+obtained with Graph Neural Networks (GNNs) [10, 3, 12]. Although this is a global method while we
+favor local and adaptive methods, we used these performances as a benchmark.
+We present related works in section 2. Then we present in section 3 our main results from [6]. Instead
+of using high performances global and high-costs methods based on Graph Neural Networks (GNNs)
+[10, 3, 12], we proposed to use standard machine learning regression methods. We showed that a careful
+feature engineering and feature selection (based on queue theory and the approach in [2]), as well as
+the use of a single feature with curve-fitting methods, allows to obtain near state-of-the-art performances
+with both a very low number of parameters, significantly lower learning and inference times compared to
+GNNs, and the with the ability to operate at the link level instead of a whole-graph level. In section 4,
+we show how these block algorithms can be transformed into versions implementable in an iterative way
+(i.e. by taking into account the data one by one as they become available), with the originality of using a
+regularization term. Then, time dependent estimations, or the addition of forgetting factor will give them
+an adaptive character. In section 5 we describe the validation dataset we built from a public generator and
+then the results of our experimentation. Finally, we conclude, discuss the overall results and draw some
+perspectives.
+2
+Related work
+[4] present an heuristic and an Mixed Integer Programming approach to optimize Service Functions Chain
+provisioning when using Network Functions Virtualization for a service provider. Their approach relies
+on minimizing a trade-off between the expected latency and infrastructures resources.
+Such optimization routing flow in SDN may need additional information to be exchanged between the
+nodes of a network. This results in an increase of the volume of signalization, by performing some
+measurements such as in [7]. This is not a consequent problem in unconstrained networks, i.e. static
+wired networks with near-infinite bandwidth but may decrease performance of wireless network with
+poor capacity. An interesting solution to save bandwidth would be to predict some of the KPIs from other
+KPIs and data exchanged globally between nodes.
+In [8, 9], authors proposed a MANETs application of SDN in the domain of tactical networks. They
+proposed a multi-level SDN controllers architecture to build both secure and resilient networking. While
+orchestrating communication efficiently under military constraints such as: high-level of dynamism, fre-
+quent network failures, resources-limited devices. The proposed architecture is a trade-off between tra-
+ditional centralized architecture of SDN and a decentralized architecture to meet dynamic in-network
+constraints.
+[5] proposed a Quality of Experience (QoE) management strategy in a SDN to optimize the loading time
+of all the tile of a mapping application. They have shown the impact of several KPIs on their application
+using a Generalized Linear Model (GLM). This mechanism make the application aware of the current
+network state.
+[10] used GNNs for predicting KPIs such as latency, error-rate and jitter. They relied on the Routenet
+architecture of Figure 1. The idea is to model the problem as a bipartite hypergraph mapping flows to
+links as depicted on Figure 2. Aggregating messages in such graph may result in predicting KPIs of the
+network in input. The model needs to know the routing scheme, traffic and links properties. Their result
+is very promising and has been the subject of two ITU Challenge in 2020 and 2021 [3, 12]. These ITU
+challenges have very good results since the top-3 teams are around 2% error in delay prediction in the
+sense of Mean-Absolute Percentage Error (MAPE).
+In [2], very promising results were obtained with a a near 1% GNN model error (in the sense of MAPE)
+on the test set. The model mix analytical M/M/1/K queueing theory used to create extra-features to
+feed GNN model. In order to satisfy the constraint of scalability proposed by the challenge, the first part
+of model operates at the link level.
+Figure 1: Routenet Architecture [10]
+2
+
+Topology
+Per-flowperformance
+Traffic matrix
+RouteNet
+metrics
+(e.g., delay, jitter, loss)
+Routing schemeA
+B
+C
+D
+E
+F3
+F2
+F1
+(a) Simple topology
+F1
+F2
+F3
+LAD
+LDE
+LAB
+LBE
+LCB
+(b) Paths-links Hypergraph of (a)
+Figure 2: Routenet [10] paths-links hypergraph transformation applied on a simple topology graph
+carrying 3 flows.
+(a) Black circles represents communication node, double headed arrows between them denotes available symmetric
+communications links and dotted arrows shows flows path. (b) Circle (resp. dotted) represents links (resp. flows)
+entities defined in the first graph (Lij is the symmetric link between node i and node j.). Unidirectional arrows
+encode the relation "<flow> is carried by <link>".
+3
+Simple machine-learning approaches for latency prediction
+Our first problem is to define an estimator ˆy of the occupancy y as a function of the different available
+“features” of the system, with a joint objective of low complexity and performance. To do so, we will
+look for an approximation function fθ(u) that allows to estimate y from the features u and parameters θ.
+ˆy = fθ(u)
+(1)
+Here u and θ are vectors that collect the different features or parameters. Once an estimate of occupancy
+is obtained, it is possible to get the latency prediction ˆdn for a specific link n by the simple relation
+ˆdn = ˆyn
+E(|Pn|)
+cn
+(2)
+where E [|Pn|] is the observed average packet size on link n and cn the capacity of this link.
+For analytical simplicity, the parameters θ will be sought by minimizing the minimum mean square error
+E
+�
+(y − ˆy)2�
+= E
+�
+(y − fθ(u))2�
+,
+(3)
+although the performances are also often evaluated in the MAPE sense
+L (ˆy, y) = 100%
+N
+N
+�
+n=1
+����
+ˆyn − yn
+yn
+����
+(4)
+which is preferred to Mean Squared Error (MSE) because of its scale-invariant property.
+We will focus here on two very simple models, although other machine learning models have also been
+considered in [6]. Indeed, these two models lend themselves very easily to an adaptive formulation. In
+this section, we will first describe these two approaches and their performances, before giving the general
+adaptive formulation, which we will particularize in both cases.
+3.1
+Feature Engineering and Linear Regression
+Based on the assumption that the system may be approximated by a model whose essential features come
+from M/M/1/K and M/G/1/K queue theory, we took essential parameters characterizing queueing
+systems, such as: ρ, ρe, π0, πK, etc. and built further features by applying interactions and various
+non-linearities (powers, log, exponential, square root). Then, we selected features in this set by a forward
+step-wise selection method; i.e. by adding in turn each feature to potential models and keeping the feature
+with best performance. Finally, we selected the model with best MAPE error. For a linear regression
+model, this led us to select and keep a set of 4 simple features, which interestingly enough, have simple
+interpretations:
+�
+�
+�
+�
+�
+�
+�
+�
+�
+π0 =
+1−ρ
+1−ρK+1
+L = ρ + π0
+�
+k kρk
+ρe = λe
+λ ρ = λe
+µ
+Se = �
+k kρk
+e
+(5)
+where L is the expected number of packets in the queue according to M/M/1/K, π0 the probability
+that the queue is empty according to M/M/1/K theory, ρe the effective queue utilization, and Se the
+3
+
+unnormalized expected value of the effective number of packet in the queue buffer. These features can
+be thought as a kind of data preprocessing, before applying ML algorithms, and this turns out to be a key
+to achieving good performances. The 4 previous features have been used as input for several machine
+learning models like Multi-Layer Perceptron model (MLP), Linear Regression, SVM, Random Forest,
+Gradient Boosting Regression Tree. We only describe here the case of linear regression, since it is a
+method for which an adaptive version is readily obtained. In this case, model (1) is simply
+ˆy = θ0 + θ1π0 + θ2L + θ3ρe + θ4Se = θT u
+(6)
+with θT = [θ0, . . . θ4] and uT = [1, π0, L, ρe, Se]. For the linear regression model in ((6), it is well
+known that the regularized minimum mean squared error
+J(θ) = E
+�
+(y − θT u)2�
++ α||θ||2
+(7)
+is obtained for
+θ : (Ruu + α1) θ = Ryu
+(8)
+where we denoted
+�Ruu = E
+�
+uuT �
+,
+the correlation matrix of u
+Ryu = E [yu] ,
+the correlation vector of y and u
+and 1 the identity matrix, α the regularization parameter.
+As far as performance is concerned with this approach, it was evaluated using static data from the GNN
+ITU Challenge 2021 [12]. Compared to the state-of-the-art, our linear regression with carefully selected
+features shows a very slight performance degradation: 1.74% in MAPE while the best state-of-the-art
+method is at 1.27%. One strong advantage is in term of training and inference time. It has a training
+time of less than a second when GNN requires more than 8 hours. Moreover, the inference time for the
+complete network is also much lower, by a factor of almost 1000 (0.296s vs 214s).
+3.2
+Curve Regression by Bernstein polynomials
+There is a high interdependence of the features we selected in Equation (5), since all these features can be
+expressed in term of ρe. Furthermore, it is confirmed by data exploration that ρe is the prominent feature
+for occupancy prediction (and in turn latency prediction), as exemplified in Figure 3.
+It is then tempting to try to further simplify our features space and estimate the occupancy from a non-
+linear transformation of the single feature ρe, as:
+ˆy = g(ρe)
+(9)
+where ˆy is the estimate of the occupancy y. The concerns are of course to define simple and efficient
+functions g, with a low number of parameters, that can model the kind of growth shown in Figure 3, and
+of course to check that the performance remains interesting.
+Figure 3: Data of ITU Challenge 2021 [12], ρe vs queue occupancy. Color-scale is an indicator of points
+cloud density.
+The estimator g is defined as a linear combination of simple functions fn:
+ˆy = g(ρe) =
+�
+n
+θn · fn(ρe)
+(10)
+which is also a linear model in terms of function fn(ρe).
+4
+
+1.0
+2988
+2000
+1500
+1000
+500
+0.8
+0.6 -
+0.4 -
+0.2 -
+0.0-
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0Several solutions were considered in [6] to define or choose the functions fn. Since we know that the
+Bernstein polynomials form a basis in the set of polynomial in the interval [0; 1]; and that the approxima-
+tion of any continuous function on [0; 1[ by a Bernstein polynomial converges uniformly, we were led to
+these polynomials:
+f K
+n (x) =
+�K
+n
+�
+xn(1 − x)K−n
+(11)
+where K is maximum order of polynomials.
+As mentioned, (10) can be rewritten as the linear model
+��y = g(ρe) =
+�
+n
+θn · fn(ρe) = θT u
+(12)
+with θT = [θ0, . . . , θK] and uT = [f K
+0 (ρe), f K
+1 (ρe), . . . , f K
+K (ρe)]. Hence, we have the same form as in
+(8) for the solution.
+In term of performances, we also obtained a minor degradation in MAPE (1.68%) compared to state-of-
+the-art (1.29%), while improving by several orders the wall training and inference times (2min/3.14s vs
+8hrs/214s); though a bit less than the simple linear regression.
+4
+Adaptive versions
+We place ourselves in the context where we have regular snapshots of the state of the network, which
+allows us to both monitor the quality of predictions, and to track changes in the network. For the n-th
+series of measurements, let us denote y(n) the measured latency and u(n) the features. We can also
+group several snapshots or several links into a vector of latencies y(n) and matrix U(n). In the following
+we will derive equations for this block case, which includes immediately the scalar case.
+The minimum mean square error (7) which has the explicit solution (8) can also be solved by a gradient
+algorithm as
+θk+1 = θk − µ ∇J(θ)|θ=θk ,
+(13)
+= θk − µ ((Ruu + α1) θk − Ryu) .
+(14)
+In (14), we can substitute the true values with estimated ones. In order to introduce adaptivity to context
+changes in the network, these estimates will preserve the temporal dimension. We thus use either a sliding
+average
+�
+ˆRuu(n) = �L
+l=0 U(n − l)U(n − l)T
+ˆRyu(n) = �L
+l=0 U(n − l)y(n − l)
+(15)
+or an exponential mean
+�
+ˆRuu(n) = �n
+l=0 λl−nU(l)U(l)T = λ ˆRuu(n − 1) + U(n)U(n)T
+ˆRyu(n) = λ ˆRyu(n − 1) + U(n)y(n).
+(16)
+where λ ≤ 1 is the forgetting factor.
+In the limit case where we take either L = 0 or λ = 0 in the previous formulas, we get the ‘instantaneous
+estimates‘
+�
+ˆRuu(n) = U(n)U(n)T
+ˆRdu(n) = U(n)y(n).
+(17)
+we obtain
+θ(n + 1) = (1 − µα)θ(n) − µU(n)
+�
+U(n)T θ(n) − y(n)
+�
+(18)
+which reduces to the well known LMS algorithm [14] in the scalar case and no regularization, α = 0.
+Alternatively, one can try to solve the normal equation (8), using the time dependent estimates as the
+exponential mean (16). The difficulty with the solution
+ˆθ(n + 1) =
+�
+ˆRuu(n + 1) + α1
+�−1 ˆRyu(n + 1)
+(19)
+is the inversion, for each n, of the correlation matrix. Let us denote
+K(n + 1) =
+�
+ˆRuu(n + 1) + α1
+�−1
+.
+(20)
+Using (16), we have
+K(n + 1)−1 = λ ˆRuu(n) + U(n + 1)U(n + 1)T + α1
+(21)
+= λ
+�
+ˆRuu(n) + α1
+�
++ U(n + 1)U(n + 1)T + α(1 − λ)1
+(22)
+= λK(n)−1 + U(n + 1)U(n + 1)T + α(1 − λ)1
+(23)
+5
+
+and
+K(n + 1) =
+��
+λK(n)−1 + U(n + 1)U(n + 1)T �
++ α(1 − λ)1
+�−1
+(24)
+= [Q(n + 1) + δ1]−1
+(25)
+with
+Q(n + 1) =
+�
+λK(n)−1 + U(n + 1)U(n + 1)T �
+(26)
+and δ = α(1 − λ) The matrix inversion lemma enables to reduce the inversion of Q(n + 1) to
+Q(n + 1)−1 = 1
+λK(n) − 1
+λ2 K(n)U(n + 1)×
+�
+1 + 1
+λU(n + 1)T K(n)U(n + 1)
+�−1
+U(n + 1)T K(n),
+(27)
+which simplifies to
+Q(n + 1)−1 = 1
+λK(n) − 1
+λ2
+K(n)u(n + 1)u(n + 1)T K(n)
+1 + 1
+λu(k + 1)T K(n)u(k + 1),
+(28)
+for scalar observations. Now, we can use the Taylor expansion to get
+K(n + 1) = [Q(n + 1) + δ1]−1 = Q(n + 1)−1 − δQ(n + 1)−2 + δ2Q(n + 1)−3 + . . .
+(29)
+This gives us a way to compute recursively the inverse of the regularized estimate of the correlation matrix
+by combining (27) and (29) into
+K(n + 1) ≈ Q(n + 1)−1 − δQ(n + 1)−2
+(30)
+which, by (27), does not require the inversion of K(n).
+In both cases, we have the updating formula
+θ(n + 1) = θ(n) + K(n + 1)U(n + 1)[y(n + 1) − θ(n)T U(n + 1)].
+(31)
+5
+Experiments and results
+5.1
+Dataset
+We generate a dataset thanks to a public challenge data generator [12]. This data generator is based on the
+well-known OMNET++ discrete event network simulator[13]. The published simulator is available as a
+docker image. However, due to the rules of the 2022 edition of the challenge, it is not possible to generate
+large topologies, i.e. no more than 10 nodes. Since our models are link-based, the use of small topologies
+does not seem to be a problem. The simulator is parameterized by a traffic matrix and a topological graph
+that are easy to generate thanks to the provided API.
+Our generated dataset, used is this paper, is the result of 11900 simulations of the same topology graph
+of 10 nodes and 30 links, subject to 100 different traffic matrices. In order to get complex results of
+simulations but at low cost, we made the choice to model a network with small queue buffers (8000 bits)
+and possibly subject of high traffic intensities (maximum traffic rate set to 4000 bits/s for each flow). Then
+for each traffic matrices, we alter the capacity of the network according to a sigmoid, in order to model
+a network subject to jamming, with 2 stationary states. The proposed jamming may cause a decrease
+in the capacity of the network links by up to a factor of 5, as depicted on Figure 4a. For simplification
+purposes, we have considered that each link of the network has the same capacity. This result in a U-
+shaped distribution of our link data samples according to the link capacity as shown in Figure 4b.
+5.2
+Results
+From the generated data, we validate our approach along several axes.
+5.2.1
+Global performances
+First, we establish the benchmark performances based on the global methods presented in the paper [6]
+and Sections 3.1 and 3.2.
+For the linear regression method described in Section 3.1, we obtain an MSE of 5.86e-4 and a MAPE
+of 9.58% for the queue occupancy prediction and an MSE of 1.10e-3 and a MAPE of 10.26%. for the
+end-to-end latency prediction.
+Concerning the curve regression using Bernstein polynomials (of degree 8) described in in Section 3.2,
+we obtain an MSE of 4.52e-4 and a MAPE of 8.72% for the queue occupancy prediction and an MSE of
+9.35e-4 and a MAPE of 9.95% for the end-to-end latency prediction.
+Note that these benchmark performances are below the performances obtained in [6], but the dataset we
+use here is probably more severe since using ground truth value for occupancy results in an MSE 6.03e-4
+of a MAPE 9.34% for the flow delay prediction. That is indeed very close of the obtained results.
+6
+
+0
+2000
+4000
+6000
+8000
+10000
+12000
+Sample ID
+10000
+15000
+20000
+25000
+30000
+35000
+40000
+45000
+50000
+Capacity of the links (bits/s)
+(a) Capacity alteration to model jamming with a decrease of the capacity up to a factor of 5.
+10000
+#40
+15000
+#35
+20000
+#30
+25000
+#25
+30000
+#20
+35000
+#15
+40000
+#10
+45000
+#5
+50000
+#0
+capacity of links (bits/s)
+#scenario ID
+0
+5000
+10000
+15000
+20000
+25000
+30000
+#links data
+(b) Link capacity distribution of the generated dataset.
+Figure 4: Overview of the generated dataset
+7
+
+5.2.2
+Behavior of iterative algorithms
+In a second step, we verify that the algorithms presented in section 4 converge and allow us to recover
+these performances. With a forgetting factor of 1 (use of all data with the same weight) and a block size
+of 10, we observe, for example in Figure 5, that the model coefficients converge towards a stable value,
+and that MAPE recovers the value obtained with the global method using all data. The convergence is
+obtained in less than 10,000 operations. It is thus possible to replace the global method, which is already
+low-cost, with an approach where the calculations are carried out recursively.
+5.2.3
+Adaptivity
+In a third step, we compare the algorithms to the case of network modifications We consider an abrupt
+change in the network capacity, which could correspond to a jamming scenario. We then examine how
+the two adaptive algorithms presented (linear regression with judiciously chosen features; and Bernstein
+polynomial model) can detect and adapt to these modifications. In this context, we examine the role of
+the forgetting factor and the regularization parameter. Figure 6 and Figure 7 present the results for the
+case of a capacity change. We observe that (i) the square of the residual error, smoothed over 100 points,
+is a remarkable indicator of a change in the network; and (ii) that the model coefficients readjust over the
+iterations after this change.
+5.2.4
+Discussion
+These experiments show the effectiveness and relevance of our iterative and adaptive versions of end-
+to-end latency estimation procedures. The iterative versions have the same performance as their global
+counterparts; an even lower cost since they can be implemented iteratively as the data is received or made
+available. The convergence time for the model coefficients is a few thousand samples while the global
+model used around 350,000 samples for training, while the GNN models require several million samples.
+Moreover, we observe that the residual error converges very quickly, in some tens of samples, which
+means that although the convergence of the models’ coefficients does not seem to be complete, they are
+equivalent from the point of view of performance for occupancy prediction. From an operational point of
+view, the model can be refreshed regularly, and the predicted KPIs between these updates can be used for
+intelligent routing. As we have observed, residual error monitoring is an excellent indicator of changes
+in the network state.
+6
+Conclusion
+In this paper, we considered the problem of designing efficient and low-cost algorithms for KPI predic-
+tion that are locally implementable and adaptive to network changes. Based on a previous work, we
+have argued and developed adaptive solutions, introducing in addition a regularization term in order to
+stabilize the results. We used a public domain simulator to simulate networks and generate relevant data.
+The experiments demonstrate the effectiveness and relevance of these algorithms. Thus, we now have
+low-complexity models that can be implemented iteratively at the level of local links. We have the possi-
+bility to predict the occupancy of the different links, and the end-to-end latencies (the models predict the
+occupancy of the queues, then compute analytically the delay for each link and finally aggregate along the
+path). Moreover, the adaptability of the solution allows to follow changes in the network state, always at
+a minimal cost, by re-adapting from the current solution and new data. The continuation of the work will
+focus on the choice criteria of the forgetting factor, on the impact of the regularization factor, in order to
+find automatic selection methods. Of course, the approaches considered here will have to be considered
+and adapted for other types of KPI, such as error rate or jitter.
+8
+
+0
+5000
+10000
+15000
+20000
+25000
+30000
+samples
+−1.0
+−0.5
+0.0
+0.5
+1.0
+Weight value
+θ_0
+θ_1
+θ_2
+θ_3
+θ_4
+θ_5
+θ_6
+θ_7
+θ_8
+(a) Iterative curve-fitting based on Bernstein polynomials of degree 8.
+0
+5000
+10000
+15000
+20000
+25000
+30000
+samples
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Weight value
+θ_0
+θ_1
+θ_2
+θ_3
+θ_4
+(b) Iterative Linear Regression
+Figure 5: Evolution of weights for our iterative methods without forgetting (non-adaptive) while fitting
+the whole dataset.
+9
+
+0
+10000
+20000
+30000
+40000
+samples
+−2
+−1
+0
+1
+2
+Weight value
+#39
+#10
+#1
+#10
+#39
+θ_0
+θ_1
+θ_2
+θ_3
+θ_4
+θ_5
+θ_6
+θ_7
+θ_8
+(a) Evolution of weights along the scenario.
+0
+10000
+20000
+30000
+40000
+samples
+0.015
+0.020
+0.025
+0.030
+0.035
+RMSE
+#39
+#10
+#1
+#10
+#39
+(b) Evolution of the RMSE along the scenario.
+Figure 6: Evolution of weights and
+√
+MSE (RMSE) for our adaptive approach of Bernstein polynomial
+curve regression of degree 8,λ = 0.9, α = 0.08. Scenario describes a nominal period between 2 periods
+of jamming. #39 corresponds to a link capacity of 11 Kbits/s, #10 40 Kbits/s and #1 49 Kbts/s. Figure is
+smoothed over 100 points.
+10
+
+0
+10000
+20000
+30000
+40000
+samples
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
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+#39
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+(a) Evolution of weights along the scenario.
+0
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+(b) Evolution of the RMSE along the scenario.
+Figure 7: Evolution of weights and
+√
+MSE (RMSE) for our adaptive approach of Linear
+Regression,λ = 0.9, α = 0.08. Scenario describes a nominal period between 2 periods of jamming. #39
+corresponds to a link capacity of 11 Kbits/s, #10 40 Kbits/s and #1 49 Kbts/s. Figure is smoothed over
+100 points.
+11
+
+References
+[1] Amin, R., Reisslein, M., Shah, N.: Hybrid SDN networks: A survey of existing approaches. IEEE
+Communications Surveys & Tutorials 20(4), 3259–3306 (2018)
+[2] de Aquino Afonso, B.K.: GNNet challenge 2021 report (1st place). https://github.com/
+ITU-AI-ML-in-5G-Challenge/ITU-ML5G-PS-001-PARANA (2021)
+[3] Barcelona Neural Networking Center: The graph neural networking challenge 2020. https://
+bnn.upc.edu/challenge/gnnet2020
+[4] Chua, F.C., Ward, J., Zhang, Y., Sharma, P., Huberman, B.A.: Stringer: Balancing latency and re-
+source usage in service function chain provisioning. IEEE Internet Computing 20(6), 22–31 (2016)
+[5] Jahromi, H.Z., Hines, A., Delanev, D.T.: Towards application-aware networking: Ml-based end-to-
+end application KPI/QoE metrics characterization in SDN. In: 2018 Tenth International Conference
+on Ubiquitous and Future Networks (ICUFN). pp. 126–131. IEEE (2018)
+[6] Larrenie, P., Bercher, J.F., Lahsen-Cherif, I., Venard, O.: Low complexity approaches for end-to-
+end latency prediction. In: Proceedings of the 13th IEEE International Conference On Computing,
+Communication and Networking Technologies. IEEE (2022)
+[7] Pasca, S.T.V., Kodali, S.S.P., Kataoka, K.: AMPS: Application aware multipath flow routing using
+machine learning in SDN. In: 2017 Twenty-third National Conference on Communications (NCC).
+pp. 1–6. IEEE (2017)
+[8] Poularakis, K., Iosifidis, G., Tassiulas, L.: SDN-enabled tactical ad hoc networks: Extending pro-
+grammable control to the edge. IEEE Communications Magazine 56(7), 132–138 (2018)
+[9] Poularakis, K., Qin, Q., Nahum, E.M., Rio, M., Tassiulas, L.: Flexible SDN control in tactical ad
+hoc networks. Ad Hoc Networks 85, 71–80 (2019)
+[10] Rusek, K., Suárez-Varela, J., Mestres, A., Barlet-Ros, P., Cabellos-Aparicio, A.: Unveiling the
+potential of graph neural networks for network modeling and optimization in SDN. In: Proceedings
+of the 2019 ACM Symposium on SDN Research. pp. 140–151 (2019)
+[11] Singh, S., Jha, R.K.: A survey on Software Defined Networking: Architecture for next generation
+network. Journal of Network and Systems Management 25(2), 321–374 (2017)
+[12] Suárez-Varela, J., et al.: The graph neural networking challenge: a worldwide competition for
+education in AI/ML for networks. ACM SIGCOMM Computer Communication Review 51(3), 9–16
+(2021)
+[13] Varga, A., Hornig, R.: An overview of the omnet++ simulation environment. In: 1st International
+ICST Conference on Simulation Tools and Techniques for Communications, Networks and Systems
+(2010)
+[14] Widrow, B., Stearns, S.: Adaptive Signal Processing. Edited by Alan V. Oppenheim, Prentice-Hall
+(1985)
+12
+

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf,len=415
+page_content='LOW COMPLEXITY ADAPTIVE MACHINE LEARNING  APPROACHES FOR END-TO-END LATENCY PREDICTION ∗  Pierre Larrenie  Thales SIX & LIGM  Université Gustave Eiffel, CNRS  Marne-la-Vallée, France  pierre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='larrenie@esiee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='fr  Jean-François Bercher  LIGM  Université Gustave Eiffel, CNRS  Marne-la-Vallée, France  jean-francois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='bercher@esiee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='fr  Olivier Venard  ESYCOM  Université Gustave Eiffel, CNRS  Marne-la-Vallée, France  olivier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='venard@esiee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='fr  Iyad Lahsen-Cherif  Institut National des Postes et Télécommunications (INPT)  Rabat, Morocco  lahsencherif@inpt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='ma  ABSTRACT  Software Defined Networks have opened the door to statistical and AI-based techniques  to improve efficiency of networking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Especially to ensure a certain Quality of Service  (QoS) for specific applications by routing packets with awareness on content nature  (VoIP, video, files, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=') and its needs (latency, bandwidth, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=') to use efficiently  resources of a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  Monitoring and predicting various Key Performance Indicators (KPIs) at any level may  handle such problems while preserving network bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  The question addressed in this work is the design of efficient, low-cost adaptive algo-  rithms for KPI estimation, monitoring and prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' We focus on end-to-end latency  prediction, for which we illustrate our approaches and results on data obtained from a  public generator provided after the recent international challenge on GNN [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  In this paper, we improve our previously proposed low-cost estimators [6] by adding  the adaptive dimension, and show that the performances are minimally modified while  gaining the ability to track varying networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  Keywords KPI Prediction · Machine Learning · Adaptivity · General Regression ·  SDN · Networking  1  Introduction  Routing while ensuring quality of service (QoS) remains a significant challenge in all networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Whatever  the resources, their use must be optimized to satisfy both throughput and QoS to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' This is true for  static wide area networks, but even more so for mobile networks with dynamic topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  The emergence of software-defined networks (SDNs) [11, 1] has enabled data to be shared more ef-  ficiently across communication layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Services can provide network requirements to routers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' routers  acquire data about network performance and allocate resources to meet those requirements as best as pos-  sible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' However, acquiring overall network performance can result in high network bandwidth consump-  tion for signaling, degrading the available resources, and is particularly limiting for resource-constrained  networks such as mobile networks (MANETs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  We consider a network for which we wish to reduce signaling and perform intelligent routing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' In order to  limit the amount of signaling, the first axis is to estimate some key performance indicators (KPIs) from  other KPIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' A second axis would be to perform this prediction locally, at the node level, rather than a  global estimation in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Finally, if predictions are to be performed locally, the complexity of  the algorithms will need to be low while preserving good prediction quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' The last point is to be able  to detect and track changes in the state of the network, which implies that the predictors will have to use  only a small number of the previous states of the network and be able to readapt continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  ∗Note: Paper accepted at the 5th International Conference on Machine Learning for Networking (MLN’2022)  and will be published as a post-proceedings in Springer’s LNCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='13536v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='NI]  31 Jan 2023  The question addressed in this work is the design of efficient, low-cost adaptive algorithms for KPI  estimation, monitoring and prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' In the present paper, we improve our previously proposed low-  cost estimators [6] by adding the adaptive dimension and show that the performances are minimally  modified while gaining the ability to track varying networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' We focus on end-to-end latency prediction,  for which we illustrate our approaches and results on data we generated using a public generator made  available after the recent international challenge [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' The best performances of the state-of-the-art are  obtained with Graph Neural Networks (GNNs) [10, 3, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Although this is a global method while we  favor local and adaptive methods, we used these performances as a benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  We present related works in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Then we present in section 3 our main results from [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Instead  of using high performances global and high-costs methods based on Graph Neural Networks (GNNs)  [10, 3, 12], we proposed to use standard machine learning regression methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' We showed that a careful  feature engineering and feature selection (based on queue theory and the approach in [2]), as well as  the use of a single feature with curve-fitting methods, allows to obtain near state-of-the-art performances  with both a very low number of parameters, significantly lower learning and inference times compared to  GNNs, and the with the ability to operate at the link level instead of a whole-graph level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' In section 4,  we show how these block algorithms can be transformed into versions implementable in an iterative way  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' by taking into account the data one by one as they become available), with the originality of using a  regularization term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Then, time dependent estimations, or the addition of forgetting factor will give them  an adaptive character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' In section 5 we describe the validation dataset we built from a public generator and  then the results of our experimentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Finally, we conclude, discuss the overall results and draw some  perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  2  Related work  [4] present an heuristic and an Mixed Integer Programming approach to optimize Service Functions Chain  provisioning when using Network Functions Virtualization for a service provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Their approach relies  on minimizing a trade-off between the expected latency and infrastructures resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  Such optimization routing flow in SDN may need additional information to be exchanged between the  nodes of a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' This results in an increase of the volume of signalization, by performing some  measurements such as in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' This is not a consequent problem in unconstrained networks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' static  wired networks with near-infinite bandwidth but may decrease performance of wireless network with  poor capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' An interesting solution to save bandwidth would be to predict some of the KPIs from other  KPIs and data exchanged globally between nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  In [8, 9], authors proposed a MANETs application of SDN in the domain of tactical networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' They  proposed a multi-level SDN controllers architecture to build both secure and resilient networking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' While  orchestrating communication efficiently under military constraints such as: high-level of dynamism, fre-  quent network failures, resources-limited devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' The proposed architecture is a trade-off between tra-  ditional centralized architecture of SDN and a decentralized architecture to meet dynamic in-network  constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  [5] proposed a Quality of Experience (QoE) management strategy in a SDN to optimize the loading time  of all the tile of a mapping application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' They have shown the impact of several KPIs on their application  using a Generalized Linear Model (GLM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' This mechanism make the application aware of the current  network state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  [10] used GNNs for predicting KPIs such as latency, error-rate and jitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' They relied on the Routenet  architecture of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' The idea is to model the problem as a bipartite hypergraph mapping flows to  links as depicted on Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Aggregating messages in such graph may result in predicting KPIs of the  network in input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' The model needs to know the routing scheme, traffic and links properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Their result  is very promising and has been the subject of two ITU Challenge in 2020 and 2021 [3, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' These ITU  challenges have very good results since the top-3 teams are around 2% error in delay prediction in the  sense of Mean-Absolute Percentage Error (MAPE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  In [2], very promising results were obtained with a a near 1% GNN model error (in the sense of MAPE)  on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' The model mix analytical M/M/1/K queueing theory used to create extra-features to  feed GNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' In order to satisfy the constraint of scalability proposed by the challenge, the first part  of model operates at the link level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  Figure 1: Routenet Architecture [10]  2  Topology  Per-flowperformance  Traffic matrix  RouteNet  metrics  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=', delay, jitter, loss)  Routing schemeA  B  C  D  E  F3  F2  F1  (a) Simple topology  F1  F2  F3  LAD  LDE  LAB  LBE  LCB  (b) Paths-links Hypergraph of (a)  Figure 2: Routenet [10] paths-links hypergraph transformation applied on a simple topology graph  carrying 3 flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  (a) Black circles represents communication node, double headed arrows between them denotes available symmetric  communications links and dotted arrows shows flows path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' (b) Circle (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' dotted) represents links (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' flows)  entities defined in the first graph (Lij is the symmetric link between node i and node j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Unidirectional arrows  encode the relation "<flow> is carried by <link>".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  3  Simple machine-learning approaches for latency prediction  Our first problem is to define an estimator ˆy of the occupancy y as a function of the different available  “features” of the system, with a joint objective of low complexity and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' To do so, we will  look for an approximation function fθ(u) that allows to estimate y from the features u and parameters θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  ˆy = fθ(u)  (1)  Here u and θ are vectors that collect the different features or parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Once an estimate of occupancy  is obtained, it is possible to get the latency prediction ˆdn for a specific link n by the simple relation  ˆdn = ˆyn  E(|Pn|)  cn  (2)  where E [|Pn|] is the observed average packet size on link n and cn the capacity of this link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  For analytical simplicity, the parameters θ will be sought by minimizing the minimum mean square error  E  �  (y − ˆy)2�  = E  �  (y − fθ(u))2�  ,  (3)  although the performances are also often evaluated in the MAPE sense  L (ˆy, y) = 100%  N  N  �  n=1  ����  ˆyn − yn  yn  ����  (4)  which is preferred to Mean Squared Error (MSE) because of its scale-invariant property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  We will focus here on two very simple models, although other machine learning models have also been  considered in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Indeed, these two models lend themselves very easily to an adaptive formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' In  this section, we will first describe these two approaches and their performances, before giving the general  adaptive formulation, which we will particularize in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='1  Feature Engineering and Linear Regression  Based on the assumption that the system may be approximated by a model whose essential features come  from M/M/1/K and M/G/1/K queue theory, we took essential parameters characterizing queueing  systems, such as: ρ, ρe, π0, πK, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' and built further features by applying interactions and various  non-linearities (powers, log, exponential, square root).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Then, we selected features in this set by a forward  step-wise selection method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' by adding in turn each feature to potential models and keeping the feature  with best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Finally, we selected the model with best MAPE error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' For a linear regression  model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' this led us to select and keep a set of 4 simple features,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' which interestingly enough,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' have simple  interpretations:  �  �  �  �  �  �  �  �  �  π0 =  1−ρ  1−ρK+1  L = ρ + π0  �  k kρk  ρe = λe  λ ρ = λe  µ  Se = �  k kρk  e  (5)  where L is the expected number of packets in the queue according to M/M/1/K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' π0 the probability  that the queue is empty according to M/M/1/K theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' ρe the effective queue utilization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' and Se the  3  unnormalized expected value of the effective number of packet in the queue buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' These features can  be thought as a kind of data preprocessing, before applying ML algorithms, and this turns out to be a key  to achieving good performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' The 4 previous features have been used as input for several machine  learning models like Multi-Layer Perceptron model (MLP), Linear Regression, SVM, Random Forest,  Gradient Boosting Regression Tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' We only describe here the case of linear regression, since it is a  method for which an adaptive version is readily obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' In this case, model (1) is simply  ˆy = θ0 + θ1π0 + θ2L + θ3ρe + θ4Se = θT u  (6)  with θT = [θ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' θ4] and uT = [1, π0, L, ρe, Se].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' For the linear regression model in ((6), it is well  known that the regularized minimum mean squared error  J(θ) = E  �  (y − θT u)2�  + α||θ||2  (7)  is obtained for  θ : (Ruu + α1) θ = Ryu  (8)  where we denoted  �Ruu = E  �  uuT �  ,  the correlation matrix of u  Ryu = E [yu] ,  the correlation vector of y and u  and 1 the identity matrix, α the regularization parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  As far as performance is concerned with this approach, it was evaluated using static data from the GNN  ITU Challenge 2021 [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Compared to the state-of-the-art, our linear regression with carefully selected  features shows a very slight performance degradation: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='74% in MAPE while the best state-of-the-art  method is at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='27%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' One strong advantage is in term of training and inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' It has a training  time of less than a second when GNN requires more than 8 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Moreover, the inference time for the  complete network is also much lower, by a factor of almost 1000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='296s vs 214s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='2  Curve Regression by Bernstein polynomials  There is a high interdependence of the features we selected in Equation (5), since all these features can be  expressed in term of ρe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Furthermore, it is confirmed by data exploration that ρe is the prominent feature  for occupancy prediction (and in turn latency prediction), as exemplified in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  It is then tempting to try to further simplify our features space and estimate the occupancy from a non-  linear transformation of the single feature ρe, as:  ˆy = g(ρe)  (9)  where ˆy is the estimate of the occupancy y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' The concerns are of course to define simple and efficient  functions g, with a low number of parameters, that can model the kind of growth shown in Figure 3, and  of course to check that the performance remains interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  Figure 3: Data of ITU Challenge 2021 [12], ρe vs queue occupancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Color-scale is an indicator of points  cloud density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  The estimator g is defined as a linear combination of simple functions fn:  ˆy = g(ρe) =  �  n  θn · fn(ρe)  (10)  which is also a linear model in terms of function fn(ρe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='0  2988  2000  1500  1000  500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='6 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='4 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='2 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='0-  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='0Several solutions were considered in [6] to define or choose the functions fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Since we know that the  Bernstein polynomials form a basis in the set of polynomial in the interval [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' and that the approxima-  tion of any continuous function on [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' 1[ by a Bernstein polynomial converges uniformly, we were led to  these polynomials:  f K  n (x) =  �K  n  �  xn(1 − x)K−n  (11)  where K is maximum order of polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  As mentioned, (10) can be rewritten as the linear model  ˆy = g(ρe) =  �  n  θn · fn(ρe) = θT u  (12)  with θT = [θ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' , θK] and uT = [f K  0 (ρe), f K  1 (ρe), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' , f K  K (ρe)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Hence, we have the same form as in  (8) for the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  In term of performances, we also obtained a minor degradation in MAPE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='68%) compared to state-of-  the-art (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='29%), while improving by several orders the wall training and inference times (2min/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='14s vs  8hrs/214s);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' though a bit less than the simple linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  4  Adaptive versions  We place ourselves in the context where we have regular snapshots of the state of the network, which  allows us to both monitor the quality of predictions, and to track changes in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' For the n-th  series of measurements, let us denote y(n) the measured latency and u(n) the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' We can also  group several snapshots or several links into a vector of latencies y(n) and matrix U(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' In the following  we will derive equations for this block case, which includes immediately the scalar case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  The minimum mean square error (7) which has the explicit solution (8) can also be solved by a gradient  algorithm as  θk+1 = θk − µ ∇J(θ)|θ=θk ,  (13)  = θk − µ ((Ruu + α1) θk − Ryu) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  (14)  In (14), we can substitute the true values with estimated ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' In order to introduce adaptivity to context  changes in the network, these estimates will preserve the temporal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' We thus use either a sliding  average  �  ˆRuu(n) = �L  l=0 U(n − l)U(n − l)T  ˆRyu(n) = �L  l=0 U(n − l)y(n − l)  (15)  or an exponential mean  �  ˆRuu(n) = �n  l=0 λl−nU(l)U(l)T = λ ˆRuu(n − 1) + U(n)U(n)T  ˆRyu(n) = λ ˆRyu(n − 1) + U(n)y(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  (16)  where λ ≤ 1 is the forgetting factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  In the limit case where we take either L = 0 or λ = 0 in the previous formulas, we get the ‘instantaneous  estimates‘  �  ˆRuu(n) = U(n)U(n)T  ˆRdu(n) = U(n)y(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  (17)  we obtain  θ(n + 1) = (1 − µα)θ(n) − µU(n)  �  U(n)T θ(n) − y(n)  �  (18)  which reduces to the well known LMS algorithm [14] in the scalar case and no regularization, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  Alternatively, one can try to solve the normal equation (8), using the time dependent estimates as the  exponential mean (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' The difficulty with the solution  ˆθ(n + 1) =  �  ˆRuu(n + 1) + α1  �−1 ˆRyu(n + 1)  (19)  is the inversion, for each n, of the correlation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Let us denote  K(n + 1) =  �  ˆRuu(n + 1) + α1  �−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  (20)  Using (16),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' we have  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='K(n + 1)−1 = λ ˆRuu(n) + U(n + 1)U(n + 1)T + α1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='(21)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='= λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='ˆRuu(n) + α1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='+ U(n + 1)U(n + 1)T + α(1 − λ)1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='(22)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='= λK(n)−1 + U(n + 1)U(n + 1)T + α(1 − λ)1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='(23)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='and  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='K(n + 1) =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='λK(n)−1 + U(n + 1)U(n + 1)T �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='+ α(1 − λ)1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='�−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='(24)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='= [Q(n + 1) + δ1]−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='(25)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='with  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='Q(n + 1) =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='λK(n)−1 + U(n + 1)U(n + 1)T �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='(26)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='and δ = α(1 − λ) The matrix inversion lemma enables to reduce the inversion of Q(n + 1) to  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='Q(n + 1)−1 = 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='λK(n) − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='λ2 K(n)U(n + 1)×  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='1 + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='λU(n + 1)T K(n)U(n + 1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='�−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='U(n + 1)T K(n),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  (27)  which simplifies to  Q(n + 1)−1 = 1  λK(n) − 1  λ2  K(n)u(n + 1)u(n + 1)T K(n)  1 + 1  λu(k + 1)T K(n)u(k + 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  (28)  for scalar observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Now, we can use the Taylor expansion to get  K(n + 1) = [Q(n + 1) + δ1]−1 = Q(n + 1)−1 − δQ(n + 1)−2 + δ2Q(n + 1)−3 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  (29)  This gives us a way to compute recursively the inverse of the regularized estimate of the correlation matrix  by combining (27) and (29) into  K(n + 1) ≈ Q(n + 1)−1 − δQ(n + 1)−2  (30)  which, by (27), does not require the inversion of K(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  In both cases, we have the updating formula  θ(n + 1) = θ(n) + K(n + 1)U(n + 1)[y(n + 1) − θ(n)T U(n + 1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  (31)  5  Experiments and results  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='1  Dataset  We generate a dataset thanks to a public challenge data generator [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' This data generator is based on the  well-known OMNET++ discrete event network simulator[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' The published simulator is available as a  docker image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' However, due to the rules of the 2022 edition of the challenge, it is not possible to generate  large topologies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' no more than 10 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Since our models are link-based, the use of small topologies  does not seem to be a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' The simulator is parameterized by a traffic matrix and a topological graph  that are easy to generate thanks to the provided API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  Our generated dataset, used is this paper, is the result of 11900 simulations of the same topology graph  of 10 nodes and 30 links, subject to 100 different traffic matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' In order to get complex results of  simulations but at low cost, we made the choice to model a network with small queue buffers (8000 bits)  and possibly subject of high traffic intensities (maximum traffic rate set to 4000 bits/s for each flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Then  for each traffic matrices, we alter the capacity of the network according to a sigmoid, in order to model  a network subject to jamming, with 2 stationary states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' The proposed jamming may cause a decrease  in the capacity of the network links by up to a factor of 5, as depicted on Figure 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' For simplification  purposes, we have considered that each link of the network has the same capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' This result in a U-  shaped distribution of our link data samples according to the link capacity as shown in Figure 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='2  Results  From the generated data, we validate our approach along several axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='1  Global performances  First, we establish the benchmark performances based on the global methods presented in the paper [6]  and Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  For the linear regression method described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='1, we obtain an MSE of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='86e-4 and a MAPE  of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='58% for the queue occupancy prediction and an MSE of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='10e-3 and a MAPE of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='26%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' for the  end-to-end latency prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  Concerning the curve regression using Bernstein polynomials (of degree 8) described in in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='2,  we obtain an MSE of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='52e-4 and a MAPE of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='72% for the queue occupancy prediction and an MSE of  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='35e-4 and a MAPE of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='95% for the end-to-end latency prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  Note that these benchmark performances are below the performances obtained in [6], but the dataset we  use here is probably more severe since using ground truth value for occupancy results in an MSE 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='03e-4  of a MAPE 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='34% for the flow delay prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' That is indeed very close of the obtained results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  6  0  2000  4000  6000  8000  10000  12000  Sample ID  10000  15000  20000  25000  30000  35000  40000  45000  50000  Capacity of the links (bits/s)  (a) Capacity alteration to model jamming with a decrease of the capacity up to a factor of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  10000  #40  15000  #35  20000  #30  25000  #25  30000  #20  35000  #15  40000  #10  45000  #5  50000  #0  capacity of links (bits/s)  #scenario ID  0  5000  10000  15000  20000  25000  30000  #links data  (b) Link capacity distribution of the generated dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  Figure 4: Overview of the generated dataset  7  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='2  Behavior of iterative algorithms  In a second step, we verify that the algorithms presented in section 4 converge and allow us to recover  these performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' With a forgetting factor of 1 (use of all data with the same weight) and a block size  of 10, we observe, for example in Figure 5, that the model coefficients converge towards a stable value,  and that MAPE recovers the value obtained with the global method using all data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' The convergence is  obtained in less than 10,000 operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' It is thus possible to replace the global method, which is already  low-cost, with an approach where the calculations are carried out recursively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='3  Adaptivity  In a third step, we compare the algorithms to the case of network modifications We consider an abrupt  change in the network capacity, which could correspond to a jamming scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' We then examine how  the two adaptive algorithms presented (linear regression with judiciously chosen features;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' and Bernstein  polynomial model) can detect and adapt to these modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' In this context, we examine the role of  the forgetting factor and the regularization parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Figure 6 and Figure 7 present the results for the  case of a capacity change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' We observe that (i) the square of the residual error, smoothed over 100 points,  is a remarkable indicator of a change in the network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' and (ii) that the model coefficients readjust over the  iterations after this change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='4  Discussion  These experiments show the effectiveness and relevance of our iterative and adaptive versions of end-  to-end latency estimation procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' The iterative versions have the same performance as their global  counterparts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' an even lower cost since they can be implemented iteratively as the data is received or made  available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' The convergence time for the model coefficients is a few thousand samples while the global  model used around 350,000 samples for training, while the GNN models require several million samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  Moreover, we observe that the residual error converges very quickly, in some tens of samples, which  means that although the convergence of the models’ coefficients does not seem to be complete, they are  equivalent from the point of view of performance for occupancy prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' From an operational point of  view, the model can be refreshed regularly, and the predicted KPIs between these updates can be used for  intelligent routing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' As we have observed, residual error monitoring is an excellent indicator of changes  in the network state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  6  Conclusion  In this paper, we considered the problem of designing efficient and low-cost algorithms for KPI predic-  tion that are locally implementable and adaptive to network changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Based on a previous work, we  have argued and developed adaptive solutions, introducing in addition a regularization term in order to  stabilize the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' We used a public domain simulator to simulate networks and generate relevant data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  The experiments demonstrate the effectiveness and relevance of these algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Thus, we now have  low-complexity models that can be implemented iteratively at the level of local links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' We have the possi-  bility to predict the occupancy of the different links, and the end-to-end latencies (the models predict the  occupancy of the queues, then compute analytically the delay for each link and finally aggregate along the  path).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Moreover, the adaptability of the solution allows to follow changes in the network state, always at  a minimal cost, by re-adapting from the current solution and new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' The continuation of the work will  focus on the choice criteria of the forgetting factor, on the impact of the regularization factor, in order to  find automatic selection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Of course, the approaches considered here will have to be considered  and adapted for other types of KPI, such as error rate or jitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  8  0  5000  10000  15000  20000  25000  30000  samples  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
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+page_content='035  RMSE  #39  #10  #1  #10  #39  (b) Evolution of the RMSE along the scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  Figure 6: Evolution of weights and  √  MSE (RMSE) for our adaptive approach of Bernstein polynomial  curve regression of degree 8,λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
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+page_content=' Scenario describes a nominal period between 2 periods  of jamming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' #39 corresponds to a link capacity of 11 Kbits/s, #10 40 Kbits/s and #1 49 Kbts/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Figure is  smoothed over 100 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
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+page_content='7  Weight value  #39  #10  #1  #10  #39  θ_0  θ_1  θ_2  θ_3  θ_4  (a) Evolution of weights along the scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
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+page_content='05  RMSE  #39  #10  #1  #10  #39  (b) Evolution of the RMSE along the scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  Figure 7: Evolution of weights and  √  MSE (RMSE) for our adaptive approach of Linear  Regression,λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='9, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Scenario describes a nominal period between 2 periods of jamming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' #39  corresponds to a link capacity of 11 Kbits/s, #10 40 Kbits/s and #1 49 Kbts/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Figure is smoothed over  100 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  11  References  [1] Amin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=', Reisslein, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=', Shah, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=': Hybrid SDN networks: A survey of existing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' IEEE  Communications Surveys & Tutorials 20(4), 3259–3306 (2018)  [2] de Aquino Afonso, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' : GNNet challenge 2021 report (1st place).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='com/  ITU-AI-ML-in-5G-Challenge/ITU-ML5G-PS-001-PARANA (2021)  [3] Barcelona Neural Networking Center: The graph neural networking challenge 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' https://  bnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='upc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='edu/challenge/gnnet2020  [4] Chua, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=', Ward, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=', Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=', Sharma, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=', Huberman, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' : Stringer: Balancing latency and re-  source usage in service function chain provisioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' IEEE Internet Computing 20(6), 22–31 (2016)  [5] Jahromi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=', Hines, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=', Delanev, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' : Towards application-aware networking: Ml-based end-to-  end application KPI/QoE metrics characterization in SDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' In: 2018 Tenth International Conference  on Ubiquitous and Future Networks (ICUFN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' 126–131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' IEEE (2018)  [6] Larrenie, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=', Bercher, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=', Lahsen-Cherif, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=', Venard, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=': Low complexity approaches for end-to-  end latency prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' In: Proceedings of the 13th IEEE International Conference On Computing,  Communication and Networking Technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' IEEE (2022)  [7] Pasca, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=', Kodali, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=', Kataoka, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=': AMPS: Application aware multipath flow routing using  machine learning in SDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' In: 2017 Twenty-third National Conference on Communications (NCC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' IEEE (2017)  [8] Poularakis, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=', Iosifidis, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=', Tassiulas, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=': SDN-enabled tactical ad hoc networks: Extending pro-  grammable control to the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' IEEE Communications Magazine 56(7), 132–138 (2018)  [9] Poularakis, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=', Qin, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=', Nahum, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=', Rio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=', Tassiulas, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=': Flexible SDN control in tactical ad  hoc networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Ad Hoc Networks 85, 71–80 (2019)  [10] Rusek, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=', Suárez-Varela, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=', Mestres, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=', Barlet-Ros, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=', Cabellos-Aparicio, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=': Unveiling the  potential of graph neural networks for network modeling and optimization in SDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' In: Proceedings  of the 2019 ACM Symposium on SDN Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' 140–151 (2019)  [11] Singh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=', Jha, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=': A survey on Software Defined Networking: Architecture for next generation  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Journal of Network and Systems Management 25(2), 321–374 (2017)  [12] Suárez-Varela, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' : The graph neural networking challenge: a worldwide competition for  education in AI/ML for networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' ACM SIGCOMM Computer Communication Review 51(3), 9–16  (2021)  [13] Varga, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=', Hornig, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=': An overview of the omnet++ simulation environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' In: 1st International  ICST Conference on Simulation Tools and Techniques for Communications, Networks and Systems  (2010)  [14] Widrow, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=', Stearns, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=': Adaptive Signal Processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Edited by Alan V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}
+page_content=' Oppenheim, Prentice-Hall  (1985)  12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf'}

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+Sentence Identification with BOS and EOS Label Combinations
+Takuma Udagawa, Hiroshi Kanayama, Issei Yoshida
+IBM Research - Tokyo, Japan
+Takuma.Udagawa@ibm.com, {hkana,issei}@jp.ibm.com
+Abstract
+The sentence is a fundamental unit in many
+NLP applications. Sentence segmentation is
+widely used as the first preprocessing task,
+where an input text is split into consecutive
+sentences considering the end of the sentence
+(EOS) as their boundaries. This task formula-
+tion relies on a strong assumption that the in-
+put text consists only of sentences, or what we
+call the sentential units (SUs). However, real-
+world texts often contain non-sentential units
+(NSUs) such as metadata, sentence fragments,
+nonlinguistic markers, etc. which are unrea-
+sonable or undesirable to be treated as a part
+of an SU. To tackle this issue, we formulate a
+novel task of sentence identification, where the
+goal is to identify SUs while excluding NSUs
+in a given text.
+To conduct sentence iden-
+tification, we propose a simple yet effective
+method which combines the beginning of the
+sentence (BOS) and EOS labels to determine
+the most probable SUs and NSUs based on
+dynamic programming. To evaluate this task,
+we design an automatic, language-independent
+procedure to convert the Universal Dependen-
+cies corpora into sentence identification bench-
+marks. Finally, our experiments on the sen-
+tence identification task demonstrate that our
+proposed method generally outperforms sen-
+tence segmentation baselines which only uti-
+lize EOS labels.
+1
+Introduction
+The sentence, which we refer to as the sentential
+unit (SU), is a fundamental unit of processing in
+many NLP applications including syntactic pars-
+ing (Dozat and Manning, 2017), semantic parsing
+(Dozat and Manning, 2018), and machine transla-
+tion (Liu et al., 2020). Existing works mostly rely
+on sentence segmentation (a.k.a. sentence bound-
+ary detection) as the first preprocessing task, where
+we predict the end of the sentence (EOS) to split a
+text into consecutive SUs (Kiss and Strunk, 2006;
+Gillick, 2009). This approach relies on a strong
+assumption that the text only consists of SUs; how-
+ever, real-world texts like web contents often con-
+tain non-sentential units (NSUs) such as the meta-
+data of attachments embedded in the email body,
+repetition of symbols for separating texts, irregular
+series of nouns, etc. (just to name a few). Such
+NSUs may cause detrimental or unexpected results
+in the downstream tasks if considered as parts of
+the SUs and are more desirable to be distinguished
+from SUs in the first preprocessing step.
+To tackle this problem, we formulate a novel
+task of sentence identification, where the goal is
+to identify SUs while excluding NSUs in a given
+text (§3). This can be regarded as an SU span ex-
+traction task, where each SU span is represented
+by the beginning of the sentence (BOS) and the
+EOS labels.1 We illustrate the difference between
+sentence segmentation and sentence identification
+in Table 1. In sentence segmentation, the text frag-
+ment of an embedded file (“- TEXT.htm << File:
+TEXT.htm >>”) needs to be considered as a part
+of an SU. In contrast, sentence identification can
+regard it as an NSU and exclude it for downstream
+applications such as dependency parsing.
+To conduct sentence identification, we propose
+a simple method which effectively combines the
+BOS and EOS probabilities to determine both SUs
+and NSUs (§4).
+To be specific, we first train
+the BOS and EOS labeling models based on ei-
+ther the sentence identification dataset (with SUs
+and NSUs) or sentence segmentation dataset (only
+SUs). Then, we search for the most probable spans
+of SUs and NSUs using a simple dynamic program-
+ming framework. Theoretically, our method can be
+considered as a natural generalization of existing
+sentence segmentation algorithms.
+To evaluate this task, we design an automatic pro-
+1For simplicity, we assume that the input text can be seg-
+mented into consecutive, non-overlapping units of SUs and
+NSUs. This way, we can also represent and evaluate SU
+extraction as an equivalent BIO labeling task (§5-§7).
+arXiv:2301.13352v1  [cs.CL]  31 Jan 2023
+
+Input Text
+Thank you. - TEXT.htm << File: TEXT.htm >> I was thinking of converting it to a hover
+(from EWT)
+vehicle. I might just sell the car and get you to drive me around all winter.
+E
+Sentence
+Thank you. - TEXT.htm << File: TEXT.htm >> I was thinking of converting it to a hover
+Segmentation
+E
+E
+vehicle. I might just sell the car and get you to drive me around all winter.
+B
+E
+B
+Sentence
+Thank you. - TEXT.htm << File: TEXT.htm >> I was thinking of converting it to a hover
+Identification
+E B
+E
+vehicle. I might just sell the car and get you to drive me around all winter.
+Table 1: Illustration of sentence segmentation and sentence identification. In sentence segmentation, EOS labels
+(E) are used to segment the input text into consecutive SUs (in blue). In sentence identification, only the spans
+bracketed by the BOS (B) and EOS labels are extracted as SUs, while the rest can be excluded as NSUs.
+cedure to convert the Universal Dependencies (UD)
+corpora (de Marneffe et al., 2021) into sentence
+identification benchmarks (§5). To be specific, (i)
+we use the original sentence boundaries in UD as
+the unit (SU and NSU) boundaries and (ii) classify
+each unit as an SU iff it contains at least one clausal
+predicate with a core/non-core argument. Impor-
+tantly, our classification rule follows the definition
+of lexical sentence in linguistics (Nunberg, 1990),
+is easily customizable with language-independent
+rules, and makes reasonable classification within
+the scope of our experiments.
+To conduct our experiments, we focus on the
+English Web Treebank (Silveira et al., 2014) as the
+primary benchmark for sentence identification and
+train the BOS/EOS labeling models by finetuning
+RoBERTa (Liu et al., 2019) (§6). We also propose
+techniques to develop these models using a stan-
+dard sentence segmentation dataset, i.e. the Wall
+Street Journal corpus (Marcus et al., 1993), which
+only contains clean, edited SUs without any NSUs.
+Based on our experimental results, we demon-
+strate that our proposed method generally outper-
+forms sentence segmentation baselines which only
+utilize EOS labels (§7). These results highlight
+the importance of combining the BOS labels in
+addition to the EOS labels for accurate sentence
+identification under various conditions.
+2
+Background
+Sentence segmentation, a.k.a. sentence boundary
+detection, is the task of segmenting an input text
+into the unit of sentences. Despite the long his-
+tory of study (Riley, 1989) and its importance in
+the entire NLP pipeline (Walker et al., 2001), this
+area has received relatively little attention. For
+one reason, the task has been recognized as “long
+solved” (Read et al., 2012) with the most recent
+approach reporting 99.8% F1 score on the standard
+English Wall Street Journal (WSJ) dataset (Wicks
+and Post, 2021). Their state-of-the-art method ER-
+SATZ combines (i) a regular-expression based de-
+tector of candidate sentence boundaries, followed
+by (ii) a Transformer-based (Vaswani et al., 2017)
+binary classifier which predicts whether the can-
+didate boundary is EOS based on the local con-
+text, i.e. surrounding few words. This modern
+context-based approach has been shown to outper-
+form competitive, widely used baselines such as
+SPLITTA (Gillick, 2009), PUNKT (Kiss and Strunk,
+2006), and MOSES (Koehn et al., 2007).
+However, two important aspects are not fully ad-
+dressed in the current literature. First is the cover-
+age of diverse domains, genres, and writing styles.
+Existing works (including Wicks and Post, 2021)
+focus on formal/edited text and assume the exis-
+tence of sentence ending punctuations (e.g. full
+stops) at the sentence boundaries. However, social
+media texts often lack such punctuations and con-
+tain various types of non-linguistic noise, which
+can lead to a substantial degradation in the seg-
+mentation performance (Read et al., 2012; Rudra-
+pal et al., 2015). Speech transcription texts also
+usually contain disfluent, ungrammatical, or frag-
+mented structures and lack both punctuations and
+casing (Wang et al., 2019; Rehbein et al., 2020).
+Considering the amount of such informal or non-
+standard texts in the real world, it is compelling
+to expand the capability of sentence segmentation
+beyond formal, standardized text.
+The second aspect is the coverage of multiple
+languages. Different languages involve different
+complexities in sentence segmentation, e.g. Chi-
+nese requires the disambiguation of commas as the
+
+sentence ending punctuation (Xue and Yang, 2011)
+and Thai does not mark EOS with any type of punc-
+tations (Aroonmanakun et al., 2007; Zhou et al.,
+2016). To advance NLP from a multilingual per-
+spective, it is crucial to develop and evaluate mod-
+els in multiple languages: Wicks and Post (2021)
+make an important step in this direction, proposing
+a language-agnostic, unified sentence segmentation
+model covering a total of 87 languages.
+Based on these observations, we first propose to
+extend the task of sentence segmentation to sen-
+tence identification, which expands the capability
+of sentence segmentation beyond formal, standard-
+ized text (§3, §4). Secondly, we propose a cross-
+lingual method of benchmarking sentence identifi-
+cation based on the UD corpora, considering every
+word or character as the candidate boundary to
+cover diverse domains, genres, and languages that
+lack sentence ending punctuations (§5). Finally,
+we follow Wicks and Post (2021) to develop mod-
+ern neural-based models that require no language-
+specific engineering and can be developed for dif-
+ferent languages in a unified manner (§6).
+3
+Task Formulation
+3.1
+Sentence Segmentation Task
+First, we introduce a precise (re-)formulation
+of the sentence segmentation task.
+Let W =
+(w0, w1, ..., wN−1) represent the input text, where
+each wi denotes a word (but can also be a sub-
+word or character). We also define the text span
+W [i : j] = (wi, ..., wj−1), their concatenation
+W [i : j] ⊕ W [j : k] = W [i : k], and SU bound-
+ary indices B = (b0, b1, ..., bM) where b0 = 0,
+bM = N, and �M
+i=1 W [bi−1 : bi] = W (i.e. the
+concatenation of all SUs recovers the input text).
+Next, we introduce the SU probability pSU(W [i:
+j]) which corresponds to the probability of the text
+span W [i:j] being an SU. Based on this probabil-
+ity, the task of sentence segmentation can be for-
+malized as searching for the boundaries B which
+maximize the following probability:2
+arg max
+B
+M
+�
+i=1
+pSU(W [bi−1 :bi])
+(1)
+The most standard approach is to define pSU(W [i:
+j]) based on a pretrained EOS labeling model, as
+we describe in §4.1. However, our (re-)formulation
+2M is a variable and need not be fixed during the search.
+as Eq. (1) is more general and permits other defini-
+tions of SU probability as well.
+3.2
+Sentence Identification Task
+In sentence identification, we consider the input
+text W can be segmented into consecutive, non-
+overlapping units of SUs and NSUs. Hence, we
+regard B = (b0, b1, ..., bM) as the unit (SU and
+NSU) boundaries and define the unit indicators
+A = (a1, a2, ..., aM) for each unit as follows:
+ai =
+�
+1
+if W [bi−1 :bi] is an SU
+0
+if W [bi−1 :bi] is an NSU
+Next,
+we
+introduce
+the
+NSU
+probability
+pNSU(W [i : j]) which corresponds to the prob-
+ability of the text span W [i : j] being an NSU.
+Based on pSU and pNSU, we can formalize the
+task of sentence identification as searching for the
+unit boundaries B and unit indicators A which
+maximize the following probability:
+arg max
+B,A
+M
+�
+i=1
+pSU(W [bi−1 :bi])ai pNSU(W [bi−1 :bi])1−ai
+(2)
+Note that this strictly generalizes the sentence seg-
+mentation task in Eq. (1), which is a special case
+where ai = 1, ∀ai ∈ A. Based on this task formu-
+lation, we discuss how we can define pSU(W [i:j])
+and pNSU(W [i:j]) to derive our sentence identifi-
+cation algorithm in §4.2.
+4
+Methods
+4.1
+Sentence Segmentation Method
+In the most standard approach, sentence segmenta-
+tion employs an EOS labeling model pEOS to define
+the SU probability pSU in Eq. (1). To be specific,
+let pEOS(wi|W ; θ) denote the EOS labeling model,
+which computes the probability of wi being EOS
+in W (θ denotes the model parameters). Typically,
+it is straightforward to train this model in a super-
+vised learning setup using a dataset annotated with
+gold EOS boundaries (Wicks and Post, 2021). For
+brevity, we use the notation pEOS(wi) as a short-
+hand for pEOS(wi|W ; θ), i.e. we omit W and θ
+(unless required) in the rest of this paper.
+Based on the pretrained model pEOS, we can
+define the SU probability as pSU(W [i : j]) =
+pEOS(wj−1) �
+i≤k<j−1(1 − pEOS(wk)), which re-
+quires the last word wj−1 to be EOS and all other
+
+words to be non-EOS. By substituting this defini-
+tion, we can decompose Eq. (1) as follows:
+(1) = arg max
+B
+M
+�
+i=1
+log pSU(W [bi−1 :bi])
+= arg max
+B
+M
+�
+i=1
+�
+log pEOS(wbi−1) +
+�
+bi−1≤j<bi−1
+log (1 − pEOS(wj))
+�
+= arg max
+B
+�
+i∈BEOS
+log pEOS(wi) +
+�
+i/∈BEOS
+log (1 − pEOS(wi))
+(3)
+where BEOS = {bi − 1 | i ∈ (1, 2, ..., M)} repre-
+sents all the EOS indices defined by B.
+This is a trivial optimization problem where we
+can simply choose BEOS = {i ∈ (0, 1, ..., N −
+1) | pEOS(wi) ≥ 0.5} to maximize Eq. (3). This
+also shows that sentence segmentation can be con-
+ducted by predicting the EOS independently for
+each wi based on pEOS(wi). In contrast, sentence
+identification involves a more complex optimiza-
+tion problem which we solve using dynamic pro-
+gramming (§4.2).
+4.2
+Sentence Identification Method
+We extend the method of sentence segmentation
+(§4.1) to conduct sentence identification. To be spe-
+cific, we employ pretrained BOS and EOS labeling
+models pBOS, pEOS to define the SU and NSU prob-
+abilities pSU, pNSU in Eq. (2). As a first step, we
+need to train the BOS and EOS labeling models:
+this can be conducted in a supervised manner using
+a dataset containing gold BOS and EOS labels, as
+we explain in §6.1.
+Based on the pretrained BOS and EOS labeling
+models, we can define the SU and NSU probabili-
+ties as follows:
+pSU(W [i:j]) = pBOS(wi)
+�
+i<k≤j−1
+(1 − pBOS(wk))
+× pEOS(wj−1)
+�
+i≤k<j−1
+(1 − pEOS(wk))
+pNSU(W [i:j]) =
+�
+i≤k≤j−1
+(1 − pBOS(wk)) ×
+�
+i≤k≤j−1
+(1 − pEOS(wk))
+In the SU probability pSU, the first word wi is
+required to be BOS, the last word wj−1 to be EOS,
+and all other words to be neither BOS nor EOS.
+Note that this definition of pSU is a natural gener-
+alization from §4.1 which only relies on the EOS
+probability pEOS.
+In contrast, the NSU probability pNSU requires
+all words to be neither BOS nor EOS. Notably, this
+definition does not distinguish contiguous NSUs in
+the sense that pNSU(W [i:k]) = pNSU(W [i:j]) ×
+pNSU(W [j :k]) if W [i:j] ⊕ W [j :k] = W [i:k].
+This is convenient as we are only interested in the
+extraction of SUs and do not need to seek the exact
+boundaries between consecutive NSUs.
+By substituting these definitions of pSU and pNSU,
+we can decompose Eq. (2) as follows:
+(2) = arg max
+B,A
+M
+�
+i=1
+�
+ai log pSU(W [bi−1 :bi])
++ (1 − ai) log pNSU(W [bi−1 :bi])
+�
+= arg max
+B,A
+�
+i∈BA
+BOS
+log pBOS(wi) +
+�
+i/∈BA
+BOS
+log (1 − pBOS(wi))
++
+�
+i∈BA
+EOS
+log pEOS(wi) +
+�
+i/∈BA
+EOS
+log (1 − pEOS(wi))
+(4)
+where BA
+BOS = {bi−1 | i ∈ (1, 2, ..., M), ai = 1}
+denotes the BOS indices and BA
+EOS = {bi − 1 | i∈
+(1, 2, ..., M), ai = 1} denotes the EOS indices,
+both defined by B and A.
+Therefore, our goal is to choose BA
+BOS and BA
+EOS
+which maximize Eq. (4). To this end, we need
+to consider the restrictions that (i) the first label
+should be BOS, (ii) the last label should be EOS,
+and (iii) BOS and EOS labels need to appear alter-
+nately. These restrictions can be incorporated in
+our dynamic programming framework to find the
+argmax of Eq. (4). For the precise algorithm, we
+refer the readers to Appendix A.
+5
+Evaluation
+Due to the novelty of the task, currently there exists
+no benchmark for evaluating sentence identifica-
+tion. To address this issue, we propose a fully
+automatic procedure to convert the Universal De-
+pendencies (UD) corpora (de Marneffe et al., 2021)
+into sentence identification benchmarks.
+Concretely speaking, we conduct the following
+two steps based on the gold UD annotation: (i) the
+detection of unit (SU and NSU) boundaries and (ii)
+the classification of each unit into SU or NSU. As
+for (i), we simply use the original sentence bound-
+aries in the UD annotation, where UD uses the term
+sentence in a broader sense including both SUs and
+NSUs (e.g. sentence fragments). Note that the ex-
+act boundaries between consecutive NSUs (which
+we call NSU–NSU boundaries) do not need to be
+accurate or consistent, since we are only interested
+in extracting the spans of SUs. However, we do
+expect that the original boundaries are generally
+reliable in all other cases (SU–SU and SU–NSU
+boundaries), which seems to be the case.
+The main problem is (ii), i.e. how to classify
+
+*∼*∼*∼*∼*∼*∼*∼*∼*∼*
+**********NOTE**********
+By video conference from _______
+Excerpt:
+02/13/2001 08:02 PM
+5:00 PT ** 6:00 MT ** 7:00 CT ** 8:00 ET
+Time: 11:30am / 1:30pm Central / 2:30pm Eastern
+Sunshine Coast, British Columbia, Canada
+- UnleadedStocks.pdf
+t r u t h o u t — Perspective
+( Answered, 2 Comments )
+The federal sites of Washington, DC.
+From Madrid to Seville to Barcelona an Valencia.
+Table 2: Examples of gold NSUs in the English Web
+Treebank (EWT) identified based on our procedure.
+Each line corresponds to one example of NSU.
+each unit as an SU or NSU. To this end, we fol-
+low the notion of lexical sentence in linguistics
+(Nunberg, 1990) which defines an SU based on the
+dependencies among the lexical items, e.g. a group
+of words that contain a subject and predicate. In
+this work, we build upon the UD dependency re-
+lations and define an SU as a unit that contains at
+least one clausal predicate with a core or non-core
+argument.3 Here, a clause expresses an event or
+proposition which we regard as an essential aspect
+of SUs. A clausal predicate and a core argument
+form the backbone of a clause, while a non-core
+argument modifies it (de Marneffe et al., 2021).
+Note that our current definition excludes noun
+phrases appearing by themselves, since they only
+consist of the nominal dependent relations. How-
+ever, we can flexibly customize the definition of
+SUs to include or exclude such phrases.
+Due to the reliance on UD, our conversion proce-
+dure can be applied to a wide variety of languages
+supported in UD (currently over 100 languages).
+However, as a first set of experiments, we focus on
+the English Web Treebank (EWT) (Silveira et al.,
+2014) as the primary benchmark of sentence identi-
+fication. This dataset comprises five genres of web
+media texts: namely weblogs, newsgroup threads,
+emails, product reviews, and Q&A websites. Con-
+sequently, the dataset contains formal SUs, infor-
+mal SUs (e.g. without capitalization or punctua-
+tions) as well as a wide variety of NSUs.
+We show some examples of NSUs in Table 2
+(and more in Appendix B) identified based on our
+3To check this condition, we simply need to verify whether
+there is at least one core argument (e.g. nsubj, obj, ccomp) or
+non-core dependent (e.g. obl, advcl, aux). For a full list of the
+UD relations, see https://universaldependencies.org/u/dep/.
+Train
+Dev
+Test
+Total SUs
+10,356
+1,523
+1,490
+Total NSUs
+2,187
+478
+587
+Word-
+Level
+B-Label
+10,356
+1,523
+1,490
+I-Label
+160,127
+18,791
+18,222
+O-Label
+6,939
+1,302
+1,822
+Character-
+Level
+B-Label
+10,356
+1,523
+1,490
+I-Label
+773,223
+92,309
+88,441
+O-Label
+47,107
+9,925
+13,232
+Table 3: EWT dataset statistics.
+procedure. As shown by the results, our procedure
+can identify various NSUs including nonlinguistic
+markers, timestamps, lists, contact information, etc.
+We can also see that noun/prepositional phrases
+are classified as NSUs based on our criteria. By
+excluding such NSUs and identifying SUs, we can
+clearly separate the portions of the text that are
+worth sophisticated linguistic analyses, e.g. based
+on dependency parsing or manual inspection.
+Finally, we summarize the dataset statistics of
+EWT in Table 3. Overall, 17∼28% of the units
+were classified as NSUs, with the test set containing
+the highest proportion of NSUs. We also regard SU
+extraction as a word-level or character-level BIO
+labeling task and report the number of gold BIO
+labels in Table 3.4 At the word-level, we can see
+that the proportion of O-labels (indicating NSUs) is
+only 4∼8% and much smaller than the proportion
+of NSUs in terms of units: this is because NSUs
+are usually short and contain only a few words.
+At the character-level, the proportion of O-labels
+is slightly larger (6∼13%): this is because NSUs
+often contain extraordinarily long words like URLs
+and long sequences of nonlinguistic symbols.
+Overall, we could verify that there exists a non-
+negligible amount of NSUs in the EWT dataset,
+which we aim to exclude with sentence identifica-
+tion in our experiments.
+6
+Experimental Setup
+6.1
+Model Setup
+As we discussed in §4.2, our sentence identification
+method requires pretrained BOS and EOS labeling
+models to identify SUs and NSUs. To develop
+these models, we simply finetune RoBERTaBASE
+4B = Beginning of SU, I = Inside of SU, and O = Outside
+of SU. Details of how we assign the gold BIO labels (at the
+word-level and character-level) are provided in Appendix C.
+
+by adding a binary BOS/EOS classifier on top of
+the encoder.
+To enable our models to handle various lengths
+of the input texts, we concatenate the consecutive
+L units of gold SUs and NSUs as the input during
+training, where L is sampled from a geometric
+distribution with parameter pCC.5 However, the
+RoBERTa encoder has the restriction that the input
+text size cannot exceed 512 subwords. Therefore, if
+the input text size is too large, we replace L with the
+maximum L′ < L which satisfies this restriction.
+Note that this is a common procedure to sample
+variable (instead of fixed) lengths of concatenated
+units (Joshi et al., 2020).
+Assuming the existence of the in-domain sen-
+tence identification dataset (EWT Train/Dev), it
+is straightforward to train the BOS/EOS labeling
+models based on our unit concatenation procedure.
+However, we may not always have the gold anno-
+tation of SUs and NSUs for the target domain. To
+take such cases into account, we also consider a
+setup where we only have the standard sentence
+segmentation dataset (WSJ Train/Dev) to train the
+BOS/EOS labeling models.
+When using the sentence segmentation dataset
+(WSJ), we need to apply the unit concatenation
+procedure using only clean, edited SUs. Unfor-
+tunately, this can yield the following data priors
+which do not actually hold in a sentence identifi-
+cation dataset (EWT): (i) an SU (almost) always
+starts with a capitalization and ends with punctua-
+tion, (ii) the first word of the input is always BOS
+and the last word is always EOS, and (iii) BOS
+always directly follows EOS.
+To address (i) and (ii), we propose a simple data
+augmentation technique to alleviate the discrepancy
+in the data priors. To address (iii), we propose
+an ensembling technique with the unidirectional
+(instead of bidirectional) models which are agnostic
+to this data prior.
+6.1.1
+Data Augmentation (+AUG)
+To address (i), we conduct a unit-level data aug-
+mentation, i.e. we modify each unit based on the
+following rules with a small probability pDA:
+• Convert all words in the unit to lower-case,
+upper-case, or title-case (e.g. “hello world”,
+5With parameter pCC ∈ (0, 1], the probability mass func-
+tion of the geometric distribution is p(L = l) = (1 −
+pCC)l−1pCC where l ∈ {1, 2, 3, ...}. As pCC decreases,
+the distribution gets more skewed towards larger L. With
+pCC = 0, we consider p(L = ∞) = 1.
+Orig.
+B
+E B
+Joe went to school. After that he ...
+(i) Unit
+B
+E
+B
+Aug.
+Joe went to school
+AFTER THAT HE ...
+(ii) Unit
+B
+E
+B
+Trunc.
+Joe went to school
+AFTER THAT HE ...
+Table 4: Illustration of our data augmentation tech-
+nique.
+In (i) unit-level augmentation, we randomly
+change the casing or remove the last punctuations of
+each unit. In (ii) unit truncation, we randomly truncate
+the first and last units of the input (and regard them as
+NSUs).
+“HELLO WORLD”, or “Hello World”).
+• Remove sentence ending punctuations based
+on a regular-expression matcher (following
+ERSATZ, Wicks and Post, 2021).
+After the unit-level augmentation, we can apply the
+unit concatenation in the exact same manner.
+Finally, to address (ii), we randomly apply a unit
+truncation to the first and last units of the concate-
+nated input. To be specific, we choose a random
+word in the first (last) unit and remove all words
+prior (posterior) to it with a small probability pT R.
+If the truncation is conducted, we regard the unit as
+an NSU and fix the gold BOS/EOS labels accord-
+ingly. See Table 4 for an illustration.
+Based on this procedure, we can expect to allevi-
+ate the data priors (i) and (ii). For more details, we
+refer the readers to Appendix D.
+6.1.2
+Unidirectional Model (+UNI)
+Simply concatenating SUs (without NSUs) yields
+the data prior (iii), i.e. BOS always directly fol-
+lows EOS. This prior can be easily captured by the
+bidirectional models pBOS(wi|W ), pEOS(wi|W )
+conditioned on the whole input W , including our
+RoBERTa-based models. For instance, as shown in
+Figure 1, the model may predict EOS at the end of
+the first unit (w2 = #) just because the next word
+(w3 = This) is likely predicted as BOS.
+To alleviate this issue, we propose to combine
+the predictions of the unidirectional models for
+BOS and EOS labeling. To be precise, let W ≤i =
+(w0, ..., wi) and W ≥i = (wi, ..., wN−1). Then,
+we can represent the unidirectional BOS model as
+pUni
+BOS(wi|W ≥i) (looking the context right-to-left)
+and EOS model as pUni
+EOS(wi|W ≤i) (looking left-to-
+right). As illustrated in Figure 1, these models are
+agnostic to the data prior (iii). In practice, we can
+
+Figure 1: Illustration of the bidirectional EOS model
+(left) and the unidirectional EOS model (right).
+simply use different attention masks and share the
+encoder parameters (except the last classifier) for
+the unidirectional and bidirectional models.
+We can utilize these unidirectional models by
+taking a linear intepolation with the bidirectional
+models as follows:
+p+Uni
+BOS (wi|W ) = λ · pUni
+BOS(wi|W ≥i) + (1−λ) · pBOS(wi|W )
+p+Uni
+EOS (wi|W ) = λ · pUni
+EOS(wi|W ≤i) + (1−λ) · pEOS(wi|W )
+Then, we can use p+Uni
+BOS and p+Uni
+EOS in place of
+pBOS and pEOS (respectively) to conduct sentence
+identification, as described in §4.2.
+Finally, we compare our proposed methods
+against sentence segmentation baselines which only
+utilize EOS labels.6 As for the baselines, we use
+the EOS labeling model developed in the same
+manner to segment the input text based on EOS.
+Note that we can optionally force the last word in
+the input to be EOS: in this case, the result will
+only contain SUs since all segments will end with
+EOS. By default, we do not force the last EOS: in
+this case, the segment after the last EOS (if exists)
+is considered as an NSU.
+As a default configuration, we use pCC = 0.5,
+pDA = 0.3, pT R = 0.1, and λ = 0.5 in our ex-
+periments. To ensure reproducibility, we report
+more details on the hyperparameters and model
+setup in Appendix D. For the precise procedure on
+how we convert between the word-, character-, and
+subword-level labels (for RoBERTa), we refer the
+readers to Appendix C.
+6.2
+Evaluation Setup
+In the evaluation phase, we consider three ways of
+assembling the input texts on which we conduct
+sentence identification. Firstly, we can apply the
+same unit concatenation procedure as described in
+6This EOS-only method is the most reasonable baseline to
+quantify the precise advantage from combining BOS labels in
+addition to EOS, which is proposed in our methods.
+§6.1. To be specific, we use pCC =0.5 (same as the
+training phase) and pCC = 0 (which concatenates
+the units up to the maximal length) to simulate both
+shorter and longer lengths of the input texts.
+However, this approach is relatively synthetic in
+the sense that we take the gold unit boundaries for
+granted. They are usually unavailable at the infer-
+ence time, so we should consider a more realistic
+setting for evaluating the methods without relying
+on the gold unit boundaries.
+To this end, we propose to evaluate sentence
+identification as a postprocessing of sentence seg-
+mentation. To be specific, we first apply the state-
+of-the-art method ERSATZ (Wicks and Post, 2021)
+on the raw text of EWT and then apply sentence
+identification to each segmented text. Note that ER-
+SATZ has high precision but still predicts false EOS
+which can fragment a gold SU: in such cases, we
+consider the fragmented SUs as NSUs and fix the
+labels accordingly (just as we did in unit truncation,
+cf. §6.1 and Table 4).
+As for the evaluation metrics, we convert the
+predictions of our methods into word/character-
+level BIO labels (cf. Appendix C) and compute
+the F1 score for each label prediction. Then, we
+summarize the results as the macro average F1
+and weighted average F1. We also compute the
+F1 score of the exact SU span extraction at the
+word/character-level. Finally, we run each exper-
+iment (from model training to testing) five times
+with different random seeds and report the average
+and standard deviation as the final results.
+7
+Results
+Table 5 summarizes the word-level evaluation re-
+sults. The results for the character-level evaluation
+show similar tendencies, so we put them in Ap-
+pendix E. The F1 score for each BIO label predic-
+tion is also available in Appendix E.
+Firstly, we take a look at the results when we
+have the in-domain sentence identification dataset
+(EWT Train/Dev) for model development. In this
+setup, we can verify that our proposed method
+(BOS&EOS) significantly outperforms the base-
+lines (EOS-Only) in all metrics. For instance, our
+method achieves consistently high performance of
+84∼89% F1 for the exact SU span extraction, both
+at the word- and character-level. This is a very
+promising result that demonstrates the effective-
+ness of our method when we can leverage the gold
+SUs and NSUs from the target domain.
+
+Peos (W2/W) = 0.65
+(W2/W≤2) = 0.08
+Unidirectional Model
+Bidirectional Model
+(Left-to-Right)
+This
+#
+This
+#
+#
+#
+is
+#
+#
+is
+wo
+W1
+W2
+wo
+W1
+W2
+W3
+W4
+W3
+W4Train/Dev
+Datasets
+Model
+EWT Test (pCC = 0.5)
+EWT Test (pCC = 0)
+EWT Test (Postprocess)
+BIO
+BIO
+Span
+BIO
+BIO
+Span
+BIO
+BIO
+Span
+Macro
+Weighted
+Macro
+Weighted
+Macro
+Weighted
+EWT
+Train/Dev
+EOS-Only
+83.2±1.5
+93.9±0.6
+72.8±1.8
+59.7±0.2
+86.4±0.1
+58.2±1.1
+86.3±2.7
+94.6±1.1
+81.6±2.4
+EOS-Only (force last)
+58.6±0.1
+86.6±0.0
+60.4±0.8
+57.6±0.2
+85.9±0.1
+57.7±1.0
+59.1±0.1
+85.7±0.0
+62.3±0.3
+BOS&EOS
+93.0±1.4
+97.3±0.6
+87.3±1.6
+91.0±1.8
+96.4±0.7
+84.1±2.6
+92.3±1.0
+96.7±0.4
+88.8±0.9
+WSJ
+Train/Dev
+EOS-Only
+71.7±0.7
+88.9±0.4
+59.2±2.4
+56.9±0.6
+85.2±0.3
+48.2±2.5
+71.5±0.3
+87.8±0.3
+67.8±0.3
+EOS-Only (force last)
+57.5±0.3
+86.2±0.2
+53.6±2.1
+55.4±0.7
+85.0±0.3
+48.2±2.5
+58.9±0.1
+85.7±0.0
+61.1±0.2
+EOS-Only (+AUG)
+66.4±1.5
+88.3±0.4
+59.5±1.4
+58.3±0.5
+86.1±0.3
+54.4±2.5
+71.1±1.3
+88.5±0.6
+66.2±1.9
+BOS&EOS
+71.5±0.2
+89.1±0.2
+59.1±1.5
+57.7±0.9
+85.4±0.2
+48.8±1.6
+71.0±0.3
+87.9±0.2
+68.4±0.3
+BOS&EOS (+UNI)
+70.4±0.7
+88.2±0.3
+60.0±1.1
+63.3±0.8
+86.0±0.4
+53.0±1.3
+70.8±0.4
+87.6±0.2
+68.4±0.1
+BOS&EOS (+UNI +AUG)
+72.5±0.4
+89.5±0.1
+66.6±0.2
+72.4±1.3
+89.1±0.5
+63.7±1.0
+74.3±1.1
+89.6±0.4
+71.9±1.4
+Table 5: Overall Results (Word-Level). We report the macro/weighted average F1 of the BIO labeling task and
+the F1 score of the exact SU span extraction task. Details of our experimental setup are discussed in §6.
+Secondly, we focus on the results where we
+only utilize the standard sentence segmentation
+dataset (WSJ Train/Dev) for model development.
+In this setup, we also report the results of applying
+our data augmentation (+AUG) and unidirectional
+model (+UNI) techniques from §6.1.7
+Due to the data discrepancy between WSJ and
+EWT, we find a natural drop in performance com-
+pared to the previous setup using in-domain EWT
+Train/Dev. However, we can verify that our tech-
+niques (+AUG, +UNI) generally help to alleviate
+this issue, and our proposed method performs on
+par or slightly better than the EOS-only baselines
+when applying these techniques. It is especially
+worth noting the improvement in the exact SU
+span extraction task (reaching 64∼72% F1), where
+the advantage of our method is the most conspic-
+uous and consistent in both word- and character-
+level evaluation. This improvement can also be
+explained by the higher performance in the B-label
+prediction with our method (Appendix E), which
+is a prerequisite for accurate SU span extraction.
+Finally, we note that the EOS-only baseline with-
+out forcing the last EOS can be quite competitive
+with shorter inputs (pCC = 0.5 and postprocessing)
+but performs considerably worse when the input
+texts are longer (pCC = 0). This is because the
+baseline can only predict the last segment of the
+input as an NSU, which is less problematic when
+the input texts are shorter but becomes increasingly
+problematic with longer inputs (since most NSUs
+will not be able to be removed). In contrast, our
+proposed method performs much more robustly
+under various input lengths.
+7We did not observe any improvement from applying these
+techniques to the in-domain dataset (EWT Train/Dev), which
+is consistent with our motivation and expectation.
+Through further experiments and analyses, we
+found that (i) the results are stable across different
+hyperparameter choices, (ii) predictions are reason-
+able especially when using the in-domain dataset
+(EWT Train/Dev) for model development, and (iii)
+our methods do not sacrifice performance on the
+formal/edited texts of the sentence segmentation
+dataset (WSJ Test). These detailed evidences can
+be found in Appendix F.
+8
+Conclusion
+In this paper, we introduced a novel task of sen-
+tence identification, where we aim to identify SUs
+while excluding NSUs in a given text (§3). Through
+sentence identification, we can clearly distinguish
+the portions of the text that are appropriate (or not)
+for the prediction and evaluation of sophisticated
+linguistic analyses, such as dependency parsing,
+semantic role labeling, etc.
+To conduct sentence identification, we proposed
+a simple yet effective method of combining the
+BOS and EOS labeling models to determine the
+SUs and NSUs (§4). To evaluate sentence iden-
+tification, we designed an automatic, language-
+independent procedure to convert the UD corpora
+into sentence identification benchmarks (§5).
+In our experiments, we developed the BOS/EOS
+labeling models by finetuning pretrained RoBERTa
+(§6). Based on the experimental results, we showed
+that our proposed method combining the BOS and
+EOS labels outperforms sentence segmentation
+baselines which only utilize EOS labels in all of
+the considered settings (§7). Overall, we expect
+sentence identification to be a fundamental frame-
+work for the preprocessing of noisy, informal, or
+non-standard texts in the real world.
+
+Limitations
+Firstly, our current experiments are limited to En-
+glish and cover only five domains of web media
+texts in EWT. However, our task formulation (§3),
+method (§4), and evaluation framework (§5) are
+fully agnostic to the language and domain. Hence
+it is straightforward to conduct experiments in dif-
+ferent languages or domains (as long as they are
+supported in the UD). While we expect similar re-
+sults with different languages/domains, we leave
+further investigation as a future work.
+Secondly, while our method performs reliably
+when the in-domain dataset is available, there is
+still a huge room left for improvement without re-
+lying on such resources (e.g. only using the stan-
+dard sentence segmentation dataset). To make our
+method fully practical, we still need to improve on
+the accuracy and robustness in such cross-domain
+scenarios. One potential approach is to refine the
+definitions of SU and NSU probabilities from §4.2
+to make sentence identification more robust. For
+instance, we can incorporate span-level scores in-
+stead of only using word-level BOS/EOS probabil-
+ities to define the SU/NSU probabilities. We leave
+further improvement and extension of our approach
+as an important future work.
+Finally, our methods are currently evaluated on
+the (exact) SU span extraction task. Ideally, we
+should also evaluate the methods on downstream
+applications such as POS tagging, syntactic pars-
+ing, semantic role labeling, etc. However, we still
+expect that the (exact) SU span extraction will play
+a primary role in the evaluation, since accurate
+(say human-level) identification of SUs/NSUs will
+likely provide unprecedented benefits on a wide va-
+riety of NLP applications dealing with real-world
+texts. While we leave the precise analyses on down-
+stream applications as future work, our contribu-
+tions make the first foundational step towards ex-
+panding the capability of the long-established sen-
+tence segmentation task.
+References
+Wirote Aroonmanakun et al. 2007. Thoughts on word
+and sentence segmentation in thai. In Proceedings
+of the Seventh Symposium on Natural language Pro-
+cessing, pages 85–90.
+Marie-Catherine de Marneffe, Christopher D. Man-
+ning, Joakim Nivre, and Daniel Zeman. 2021. Uni-
+versal Dependencies.
+Computational Linguistics,
+47(2):255–308.
+Timothy Dozat and Christopher D Manning. 2017.
+Deep biaffine attention for neural dependency pars-
+ing. In Proc. of ICLR.
+Timothy Dozat and Christopher D. Manning. 2018.
+Simpler but more accurate semantic dependency
+parsing. In Proceedings of the 56th Annual Meet-
+ing of the Association for Computational Linguis-
+tics (Volume 2: Short Papers), pages 484–490, Mel-
+bourne, Australia. Association for Computational
+Linguistics.
+Dan Gillick. 2009. Sentence boundary detection and
+the problem with the U.S. In Proceedings of Human
+Language Technologies: The 2009 Annual Confer-
+ence of the North American Chapter of the Associa-
+tion for Computational Linguistics, Companion Vol-
+ume: Short Papers, pages 241–244, Boulder, Col-
+orado. Association for Computational Linguistics.
+Mandar Joshi, Danqi Chen, Yinhan Liu, Daniel S.
+Weld, Luke Zettlemoyer, and Omer Levy. 2020.
+SpanBERT: Improving pre-training by representing
+and predicting spans. Transactions of the Associa-
+tion for Computational Linguistics, 8:64–77.
+Tibor Kiss and Jan Strunk. 2006. Unsupervised mul-
+tilingual sentence boundary detection.
+Computa-
+tional Linguistics, 32(4):485–525.
+Philipp Koehn, Hieu Hoang, Alexandra Birch, Chris
+Callison-Burch, Marcello Federico, Nicola Bertoldi,
+Brooke Cowan,
+Wade Shen,
+Christine Moran,
+Richard Zens, Chris Dyer, Ondˇrej Bojar, Alexandra
+Constantin, and Evan Herbst. 2007. Moses: Open
+source toolkit for statistical machine translation. In
+Proceedings of the 45th Annual Meeting of the As-
+sociation for Computational Linguistics Companion
+Volume Proceedings of the Demo and Poster Ses-
+sions, pages 177–180, Prague, Czech Republic. As-
+sociation for Computational Linguistics.
+Yinhan Liu, Jiatao Gu, Naman Goyal, Xian Li, Sergey
+Edunov, Marjan Ghazvininejad, Mike Lewis, and
+Luke Zettlemoyer. 2020.
+Multilingual denoising
+pre-training for neural machine translation. Transac-
+tions of the Association for Computational Linguis-
+tics, 8:726–742.
+Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Man-
+dar Joshi, Danqi Chen, Omer Levy, Mike Lewis,
+Luke Zettlemoyer, and Veselin Stoyanov. 2019.
+RoBERTa: A robustly optimized BERT pretraining
+approach. arXiv preprint arXiv:1907.11692.
+
+Mitchell P. Marcus, Beatrice Santorini, and Mary Ann
+Marcinkiewicz. 1993.
+Building a large annotated
+corpus of English: The Penn Treebank. Computa-
+tional Linguistics, 19(2):313–330.
+Geoffrey Nunberg. 1990. The linguistics of punctua-
+tion. 18. Center for the Study of Language (CSLI).
+Jonathon Read, Rebecca Dridan, Stephan Oepen, and
+Lars Jørgen Solberg. 2012. Sentence boundary de-
+tection: A long solved problem? In Proceedings of
+COLING 2012: Posters, pages 985–994, Mumbai,
+India. The COLING 2012 Organizing Committee.
+Ines
+Rehbein,
+Josef
+Ruppenhofer,
+and
+Thomas
+Schmidt. 2020. Improving sentence boundary detec-
+tion for spoken language transcripts. In Proceedings
+of the Twelfth Language Resources and Evaluation
+Conference, pages 7102–7111, Marseille, France.
+European Language Resources Association.
+Michael D. Riley. 1989.
+Some applications of tree-
+based modelling to speech and language. In Speech
+and Natural Language: Proceedings of a Workshop
+Held at Cape Cod, Massachusetts, October 15-18,
+1989.
+Dwijen Rudrapal, Anupam Jamatia, Kunal Chakma,
+Amitava Das, and Björn Gambäck. 2015.
+Sen-
+tence boundary detection for social media text. In
+Proceedings of the 12th International Conference
+on Natural Language Processing, pages 254–260,
+Trivandrum, India. NLP Association of India.
+Natalia Silveira,
+Timothy Dozat,
+Marie-Catherine
+de Marneffe, Samuel Bowman, Miriam Connor,
+John Bauer, and Chris Manning. 2014. A gold stan-
+dard dependency corpus for English. In Proceedings
+of the Ninth International Conference on Language
+Resources and Evaluation (LREC’14), pages 2897–
+2904, Reykjavik, Iceland. European Language Re-
+sources Association (ELRA).
+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob
+Uszkoreit, Llion Jones, Aidan N Gomez, Ł ukasz
+Kaiser, and Illia Polosukhin. 2017. Attention is all
+you need. In Proc. of NeurIPS.
+Daniel J. Walker, David E. Clements, Maki Darwin,
+and Jan W. Amtrup. 2001. Sentence boundary de-
+tection: a comparison of paradigms for improving
+MT quality. In Proceedings of Machine Translation
+Summit VIII, Santiago de Compostela, Spain.
+Xiaolin Wang, Masao Utiyama, and Eiichiro Sumita.
+2019. Online sentence segmentation for simultane-
+ous interpretation using multi-shifted recurrent neu-
+ral network. In Proceedings of Machine Translation
+Summit XVII: Research Track, pages 1–11, Dublin,
+Ireland. European Association for Machine Transla-
+tion.
+Rachel Wicks and Matt Post. 2021. A unified approach
+to sentence segmentation of punctuated text in many
+languages. In Proceedings of the 59th Annual Meet-
+ing of the Association for Computational Linguistics
+and the 11th International Joint Conference on Nat-
+ural Language Processing (Volume 1: Long Papers),
+pages 3995–4007, Online. Association for Computa-
+tional Linguistics.
+Nianwen Xue and Yaqin Yang. 2011.
+Chinese sen-
+tence segmentation as comma classification. In Pro-
+ceedings of the 49th Annual Meeting of the Associ-
+ation for Computational Linguistics: Human Lan-
+guage Technologies, pages 631–635, Portland, Ore-
+gon, USA. Association for Computational Linguis-
+tics.
+Nina Zhou, AiTi Aw, Nattadaporn Lertcheva, and Xu-
+ancong Wang. 2016. A word labeling approach to
+Thai sentence boundary detection and POS tagging.
+In Proceedings of COLING 2016, the 26th Inter-
+national Conference on Computational Linguistics:
+Technical Papers, pages 319–327, Osaka, Japan. The
+COLING 2016 Organizing Committee.
+
+A
+Dynamic Programming Algorithm
+To find the maximum value (and the argmax) of
+Eq. 4 from §4.2, we rely on a simple dynamic
+programming framework. To be specific, we con-
+sider the partial labeling of BOS and EOS up to
+W ≤k = (w0, ..., wk), where k ≤ N − 1. Then,
+we aim to compute the maximum log probability
+of Eq. 4 based on the partial labeling, i.e. using
+W ≤k in place of W .
+Since the labeling is partial, W ≤k may end in-
+side the SU (i.e. the last label is BOS) or outside the
+SU (i.e. the last label is EOS). Let log pIS(k+1) de-
+note the maximum log probability when W ≤k ends
+inside the SU and log pOS(k + 1) the maximum
+log probability when W ≤k ends outside the SU.
+Then, we can initialize log pIS(0) = log 0 = −∞,
+log pOS(0) = log 1 = 0 (since we always start out-
+side the SU) and iteratively update the two values
+as follows:
+log p′
+IS(i) = max { log pIS(i) + log (1−pBOS(wi)),
+log pOS(i) + log pBOS(wi) }
+log p′
+OS(i) = log pOS(i) + log (1−pBOS(wi))
+log pIS(i+1) = log p′
+IS(i) + log (1−pEOS(wi))
+log pOS(i+1) = max { log p′
+IS(i) + log pEOS(wi),
+log p′
+OS(i) + log (1−pEOS(wi)) }
+(5)
+Note that we first update pIS(i) → p′
+IS(i) and
+pOS(i) → p′
+OS(i) based on the BOS probability
+pBOS(wi). Then, we update p′
+IS(i)→pIS(i+1) and
+p′
+OS(i)→pOS(i+1) based on the EOS probability
+pEOS(wi).8 The iterative procedure is illustrated in
+Figure 2.
+Finally, we can compute the log probability
+log pOS(N) (since we always end outside the SU),
+which corresponds to the maximum value of Eq. 4.
+To obtain the argmax, we can simply incorporate
+backtracking during the iterative updates of Eq. 5.
+Through this dynamic programming framework,
+we can ensure that the restrictions from §4.2 are
+satisfied: namely, (i) the first label should be BOS,
+(ii) the last label should be EOS, and (iii) BOS and
+EOS labels need to appear alternately.
+In practice, we can limit the candidates of BOS
+indices to the subset where pBOS(wi) is higher
+than a certain threshold c. This can be efficiently
+implemented by simply skipping the updates of
+p′
+IS(i) and p′
+OS(i), i.e. using p′
+IS(i) = pIS(i) and
+8Note that if a single word wi is labeled as both BOS and
+EOS at the same time, we can extract it as a single SU.
+p′
+OS(i) = pOS(i), if pBOS(wi) < c.9 Likewise, we
+can limit the candidates of EOS indices by skip-
+ping the updates of pIS(i + 1) and pOS(i + 1) if
+pEOS(wi) < c. Generally speaking, this leads to
+a more efficient algorithm: therefore, we use the
+candidate threshold of c = 0.1 for restricting both
+BOS and EOS indices throughout our experiments.
+B
+SU and NSU Examples
+In Table 6, we provide more examples of SUs and
+NSUs identified based on our procedure described
+in §5. As for the SUs, we can verify that EWT con-
+tains clean, formal SUs with appropriate capitaliza-
+tion and punctuation. We can also verify that EWT
+contains various types of informal SUs, e.g. that
+lack capitalization/punctuation, use non-standard
+casing, end with emoticons, include spelling errors,
+concatenate consecutive SUs without a space, etc.
+C
+Label Assignment and Conversion
+In this section, we explain the precise procedure
+on how we (i) assign the gold character-level
+labels, (ii) convert the character-level labels to
+word/subword-level labels, and (iii) convert the
+subword-level labels to character/word-level labels.
+We limit our explanation to BIO labels, since it is
+straightforward to convert them to the combination
+of BOS and EOS labels (and vice versa).
+Firstly, we can assign the gold character-level
+labels from the UD annotation by taking the
+character-level alignment, which determines the
+exact spans of SUs and NSUs. From the character-
+level labels, we can assign the word- or subword-
+level labels based on the following rule:
+• If the word (or subword) contains a character
+with the B-label, assign it the B-label.
+• Else if it contains a character with the I-label,
+assign the I-label.
+• Otherwise assign the O-label.
+For instance, this procedure is used to create the
+subword-level labels for training our BOS/EOS
+labeling models.
+To evaluate our methods, we need to convert
+the subword-level labels produced by our methods
+into the character-level labels, which can then be
+converted into the word-level labels (based on the
+9This is equivalent to forcing wi to be non-BOS, i.e. set-
+ting pBOS(wi) = 0 in Eq. 5.
+
+Figure 2: Illustration of the dynamic programming procedure.
+SUs
+President Bush on Tuesday nominated two individuals to replace retiring jurists on federal courts in the Washington area.
+Unfortunately, Mr. Lay will be in San Jose, CA participating in a conference, where he is a speaker, on June 14.
+“In 1972, there was an enormous glut of pilots,” Campenni says.
+PS – There is a happy hour tonight at Scudeiros on Dallas Street (just west of the Met Garage) beginning around 5:00.
+2) Your vet would not prescribe them if they didn’t think it would be helpful.
+BUT EVERYONE HAS THERE OWN WAY!!!!!!
+The motel is very well maintained, and the managers are so accomodating, it’s kind of like visiting family each year! ;-)
+where can I find the best tours to the Mekong Delta at reasonable prices?
+it seems like its healthier too, but its prolly not.
+I have wifi at my house, but thats just at my house...is there anyway i can buy some card to make the ipod itself have wifi?
+NSUs
+—->===}*{===<—-
+- Lisa_coverletter.doc << File: Lisa_coverletter.doc >>
+Thur. Sept. 28 - Paris (Versailles or Fontainbleu - half day side trip)
+9.3m - Number of US unemployed in April 2004.
+Game 1: Monday, May 28 @ 2:00PM vs. Los Angeles SPARKS
+Mixed Tempura.....................8.25 Shrimp or vegetable tempura & salad.
+Infinity stereo, bucket seats, nerf bars, tool box, bed liner, camper tow package, 5 speed manual.
+printing, printing, copies, printing, copies, printing,
+A++++ !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
+Dear Sir / Madam,
+Table 6: Examples of gold SUs and NSUs in the English Web Treebank (EWT) identified based on our procedure
+(§5). Each line corresponds to one example of SU or NSU.
+previous procedure). To convert a subword-level
+label into a sequence of character-level labels, we
+apply the following rule (where n denotes the num-
+ber of characters in the subword):
+• If the subword has the B-label, the character-
+level labels are 1 B-label followed by n − 1
+I-labels.
+• If the subword has the I-label, the character-
+level labels are n I-labels.
+• If the subword has the O-label, the character-
+level labels are n O-labels.
+D
+Details on the Model Setup
+As discussed in §6.1, we finetune the pretrained
+RoBERTaBASE publicly available on the Hugging-
+Face model hub10. We add a binary BOS/EOS
+classifier on top of the encoder, which is a single-
+layer MLP with a hidden size of 768. We share the
+10https://huggingface.co/models
+encoder parameters and use different classifiers for
+the BOS/EOS predictions. The BOS/EOS models
+are trained jointly by summing their losses.
+When we combine the unidirectional models
+(+UNI), we take the same approach and use differ-
+ent classifiers for the unidirectional/bidirectional
+models. Again, the encoder parameters are shared
+and all models are trained jointly.
+As for the training data preparation, we apply the
+unit concatenation and data augmentation (+AUG)
+on the fly, i.e. we see different concatenation and
+augmentation of the units in each iteration. The
+same procedure is applied on the validation set.
+During data augmentation, we remove the last
+sentence ending punctuation based on the following
+regular-expression, similar to the candidate bound-
+ary detector in ERSATZ (Wicks and Post, 2021):
+• (.∗PeP ∗) where P denotes the set of punctua-
+tions and Pe ⊂ P denotes the sentence ending
+punctuations.
+Since our experiments are conducted on English,
+
+Update based on PBos(wi)
+Update based on peos(wi)
+In-sentence: pis(i)
+pis(i) = max (pis(i) * (1 - pBos(wi),
+Pis(i + 1) = pis(i) * (1 -peos (wi))
+1 - PBos(wi)
+1 - PEos(wi)
+Pos(i) * pBos(wi) )
+PEos(wi)
+PBos(wi)
+Text 
+Out-of-sentence: pos(i)
+Pos(i + 1) = max (pis(i) * peos(wi)
+pos(i) = pos(i) * (1 - pBos (wi)
+1 - PBos(wi)
+1 - peos(wi)
+ps(i) * (1 - peos(wi)) )
+1-m . Tm "om Jae (ns apisul=) s u! buiaq jo K!qeoud :(2)sid
+Return: Pos(N)
+Start: pis(O) = 0, Pos(0) = 1
+Pos(i): probability of being in OS (=outside SU) after Wo, Wi, ..Wi-1we use P = {.?!")’} and Pe = {.?!}.
+Finally, all models are implemented in Pytorch
+and trained on a single Tesla V100-SXM2-32GB
+GPU. We use a batch size of 8, accumulate the
+gradients for 32 batches, and apply the gradient
+clipping at 1.0 before updating the model weights.
+As for the optimizer, we use Adam with the initial
+learning rate of 0.0001 and exponentially decay the
+learning rate by γ = 0.95 after each epoch. We
+check the validation loss every 200 batches and
+stop the training early if there is no improvement
+for 5 consecutive evaluations.
+E
+The Full Experimental Results
+In this section, we report the full results of our
+experiments which did not fit in §7. Table 7 shows
+the word-level F1 scores for each B-, I-, and O-
+label prediction. Table 8 shows the overall results
+for the character-level evaluation.
+Generally speaking, we can confirm the same
+results as observed in §7. Firstly, our proposed
+method significantly outperforms the baselines
+when we use the EWT Train/Dev dataset for model
+development.
+Secondly, our method performs
+slightly better than (or at least on par with) the
+baselines when developed on the WSJ Train/Dev
+dataset. Finally, the baseline without forcing the
+last EOS is competitive with shorter inputs (pCC =
+0.5 and postprocessing) but performs considerably
+worse when the input texts are longer (pCC = 0).
+F
+Further Experiments and Analyses
+In this section, we provide further experiments
+and analyses to complement our study. To be spe-
+cific, we provide discussions on the effect of the
+choice of hyperparameters (F.1), qualitative anal-
+yses based on example model outputs (F.2), and
+evaluation of sentence identification based on the
+sentence segmentation dataset (F.3).
+F.1
+Effect of Hyperparameters
+As a default configuration, we used pDA
+=
+0.3, pT R =0.1 for the data augmentation (+AUG)
+and λ = 0.5 for the unidirectional model ensem-
+bling (+UNI). To examine the effect of the choice
+of these hyperparameters, we conducted further
+experiments by changing these default hyperpa-
+rameters. Note that all evaluation results in this
+subsection are based on BOS&EOS (+UNI +AUG)
+developed on WSJ Train/Dev.
+Firstly, we focus on the data augmentation and
+report the results of our method trained with dif-
+ferent sets of pDA and pT R (with λ fixed at 0.5).
+Since increasing pDA leads to higher recall (and
+lower precision) of SU extraction and increasing
+pT R leads to higher precision (and lower recall),
+we used a fixed ratio of pDA : pT R = 3 : 1 which
+seemed to make a good trade-off. As shown in
+Table 9, the results are generally stable with the
+different choices of the hyperparameters. However,
+more data augmentation (with larger values of pDA
+and pT R) tends to slightly improve the performance,
+especially for the exact SU span extraction.
+Secondly, we focus on the unidirectional model
+ensembling and report the results of changing the
+linear interpolation rate λ ∈ [0, 1], where λ = 0 is
+equivalent to using only the bidirectional models
+and λ = 1 only the unidirectional models. We fix
+pDA =0.3 and pT R =0.1 and only change λ at the
+inference time without retraining the unidirectional
+or bidirectional models. As shown in Figure 3, we
+found that unidirectional and bidirectional models
+generally have complementary benefits, and choos-
+ing the intermediate value of λ leads to the best
+performance. The results also indicate that we may
+be able to obtain further improvement by tuning
+λ on the validation set, although we simply fixed
+λ = 0.5 throughout our experiments.
+F.2
+Qualitative Analyses
+In Table 10 and 11, we show the actual predictions
+made by our proposed method developed on EWT
+Train/Dev and WSJ Train/Dev. For the latter, we
+applied +UNI and +AUG with the default hyperpa-
+rameters.
+In the first example (Table 10), we can verify
+that both models identify the correct SU span while
+removing the non-sentential header as the NSU.
+This is a relatively easy example, since the start of
+the SU is capitalized and less ambiguous.
+In the second example (Table 11), we can ob-
+serve that our method using in-domain data (EWT
+Train/Dev) extracts the correct SU span, while our
+method developed on out-of-domain data (WSJ
+Train/Dev) incorrectly excludes a part of an SU.
+This seems to be a relatively difficult example,
+since the start of the SU is not capitalized and more
+ambiguous. It is worth noting that such SUs can
+be reliably extracted when we can leverage the in-
+domain annotation of gold SUs and NSUs.
+
+Train/Dev
+Datasets
+Model
+EWT Test (pCC = 0.5)
+EWT Test (pCC = 0.0)
+EWT Test (Postprocess)
+B-Label
+I-Label
+O-Label
+B-Label
+I-Label
+O-Label
+B-Label
+I-Label
+O-Label
+EWT
+Train/Dev
+EOS-Only
+85.6±0.8
+97.3±0.3
+66.6±3.5
+78.0±0.6
+95.1±0.1
+6.0±0.4
+90.2±1.2
+97.5±0.5
+71.3±6.6
+EOS-Only (force last)
+79.8±0.2
+95.9±0.0
+0.0±0.0
+77.8±0.6
+95.1±0.1
+0.0±0.0
+81.7±0.2
+95.7±0.0
+0.0±0.0
+BOS&EOS
+94.3±0.6
+98.7±0.3
+86.1±3.5
+93.0±1.0
+98.2±0.4
+81.7±4.0
+94.7±0.3
+98.4±0.2
+83.9±2.4
+WSJ
+Train/Dev
+EOS-Only
+78.7±1.3
+94.4±0.4
+42.1±1.3
+71.8±1.9
+94.3±0.2
+4.5±0.4
+83.3±0.2
+93.8±0.3
+37.3±0.9
+EOS-Only (force last)
+76.7±0.9
+95.6±0.1
+0.0±0.0
+71.7±1.9
+94.6±0.2
+0.0±0.0
+81.0±0.4
+95.6±0.0
+0.0±0.0
+EOS-Only (+AUG)
+79.4±1.0
+95.4±0.2
+24.5±4.1
+78.1±1.8
+93.3±0.2
+1.4±1.1
+82.7±1.1
+94.9±0.5
+35.6±3.2
+BOS&EOS
+79.4±0.9
+94.8±0.2
+40.5±0.6
+72.9±1.2
+94.3±0.2
+5.8±2.0
+83.9±0.2
+94.1±0.2
+35.2±0.9
+BOS&EOS (+UNI)
+79.8±0.6
+93.9±0.2
+37.5±1.5
+76.2±1.3
+93.3±0.3
+20.2±1.2
+83.8±0.1
+93.8±0.1
+34.7±0.9
+BOS&EOS (+UNI +AUG)
+83.7±0.1
+95.0±0.2
+38.7±1.2
+83.0±0.6
+94.6±0.3
+39.7±3.5
+85.9±0.6
+95.2±0.2
+41.9±2.8
+Table 7: BIO Labeling Results (Word-Level). We report the F1 scores for each B-, I- and O-label prediction.
+Train/Dev
+Datasets
+Model
+EWT Test (pCC = 0.5)
+EWT Test (pCC = 0.0)
+EWT Test (Postprocess)
+BIO
+BIO
+Span
+BIO
+BIO
+Span
+BIO
+BIO
+Span
+Macro
+Weighted
+Macro
+Weighted
+Macro
+Weighted
+EWT
+Train/Dev
+EOS-Only
+83.8±1.1
+92.7±0.5
+72.8±1.8
+58.5±0.2
+81.5±0.0
+58.2±1.1
+87.7±2.3
+93.9±1.2
+81.6±2.4
+EOS-Only (force last)
+57.7±0.1
+81.0±0.0
+60.4±0.8
+56.9±0.2
+80.9±0.0
+57.7±1.0
+58.1±0.1
+79.9±0.0
+62.3±0.3
+BOS&EOS
+94.0±1.0
+97.2±0.6
+87.3±1.6
+92.2±1.5
+96.3±0.7
+84.1±2.6
+93.5±0.6
+96.6±0.4
+88.9±0.8
+WSJ
+Train/Dev
+EOS-Only
+72.8±0.6
+86.9±0.4
+59.1±2.3
+56.0±0.6
+80.9±0.1
+48.2±2.5
+73.3±0.4
+85.6±0.2
+67.7±0.4
+EOS-Only (force last)
+56.6±0.3
+80.9±0.0
+53.5±2.0
+54.9±0.6
+80.7±0.1
+48.2±2.5
+57.8±0.2
+79.9±0.0
+61.0±0.3
+EOS-Only (+AUG)
+64.3±1.5
+83.5±0.6
+59.5±1.4
+57.4±0.5
+81.0±0.2
+54.4±2.5
+69.2±1.7
+84.0±0.8
+66.2±1.9
+BOS&EOS
+72.7±0.7
+87.1±0.2
+59.1±1.5
+57.8±1.9
+81.6±0.7
+48.8±1.6
+72.4±1.0
+85.2±0.5
+68.3±0.3
+BOS&EOS (+UNI)
+72.4±0.6
+86.3±0.3
+59.6±1.0
+65.3±1.0
+83.6±0.5
+52.9±1.3
+72.8±0.4
+85.3±0.2
+68.0±0.2
+BOS&EOS (+UNI +AUG)
+72.2±1.3
+86.1±0.6
+66.5±0.3
+72.8±1.8
+86.5±0.9
+63.6±1.0
+73.2±1.9
+85.7±0.9
+71.8±1.5
+Table 8: Overall Results (Character-Level). We report the macro/weighted average F1 of the BIO labeling task
+and the F1 score of the exact SU span extraction task.
+F.3
+Evaluation on the Sentence Segmentation
+Dataset
+Finally, we report the results of sentence identifica-
+tion on the standard sentence segmentation dataset
+(WSJ Test).
+In Table 12, we summarize the WSJ dataset
+statistics. Note that WSJ only contains SUs and
+do not contain any NSUs (O-labels). However, we
+can still evaluate the performance using the same
+metrics, i.e. the macro/weighted average F1 of the
+BIO labeling task and the F1 of the exact SU span
+extraction task.11
+Table 13 summarizes the word-level evaluation
+results. Since we are evaluating on WSJ Test, the
+performance is naturally better when the models
+are trained on WSJ Train/Dev rather than EWT
+Train/Dev (which is now out-of-domain).
+When the models are trained on EWT, we found
+that the baseline (EOS-Only) forcing the last EOS
+performs the best. This is natural, since this base-
+line better reflects the nature of the sentence seg-
+mentation dataset where all units are SUs. How-
+11Since the O-label does not exist, we report the macro
+average F1 as the average F1 scores of the B-label and I-label
+predictions.
+ever, our method (BOS&EOS) is still comparable
+to this baseline and do not (or minimally) sacrifice
+performance on such datasets.
+When the models are trained on WSJ, we found
+that our method without +UNI or +AUG performs
+the best. This is most likely because we can lever-
+age the knowledge of BOS to predict EOS. When
+we apply the data augmentation (+AUG) and uni-
+directional model ensembling (+UNI), we observe
+a slight decrease in performance compared to our
+vanilla method. However, the results are still com-
+parable and even outperforms the baselines in some
+metrics (e.g. the exact SU span extraction task).
+Overall, we can conclude that our methods do
+not sacrifice the performance on the the clean,
+edited texts of the sentence segmentation dataset.
+
+Evaluation
+Augmentation Rates
+EWT Test (pCC = 0.5)
+EWT Test (pCC = 0)
+EWT Test (Postprocess)
+BIO
+BIO
+Span
+BIO
+BIO
+Span
+BIO
+BIO
+Span
+Macro
+Weighted
+Macro
+Weighted
+Macro
+Weighted
+Word-Level
+pDA =0.15, pT R =0.05
+71.3±1.1
+89.0±0.5
+65.7±1.3
+71.5±0.9
+88.6±0.5
+62.3±1.7
+73.5±1.4
+89.2±0.6
+71.2±1.8
+pDA =0.3, pT R =0.1
+72.5±0.4
+89.5±0.1
+66.6±0.2
+72.4±1.3
+89.1±0.5
+63.7±1.0
+74.3±1.1
+89.6±0.4
+71.9±1.4
+pDA =0.45, pT R =0.15
+73.2±1.0
+90.0±0.1
+67.3±0.8
+73.0±0.9
+89.5±0.6
+64.0±1.8
+75.1±1.3
+90.0±0.4
+72.1±0.7
+Character-
+Level
+pDA =0.15, pT R =0.05
+72.3±1.9
+86.3±1.0
+65.4±1.4
+73.6±0.7
+86.8±0.3
+62.2±1.7
+73.8±1.5
+86.1±0.8
+71.0±1.8
+pDA =0.3, pT R =0.1
+72.2±1.3
+86.1±0.6
+66.5±0.3
+72.8±1.8
+86.5±0.9
+63.6±1.0
+73.2±1.9
+85.7±0.9
+71.8±1.5
+pDA =0.45, pT R =0.15
+71.9±1.1
+86.1±0.3
+67.2±0.8
+72.3±0.9
+86.3±0.6
+64.0±1.8
+73.6±1.5
+86.0±0.6
+72.1±0.7
+Table 9: Effect of Data Augmentation Rates (Word/Character-Level).
+We use different data augmentation
+rates (pDA and pT R) and evaluate BOS&EOS (+UNI +AUG) developed on WSJ Train/Dev.
+We report the
+macro/weighted average F1 of the BIO labeling task and the F1 score of the exact SU span extraction task.
+B
+Developed
+... 06/04/2001 05:54 PM Can you pass this along to Elizabeth to ensure Sanders
+on EWT
+E
+is on board as well?
+B
+Developed
+... 06/04/2001 05:54 PM Can you pass this along to Elizabeth to ensure Sanders
+on WSJ
+E
+is on board as well?
+Table 10: Example Outputs (Both Correct). We show the predictions made by our proposed method (BOS&EOS)
+developed on EWT Train/Dev (top) or WSJ Train/Dev (bottom). We can verify that both methods identify the
+correct SU span while removing the non-sentential header as the NSU.
+B
+Developed
+with my breakfast I like bacon and sausage when I having a big breakfast like
+on EWT
+E
+a grand slam with pancakes and the works.
+B
+Developed
+with my breakfast I like bacon and sausage when I having a big breakfast like
+on WSJ
+E
+a grand slam with pancakes and the works.
+Table 11: Example Output with One Incorrect Case. We show the predictions made by our proposed method
+(BOS&EOS) developed on EWT Train/Dev (top) or WSJ Train/Dev (bottom). We can verify that the former
+extracts the correct SU span, while the latter incorrectly excludes the first prepositional phrase as an NSU.
+Train
+Dev
+Test
+Total SUs
+37,447
+2,021
+7,442
+Total NSUs
+0
+0
+0
+Word-Level
+B-Label
+37,447
+2,021
+7,442
+I-Label
+805,387
+44,354
+163,132
+O-Label
+0
+0
+0
+Character-Level
+B-Label
+37,447
+2,021
+7,442
+I-Label
+4,308,729
+236,798
+876,461
+O-Label
+0
+0
+0
+Table 12: WSJ dataset statistics.
+
+(a) EWT Test (pCC =0.5), BIO Macro (b) EWT Test (pCC =0.5), BIO Weighted
+(c) EWT Test (pCC =0.5), Span
+(d) EWT Test (pCC =0.0), BIO Macro (e) EWT Test (pCC =0.0), BIO Weighted
+(f) EWT Test (pCC =0.0), Span
+(g) EWT Test (Postproc.), BIO Macro
+(h) EWT Test (Postproc.), BIO Weighted
+(i) EWT Test (Postproc.), Span
+Figure 3: Effect of the Unidirectional Model Interpolation Rate (Word-Level). We change λ ∈ [0, 1] and report
+the macro/weighted average F1 of the BIO labeling task and the F1 score of the exact SU span extraction task.
+Interpolated results are shown in blue and non-interpolated results (i.e. λ = 0) shown in red. The line shows the
+mean and the shade shows the standard deviation from the five experimental runs.
+Train/Dev
+Datasets
+Model
+WSJ Test (pCC = 0.5)
+WSJ Test (pCC = 0)
+BIO
+BIO
+Span
+BIO
+BIO
+Span
+Macro
+Weighted
+Macro
+Weighted
+EWT
+Train/Dev
+EOS-Only
+97.4±0.1
+99.5±0.0
+87.3±0.3
+97.3±0.0
+99.5±0.0
+87.2±0.2
+EOS-Only (force last)
+97.6±0.1
+99.9±0.0
+87.8±0.3
+97.3±0.0
+99.6±0.0
+87.3±0.2
+BOS&EOS
+97.1±0.2
+99.4±0.0
+86.7±0.5
+97.0±0.1
+99.3±0.0
+86.5±0.3
+WSJ
+Train/Dev
+EOS-Only
+98.4±0.6
+99.7±0.1
+92.1±2.9
+98.2±0.4
+99.7±0.1
+90.6±1.8
+EOS-Only (force last)
+98.4±0.6
+99.7±0.1
+92.1±2.9
+98.2±0.4
+99.7±0.1
+90.6±1.8
+EOS-Only (+AUG)
+98.2±1.1
+99.1±1.0
+92.6±2.5
+97.3±1.9
+99.3±0.8
+87.8±6.3
+BOS&EOS
+99.2±0.2
+99.7±0.3
+95.5±0.5
+98.7±0.1
+99.7±0.2
+93.1±0.4
+BOS&EOS (+UNI)
+98.5±0.3
+98.9±0.5
+92.9±1.0
+98.1±0.3
+98.8±0.5
+91.4±0.8
+BOS&EOS (+UNI +AUG)
+98.7±0.2
+99.3±0.4
+94.0±0.7
+98.2±0.3
+99.1±0.3
+91.8±1.1
+Table 13: Overall Results on WSJ Test (RoBERTa, Word-Level). We report the macro/weighted average F1 of
+the BIO labeling task and the F1 score of the exact SU span extraction task.
+
+0.90
+0.89
+0.88
+0.87
+0.86
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+入0.73
+0.72
+0.71
+0.70
+0.69
+0.68
+0.67
+0.0 
+0.2 
+0.4
+0.6
+0.8
+1.00.73
+0.72
+0.71
+0.70
+0.69
+0.68
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+入0.90
+0.89
+0.88
+0.87
+0.86
+0.85
+0.0
+0.2
+0.4
+0.6
+0.8
+1.00.66
+0.64
+0.62
+0.60
+0.58
+0.56
+0.54
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+入0.74
+0.72
+0.70
+0.68
+0.66
+0.64
+0.62
+0.60
+0.0
+0.2
+0.4
+0.6
+0.8
+1.00.90
+0.88
+0.86
+0.84
+0.82
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+入0.65
+0.60
+0.55
+0.50
+0.45
+0.0
+0.2
+0.4
+0.6
+0.8
+1.00.77
+0.76
+0.75
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+0.71
+0.0
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+1.0

h9AzT4oBgHgl3EQf4v4M/content/tmp_files/2301.01847v1.pdf.txt

@@ -0,0 +1,1578 @@
+Probabilistic Genotype-Phenotype Maps Reveal Mutational Robustness
+of RNA Folding, Spin Glasses, and Quantum Circuits
+Anna Sappington1, ∗ and Vaibhav Mohanty1, ∗
+1Harvard-MIT Health Sciences and Technology, Harvard Medical School, Boston, MA 02115
+and Massachusetts Institute of Technology, Cambridge, MA 02139
+(Dated: January 1, 2023)
+Recent studies of genotype-phenotype (GP) maps have reported universally enhanced phenotypic
+robustness to genotype mutations, a feature essential to evolution. Virtually all of these studies
+make a simplifying assumption that each genotype maps deterministically to a single phenotype.
+Here, we introduce probabilistic genotype-phenotype (PrGP) maps, where each genotype maps to
+a vector of phenotype probabilities, as a more realistic framework for investigating robustness. We
+study three model systems to show that our generalized framework can handle uncertainty emerging
+from various physical sources: (1) thermal fluctuation in RNA folding, (2) external field disorder in
+spin glass ground state finding, and (3) superposition and entanglement in quantum circuits, which
+are realized experimentally on a 7-qubit IBM quantum computer. In all three cases, we observe a
+novel biphasic robustness scaling which is enhanced relative to random expectation for more frequent
+phenotypes and approaches random expectation for less frequent phenotypes.
+Introduction.—Systems which take a sequence as in-
+put and nontrivially produce a structure, function, or
+behavior as output are ubiquitous throughout the sci-
+ences and engineering.
+In biological systems such as
+RNA folding [1–11], lattice protein folding [4], pro-
+tein self-assembly [12, 13], and gene regulatory net-
+works [14, 15], the relationship between genotype (stored
+biological information) and phenotype (observable or
+functional properties) can be structured as genotype-
+phenotype (GP) maps, which have a rich history of com-
+putational and analytical investigation [1–32]. Systems
+from physics and computer science have also been ana-
+lyzed as GP maps, including the spin glass ground state
+problem [30], linear genetic programming [26], and digi-
+tal circuits [31].
+Despite being completely disparate systems, all of the
+GP maps above share a number of common structural
+features, most notably an enhanced robustness of the
+phenotypes to genotype mutations. Phenotypic robust-
+ness ρn of a phenotype n is the average probability that
+a single character mutation of a genotype g which maps
+to n does not change the resultant phenotype n, averaged
+over all genotypes g mapping to n. Random assignment
+of genotype to phenotype predicts that ρn ≈ fn [4], where
+fn is the fraction of genotypes that map to phenotype
+n. However, the systems mentioned above display sub-
+stantially enhanced robustness, exhibiting the relation-
+ship ρn ≈ a + b log fn ≫ fn with system-dependent con-
+stants a and b. It has been shown that, in evolution, this
+enhanced robustness facilitates discovery of new pheno-
+types [11, 19, 20, 33] and is crucial for navigating fitness
+landscapes [5]. As a result, it is important to accurately
+quantify robustness and its relationship with phenotype
+frequency.
+All of the GP map studies referenced above make the
+assumption that a genotype maps deterministically to a
+single phenotype. However, we argue that for most of
+the above systems, this is a major simplification. For in-
+stance, within a bulk sample of ∼ N mammalian cells,
+we expect to find ∼ N copies and ∼ N ∗ 104 copies of a
+protein [34]. In vitro, such molecules often misfold [35],
+which is why cellular machinery exists to assist this fold-
+ing and to degrade misfolded structures in vivo. By map-
+ping a genotype to only the ground state energy struc-
+ture, previous studies [1–11] make an implicit zero tem-
+perature approximation for the ensemble of molecules,
+even if the Gibbs free energy of an individual molecule
+itself is calculated within the folding software at finite
+temperature. Similarly, in studies of gene regulatory net-
+works, spin glasses, linear genetic programs, and digital
+circuits, the systems investigated are small and do not
+interact with external networks or variables. These in-
+vestigations assume that the environmental effect on the
+GP mapping of the subsystem of interest is static.
+In this Letter, we introduce probabilistic genotype-
+phenotype (PrGP) maps, in contrast to the above sys-
+tems which we call deterministic genotype-phenotype
+(DGP) maps, which emerge as a limiting case of PrGP
+maps.
+The definitions of phenotypic robustness and
+transition probabilities retain the same physical mean-
+ing in PrGP maps as in DGP maps, and we empha-
+size that PrGP maps can handle disorder and uncer-
+tainty emerging from a variety of sources. To address the
+implicit zero temperature approximation in sequence-to-
+structure mappings (RNA, lattice protein folding, protein
+self-assembly), we study the folding of RNA primary se-
+quences to a canonical ensemble of secondary structures
+corresponding to low-lying local free energy minima. To
+address external variable disorder with a known distri-
+bution, we study the zero temperature mapping of a
+spin glass bond configuration to its ground state with
+quenched external field disorder, building a phenotype
+probability vector using many replicas of the disordered
+field. This has implications for viral fitness landscape in-
+arXiv:2301.01847v1  [cond-mat.stat-mech]  4 Jan 2023
+
+2
+Circuit operation U
+Exact output:
+P(sGS) = 
+= U
+Experimental output:
+     classical measurement
+RNA Folding
+AUCGAGGGGCCCCAGUCAGU
+ΔG = 0
+Spin Glass Ground State
+Quantum Circuit
+Local free energy
+minimum structure
+Genotype: Primary sequence
+ViennaRNA folding
+unfolded
+Phenotype Probability
+P(structure) = 
+{Jij} = (+1,-1,+1,+1,-1,+1,+1,-1,-1)
+Genotype: Bond configuration
+bond Jij 
+spin si 
+external field
+hi ~ N(h0, σh
+2)
+{hi} Replica 1
+Finding ground state
+{hi} Replica 2
+{hi} Replica 3
+sGS = ↑↓↑↓↑↑
+sGS = ↑↓↑↓↑↑
+sGS = ↑↓↑↓↑↓
+Phenotype Probability
+P(    ) = 
+# of times sGS occurs as GS  
+# of replicas
+{Gi} = (X, Y, H, Z)
+Genotype: Circuit gate configuration
+Z
+H
+Y
+X
+0〉
+0〉
+0〉
+0〉
+0〉
+Phenotype Probability
+0〉
+0〉
+1〉
+1〉
+1〉
+n〉
+00...0〉
+# of classical observations of 
+# of experimental shots
+n〉
+P(    ) = 
+n〉
+U 00...0〉
+〈n
+2
+Exact:
+Exp.:
+exp[-ΔGstructure/(RT)]
+exp[-ΔGstructure/(RT)]
+Σ
+structure
+Genotype Alphabet 
+{A, C, U, G} (or {C, G})
+{-1, +1}
+{Z, X, Y, H, S, S†, T, T†}
+Alphabet Size k
+4 (or 2)
+2
+8
+Phenotype
+Folded dot-bracket structure
+Ground state spin configuration 
+Classical measurement of circuit output
+Source of Uncertainty
+Thermal fluctuation, T > 0
+Disordered external field
+Superposition and entanglement
+FIG. 1. Schematic representations of the PrGP model systems studied in this work. Each system’s genotype, source of disorder,
+and method for calculating the phenotype probability vector are indicated.
+ference [36–40], where external fields, in part, model host
+immune pressure [39]. Lastly, to investigate inherent un-
+certainty in phenotypes, we introduce quantum circuit
+GP maps where uncertainty emerges from superposition
+and entanglement of classically measurable basis states.
+Our experimental realization of these quantum circuits
+on a 7-qubit IBM quantum computer also introduces
+measurement noise, which has a clear and unique effect
+on robustness. The PrGP map properties of the three
+model systems are summarized visually in Figure 1.
+We observe that PrGP maps exhibit a novel bipha-
+sic scaling of robustness versus phenotype frequency
+which, for higher frequency phenotypes, resembles the
+ρn ∝ log fn seen in DGP maps but is suppressed, and,
+for lower frequency phenotypes, settles closer to a lin-
+ear relationship between ρn and fn, suggesting that the
+lowest frequency phenotypes either appear sporadically
+throughout the GP map or are uniformly scattered at
+low probabilities throughout the genotype domain.
+Theory.—Let Ω(g) = n represent the mapping of geno-
+type g to phenotype n, where g is an element of Sℓ,k, the
+set of all kℓ sequences of length ℓ drawn from an alpha-
+bet of k characaters. A generalization of robustness is
+the transition probability φmn, the average probability
+that a single character mutation of a genotype mapping
+to phenotype n will change the phenotype to m, with the
+average taken over all genotypes mapping to n. For DGP
+maps, φmn is given by
+φmn =
+�
+g∈Sℓ,k I[Ω(g) = n] �
+h∈nn(g) I[Ω(h) = m]
+ℓ(k − 1) �
+g∈Sℓ,k I[Ω(g) = n]
+, (1)
+where I[·] is the indicator function, and nn(g) is the single
+character mutational neighborhood of sequence g. For
+PrGP maps, we weaken the indicator I[Ω(g) = n] to a
+probability pn(g) ≡ P[Ω(g) = n], which allows us to write
+φmn =
+�
+{g,h}∈∆ℓ,k[p(g) ⊗ p(h) + (p(g) ⊗ p(h))T ]mn
+ℓ(k − 1)kℓfn
+,
+(2)
+where p(g) = (p0(g), p1(g), . . .) is the phenotype prob-
+ability vector to which genotype g maps, and ∆ℓ,k
+is the set of all kℓℓ(k − 1)/2 unordered pairs of se-
+quences in Sℓ,k which differ by exactly one charac-
+ter.
+The phenotype probability vector obeys the nor-
+malization conditions kℓf
+= �
+g∈Sℓ,k p(g) and 1 =
+�
+n∈{phenotypes} pn(g) for all g ∈ Sℓ,k, and phenotype ro-
+bustnesses are given by the diagonal of the transition
+probability matrix, ρn = φnn. The phenotype entropy
+S(g) = − �
+n∈{phenotypes} pn(g) log pn(g) of a genotype g
+is also useful for quantifying how deterministic or prob-
+abilistic a PrGP map is.
+In DGP maps, a random null model [4] for robustness
+can be built by randomly assigning genotype-phenotype
+pairings while keeping the frequencies f constant.
+As
+a result, the probability of a single character mutation
+leading to a change from phenotype n to phenotype m
+is approximately φmn ≈ fm for all m. For PrGP maps,
+a naive expectation can be built by letting all phenotype
+probability vectors equal the frequency vector, p(g) = f
+for all genotypes g. From eq. (2), one finds that φmn =
+fm; thus, the two random expectations are the same,
+even though they physically represent different scenarios.
+RNA Secondary Structure Maps.—In RNA folding
+DGP map studies [1–11], the global free energy minimum
+secondary structure (reported as a “dot-bracket” string
+indicating polymer connectivity) was calculated for every
+RNA sequence of fixed length drawn from the alphabet
+of the four canonical nucleotides {A, C, G, U} (alphabet
+size k = 4).
+Here, we are interested in not only the
+global free energy minimum structures but also the low-
+lying local minima, and we additionally investigate the
+temperature-dependent behavior of the robustness. We
+use the RNAsubopt program from the ViennaRNA pack-
+
+3
+a
+b
+c
+d
+e
+f
+FIG. 2. Plots of robustness versus (a,c,e) frequency and versus
+(b,d,f) log10(frequency) for (a,b) RNA folding in, (c,d) spin
+glass ground state, and (e,f) quantum circuit PrGP maps.
+The dashed line is the random null expectation ρn = fn.
+age (version 2.4.17) [41] to calculate the secondary struc-
+tures and associated Gibbs free energies for the local free
+energy minima within 6 kcal/mol of the global free energy
+minimum (or all the nonpositive free energy local min-
+ima, if the global minimum is greater than −6 kcal/mol).
+Because of the increased computational time required to
+discover all the local minima within an energy range, we
+use a reduced alphabet of {C, G} for our main simula-
+tions with sequence length ℓ = 20. A validation study
+with ℓ = 12 and the full k = 4 alphabet is reported in the
+Supplemental Material [42]. Simulations for the ℓ = 20,
+k = 2 trials were conducted at 20 ◦C, 37 ◦C (human body
+temperature), and 70 ◦C. We take the low-lying local free
+energy minima structures to comprise a canonical ensem-
+ble at the simulation temperature, so the probability of
+RNA sequence g mapping to secondary structure n is
+determined from pn(g) = e−∆Gn/(RT )/Z, where Z nor-
+malizes the vector.
+We then calculate the robustness,
+transition probabilities, and phenotype entropy distribu-
+tions as detailed in the previous section. The DGP map
+limits of the PrGP map are also plotted for each temper-
+ature.
+In Figure 2(a-b), we plot the relationship between ro-
+bustness and frequency for the ℓ = 20, k = 2 RNA PrGP
+map and for the DGP map limiting cases for each sim-
+ulation temperature (see Supplemental Material [42] for
+Perason and Spearman correlations).
+The DGP maps
+confirm the results of refs. [3, 4], which emphasize that
+ρn ∝ log fn for most phenotypes with significant eleva-
+tion above the random null model [4] expectation. We
+find that there is little temperature dependence in DGP
+robustness calculations (see Supplemental Material [42]),
+suggesting that the effect of temperature does little to
+alter the exact ground state phenotype. However, our
+PrGP map results showcase a different robustness behav-
+ior. As the simulation temperature increases, there is a
+gradual but clear suppression of the robustness versus fre-
+quency relationship, as is apparent in both panels (a) and
+(b). We suggest this occurs due to two factors: firstly,
+though the ground state structure itself does not change
+much with temperature, the ground state becomes less
+stable relative to low-lying local minima, thereby increas-
+ing phenotype entropy, as evidenced by the entropy plots
+in the Supplemental Material [42]. As a result, for the
+corresponding p(g) ⊗ p(h) terms contributing to φmn,
+probability mass is drawn away from the diagonals to-
+ward the off-diagonal transition probabilities. Secondly,
+as temperature increases, many low frequency (higher
+∆G) phenotypes are discovered, increasing the number
+of phenotypes and drawing probability mass away from
+the more robust phenotypes.
+For high frequency phenotypes, the PrGP map robust-
+ness is suppressed relative to the DGP map robustness,
+but is nonetheless substantially elevated above the ran-
+dom null expectation like in the DGP maps. However,
+for lower frequencies, the robustness behaves more like
+the random model; in the Supplemental Material, we see
+from a log-log plot of ρn versus fn that robustness travels
+nearly parallel to the random null expectation, suggest-
+ing linear ρn ∝ fn behavior up to a constant multiplica-
+tive factor. This biphasic robustness behavior becomes
+even clearer in the spin glass and quantum circuit PrGP
+maps.
+Off-diagonal transition probabilities maintained
+an approximate relationship φmn ∝ fm for m ̸= n, in
+concordance with DGP maps (see Supplemental Mate-
+rial [42]).
+Spin Glass Ground State Maps.—In a previous spin
+glass [43, 44] DGP map study [30], a zero temperature
+±J spin glass on a random graph G(V, E) with Hamilto-
+nian H(s; J) = − �
+{i,j}∈E Jijsisj − �
+i∈V hisi was con-
+sidered. The genotype is the bond configuration where
+each Jij ∈ {−1, +1}, and the phenotype is the ground
+state configuration where each si ∈ {−1, +1}. Degen-
+eracies of the ground state were broken by the uniformly
+drawn, i.i.d. random external fields hi ∈ [−10−4, 10−4]
+which were fixed for each simulation. In our spin glass
+
+4
+PrGP map, we use a similar setup, but we are interested
+in the effect of external field disorder on robustness. We
+therefore incorporate the effects of Gaussian-distributed
+external fields hi ∼ N(h0,i, σ2
+h), where the uniformly dis-
+tributed means h0,i ∈ [−0.1, 0.1] are fixed across all re-
+alizations of the disorder for each simulation. To obtain
+accurate robustness measurements, we exactly calculate
+every ground state for spin glasses with |V | = 9, and
+|E| = 15 by exhaustive enumeration. We examine the
+effect of external field disorder by simulating 450 repli-
+cas of {hi} with variances σ2
+h = 0.001, 0.01, and 0.1 and
+fixed means {h0,i}.
+Phenotype probability vectors for
+each genotype g ≡ J were constructed by tallying and
+normalizing the number of appearances of each ground
+state across each replica. Graph topology G(V, E) cor-
+responding to data presented here, as well as validation
+trial data, are in the Supplemental Material [42].
+In Figure 2(c-d), we plot robustness versus frequency
+of each ground state for each external field variance σ2
+h
+as well as the DGP map limiting case, which qualita-
+tively reproduce the results of the earlier work [30] (see
+Supplemental Material [42] for Pearson and Spearman
+correlations). Trends similar to the RNA PrGP map are
+observed. Namely, as the disorder parameter (temper-
+ature for RNA and field variance for spin glasses) in-
+creases the uncertainty in the genotype-phenotype pair-
+ing, the phenotype entropy distribution shifts rightward
+(see Supplemental Material [42]), and the robustness ver-
+sus frequency relationship becomes suppressed relative
+to the DGP map limit. Here, the spin glass results are
+more clearly suggestive of the proposed biphasic robust-
+ness relationship, especially apparent in panel (d). For
+the highest frequencies, the ρn is substantially enhanced
+above the random null expectation and behavior close
+to the deterministic limit is observed. However, for the
+smallest frequencies, nearly linear behavior is observed;
+in the log-log plot of ρn versus fn (see Supplemental Ma-
+terial [42]), we see a strong sign that ρn ∝ fn, with the
+empirical robustness nearly parallel to the random expec-
+tation. As with the RNA folding PrGP maps, we suspect
+two causes which both contribute to this behavior: (1)
+as σ2
+h increases, there is a higher chance of changing the
+ground state, which increases phenotype entropy, and (2)
+a larger number of spin configurations appear as ground
+states, but with low frequency, drawing away probability
+mass from the more frequent phenotypes.
+Quantum Circuit Maps.—Although methods to evolve
+quantum circuits have been suggested [45], to our knowl-
+edge this work is the first to analyze the structural prop-
+erties of quantum circuit GP maps. We generate ran-
+dom quantum circuits (see Supplemental Material for al-
+gorithm) with 7 qubits and 4 layers of gates. Circuits
+are randomly seeded with CNOT gates which cannot
+participate in the genotype, and the remaining spaces
+are filled with single-qubit gates drawn from the al-
+phabet {Z, X, Y, H, S, S†, T, T †}.
+We choose ℓ = 4 of
+these gates to be variable gates which comprise the geno-
+type.
+The input to the circuit is the prepared state
+|00 . . . 0⟩ ≡ |0⟩⊗· · ·⊗|0⟩, and the exact probability of clas-
+sically measuring the basis state |n⟩ = �
+|qi⟩∈{|0⟩,|1⟩} |qi⟩
+is pn(g) = |⟨n| U(g) |00 . . . 0⟩|2, where |qi⟩ is the i-th
+qubit, and U(g) is the total circuit operation. We realize
+these quantum circuits on the ibm lagos v1.2.0 quantum
+computer [42], one of the 7-qubit IBM Quantum Falcon
+r5.11H processors. Experimental phenotype probability
+vectors are constructed from tallying classical measure-
+ments from 1000 shots for each genotype. The circuits
+from our experimental trials are depicted in the Supple-
+mental Material [42].
+In Figure 2(e-f), we plot robustness versus frequency
+for each circuit output state, using both exact and ex-
+perimental phenotype probability vectors for robustness
+calculations (see Supplemental Material [46] for Pearson
+and Spearman correlations and for data from additional
+validation trials). For the exact probabilities, the results
+in panel (f) strongly support the enhanced ρn ∝ log fn
+scaling (Pearson r = 0.998). The spread of phenotypes
+in the frequency domain is due to superposition and/or
+entanglement; moreover, we see that many of the phen-
+toypes are degenerate with identical frequency and ro-
+bustness. This degeneracy is broken in our experimen-
+tal measurements, which also exhibit measurement noise.
+Since we have finite shots, the degeneracies for the phe-
+notypes observed in the exact case end up broken. The
+frequency and robustness of these logarithmically scal-
+ing phenotypes is suppressed relative to the exact case
+as probability density is drawn towards additional phe-
+notypes, which are observed experimentally and which
+were not observed in the exact case. These appear due
+to measurement noise/decoherence effects in the physical
+system. The rightward shift of the phenotype entropy
+S(g) (see Supplemental Material [42]) further illustrates
+this effect.
+Of the three systems investigated here, the quantum
+circuit PrGP map results in panel (f) are perhaps most
+illustrative of our suggested biphasic robustness scaling.
+The low frequency phenotypes which are introduced due
+to measurement noise in the experimental trials lie much
+closer to the random null expectation than the higher
+frequency phenotypes observed in the exact calculations,
+which rather scale with enhanced robustness similar to
+what is seen in standard DGP maps.
+Discussion.—Compared to existing DGP maps, our in-
+troduction of PrGP maps not only allows for the inclu-
+sion of realistic, physical sources of disorder like thermal
+fluctuation and external variables, but it also permits the
+analysis of new systems like quantum circuits with inher-
+ent uncertainty built into the genotype-phenotype map-
+ping and from measurement disorder. We emphasize the
+broad applicability of this framework to a vast array of
+systems across biology, physics, and computer science,
+
+5
+and other disciplines for the analysis of robustness and
+stability. The proposed biphasic robustness scaling sug-
+gests that robustness of high frequency phenotypes in the
+DGP limit is suppressed in the PrGP formulation due to
+phenotype entropy increases and due to the discovery of
+new low frequency phenotypes. Moreover, low frequency
+phenotypes, which lie closer to the random null expecta-
+tion, either appear randomly throughout genotype space
+(like in the DGP random null model), or they appear
+somewhat uniformly throughout a large portion of geno-
+type space, but remain at low frequency (like in our new
+PrGP random null model). This scaling is observed in all
+three studied systems, despite being disparate, hinting at
+its universality. How this robustness trend affects navi-
+gability of (probabilistic) fitness landscapes is an impor-
+tant direction for further investigation. We also suggest
+that the mapping of genotypes to probability vectors in-
+stead of discrete phenotypes may facilitate the taking of
+gradients of, for instance, a negative loss-likelihood loss
+function in the process of learning PrGP or even DGP
+maps using statistical learning methods.
+Acknowledgements.—We acknowledge the use of IBM
+Quantum services and the MIT Engaging Cluster for
+this work.
+This work was supported by awards
+T32GM007753 and T32GM144273 from the National In-
+stitute of General Medical Sciences. The content is solely
+the responsibility of the authors and does not necessar-
+ily represent the official views of the National Institute
+of General Medical Sciences, the National Institutes of
+Health, IBM, or the IBM Quantum Team. The authors
+declare no known conflict of interest.
+∗ The authors contributed equally to this work. Cor-
+respondence:
+asappington@hms.harvard.edu and mo-
+hanty@hms.harvard.edu.
+[1] M.
+Weiß
+and
+S.
+E.
+Ahnert,
+Neutral
+components
+show a hierarchical community structure in the geno-
+type–phenotype map of RNA secondary structure, Jour-
+nal of The Royal Society Interface 17, 20200608 (2020).
+[2] M. Weiß and S. E. Ahnert, Using small samples to esti-
+mate neutral component size and robustness in the geno-
+type–phenotype map of RNA secondary structure, Jour-
+nal of The Royal Society Interface 17, 20190784 (2020).
+[3] J. Aguirre, J. M. Buld´u, M. Stich, and S. C. Manrubia,
+Topological Structure of the Space of Phenotypes: The
+Case of RNA Neutral Networks, PLoS ONE 6, e26324
+(2011).
+[4] S. F. Greenbury, S. Schaper, S. E. Ahnert, and A. A.
+Louis, Genetic Correlations Greatly Increase Muta-
+tional Robustness and Can Both Reduce and Enhance
+Evolvability, PLOS Computational Biology 12, e1004773
+(2016).
+[5] S. F. Greenbury, A. A. Louis, and S. E. Ahnert, The
+structure of genotype-phenotype maps makes fitness
+landscapes navigable, Nature Ecology & Evolution 6,
+1742 (2022).
+[6] K. Dingle, S. Schaper, and A. A. Louis, The struc-
+ture of the genotype–phenotype map strongly constrains
+the evolution of non-coding RNA, Interface Focus 5,
+20150053 (2015).
+[7] K. Dingle,
+C. Q. Camargo, and A. A. Louis, In-
+put–output maps are strongly biased towards simple out-
+puts, Nature Communications 9, 761 (2018).
+[8] K. Dingle, F. Ghaddar, P. ˇSulc, and A. A. Louis, Phe-
+notype Bias Determines How Natural RNA Structures
+Occupy the Morphospace of All Possible Shapes, Molec-
+ular Biology and Evolution 39, msab280 (2022).
+[9] K. Dingle, G. V. P´erez, and A. A. Louis, Generic pre-
+dictions of output probability based on complexities of
+inputs and outputs, Scientific Reports 10, 4415 (2020).
+[10] T. J¨org, O. C. Martin, and A. Wagner, Neutral net-
+work sizes of biological RNA molecules can be computed
+and are not atypically small, BMC Bioinformatics 9, 464
+(2008).
+[11] A. Wagner, Robustness and evolvability: a paradox re-
+solved, Proceedings of the Royal Society B: Biological
+Sciences 275, 91 (2008).
+[12] S. F. Greenbury, I. G. Johnston, A. A. Louis, and S. E.
+Ahnert, A tractable genotype–phenotype map modelling
+the self-assembly of protein quaternary structure, Jour-
+nal of The Royal Society Interface 11, 20140249 (2014).
+[13] S. Tesoro and S. E. Ahnert, Non-deterministic genotype-
+phenotype maps of biological self-assembly, EPL (Euro-
+physics Letters) 123, 38002 (2018).
+[14] C. Q. Camargo and A. A. Louis, Boolean Threshold Net-
+works as Models of Genotype-Phenotype Maps, Complex
+Networks XI , 143 (2020).
+[15] S. Kauffman, Homeostasis and Differentiation in Random
+Genetic Control Networks, Nature 224, 177 (1969).
+[16] A. Wagner, Distributed robustness versus redundancy
+as causes of mutational robustness, BioEssays 27, 176
+(2005).
+[17] A. Wagner, Robustness and evolvability in living systems,
+3rd ed., Princeton studies in complexity (Princeton Univ.
+Press, Princeton, NJ, 2007) oCLC: 845177181.
+[18] J. L. Payne and A. Wagner, Constraint and Contingency
+in Multifunctional Gene Regulatory Circuits, PLoS Com-
+putational Biology 9, e1003071 (2013).
+[19] J. L. Payne, J. H. Moore, and A. Wagner, Robustness,
+evolvability, and the logic of genetic regulation, Artificial
+Life 20, 111 (2014).
+[20] J. L. Payne and A. Wagner, The Robustness and Evolv-
+ability of Transcription Factor Binding Sites, Science
+343, 875 (2014).
+[21] S. Schaper and A. A. Louis, The Arrival of the Frequent:
+How Bias in Genotype-Phenotype Maps Can Steer Pop-
+ulations to Local Optima, PLoS ONE 9, e86635 (2014).
+[22] S. F. Greenbury and S. E. Ahnert, The organiza-
+tion of biological sequences into constrained and un-
+constrained parts determines fundamental properties of
+genotype–phenotype maps, Journal of The Royal Society
+Interface 12, 20150724 (2015).
+[23] S.
+E.
+Ahnert,
+Structural
+properties
+of
+geno-
+type–phenotype maps, Journal of The Royal Society
+Interface 14, 20170275 (2017).
+[24] M. Weiß and S. E. Ahnert, Phenotypes can be robust and
+evolvable if mutations have non-local effects on sequence
+constraints, Journal of The Royal Society Interface 15,
+20170618 (2018).
+[25] D. Nichol, M. Robertson-Tessi, A. R. A. Anderson, and
+
+6
+P. Jeavons, Model genotype–phenotype mappings and
+the algorithmic structure of evolution, Journal of The
+Royal Society Interface 16, 20190332 (2019).
+[26] T. Hu, M. Tomassini, and W. Banzhaf, A network per-
+spective on genotype–phenotype mapping in genetic pro-
+gramming, Genetic Programming and Evolvable Ma-
+chines 10.1007/s10710-020-09379-0 (2020).
+[27] S. Manrubia, J. A. Cuesta, J. Aguirre, S. E. Ahnert,
+L. Altenberg, A. V. Cano, P. Catal´an, R. Diaz-Uriarte,
+S. F. Elena, J. A. Garc´ıa-Mart´ın, P. Hogeweg, B. S. Kha-
+tri, J. Krug, A. A. Louis, N. S. Martin, J. L. Payne,
+M. J. Tarnowski, and M. Weiß, From genotypes to organ-
+isms: State-of-the-art and perspectives of a cornerstone
+in evolutionary dynamics, Physics of Life Reviews 38, 55
+(2021).
+[28] J. L. Payne and A. Wagner, The causes of evolvabil-
+ity and their evolution, Nature Reviews Genetics 20, 24
+(2019).
+[29] S. Schaper, I. G. Johnston, and A. A. Louis, Epistasis
+can lead to fragmented neutral spaces and contingency in
+evolution, Proceedings of the Royal Society B: Biological
+Sciences 279, 1777 (2012).
+[30] V. Mohanty and A. A. Louis, Robustness and Stability
+of Spin Glass Ground States to Perturbed Interactions,
+Phys. Rev. E In press (2022), arXiv: 2012.05437 [cond-
+mat.dis-nn].
+[31] A. H. Wright and C. L. Laue, Evolving Complexity is
+Hard (2022), arXiv:2209.13013 [cs].
+[32] I. G. Johnston, K. Dingle, S. F. Greenbury, C. Q. Ca-
+margo, J. P. K. Doye, S. E. Ahnert, and A. A. Louis,
+Symmetry and simplicity spontaneously emerge from the
+algorithmic nature of evolution, Proceedings of the Na-
+tional Academy of Sciences 119, e2113883119 (2022).
+[33] J. A. Draghi, T. L. Parsons, G. P. Wagner, and J. B.
+Plotkin, Mutational robustness can facilitate adaptation,
+Nature 463, 353 (2010), number: 7279 Publisher: Nature
+Publishing Group.
+[34] R. Milo and R. Phillips, Cell biology by the numbers (Gar-
+land Science, Taylor & Francis Group, New York, NY,
+2016).
+[35] R. Russell, RNA misfolding and the action of chaperones,
+Frontiers in bioscience : a journal and virtual library 13,
+1 (2008).
+[36] R. H. Y. Louie, K. J. Kaczorowski, J. P. Barton, A. K.
+Chakraborty, and M. R. McKay, Fitness landscape of
+the human immunodeficiency virus envelope protein that
+is targeted by antibodies, Proceedings of the National
+Academy of Sciences 115, E564 (2018).
+[37] T. C. Butler, J. P. Barton, M. Kardar, and A. K.
+Chakraborty, Identification of drug resistance mutations
+in HIV from constraints on natural evolution, Physical
+Review E 93, 022412 (2016).
+[38] J. P. Barton, N. Goonetilleke, T. C. Butler, B. D. Walker,
+A. J. McMichael, and A. K. Chakraborty, Relative rate
+and location of intra-host HIV evolution to evade cellu-
+lar immunity are predictable, Nature Communications 7,
+11660 (2016).
+[39] K. Shekhar, C. F. Ruberman, A. L. Ferguson, J. P. Bar-
+ton, M. Kardar, and A. K. Chakraborty, Spin models
+inferred from patient-derived viral sequence data faith-
+fully describe HIV fitness landscapes, Physical Review E
+88, 062705 (2013).
+[40] T. A. Hopf, J. B. Ingraham, F. J. Poelwijk, C. P. I.
+Sch¨arfe, M. Springer, C. Sander, and D. S. Marks, Muta-
+tion effects predicted from sequence co-variation, Nature
+Biotechnology 35, 128 (2017).
+[41] R. Lorenz, S. H. Bernhart, C. H¨oner zu Siederdissen,
+H. Tafer, C. Flamm, P. F. Stadler, and I. L. Hofacker,
+ViennaRNA Package 2.0, Algorithms for Molecular Biol-
+ogy 6, 26 (2011).
+[42] See Supplemental Material for this work.
+[43] S. F. Edwards and P. W. Anderson, Theory of spin
+glasses, Journal of Physics F: Metal Physics 5, 965
+(1975).
+[44] D. Sherrington and S. Kirkpatrick, Solvable Model of a
+Spin-Glass, Physical Review Letters 35, 1792 (1975).
+[45] D. Tandeitnik and T. Guerreiro, Evolving Quantum Cir-
+cuits (2022), arXiv:2210.05058 [quant-ph].
+[46] IBM Quantum. https://quantum-computing.ibm.com/.
+(2021).
+
+Supplemental Materials for “Probabilistic Genotype-Phenotype Maps Reveal
+Mutational Robustness of RNA Folding, Spin Glasses, and Quantum Circuits”
+Anna Sappington1, ∗ and Vaibhav Mohanty1, ∗
+1Harvard-MIT Health Sciences and Technology, Harvard Medical School, Boston, MA 02115
+and Massachusetts Institute of Technology, Cambridge, MA 02139
+CONTENTS
+I. Entropy Distributions for Main Text Systems
+2
+II. Extended Data for Main Text RNA Folding PrGP Map, GC Alphabet, ℓ = 20, k = 2
+3
+III. Validation Trial for RNA Folding PrGP Map, Full Alphabet, ℓ = 12, k = 4
+6
+IV. Extended Data for Main Text Spin Glass PrGP Map
+8
+V. Validation Trial for Spin Glass PrGP Map
+11
+VI. Quantum Circuit Generation Algorithm
+13
+VII. Extended Data for Main Text Quantum Circuit PrGP Map
+14
+VIII. Validation Trials for Quantum Circuit PrGP Map
+17
+References
+20
+∗ The authors contributed equally to this work. Correspondence: asappington@hms.harvard.edu and mohanty@hms.harvard.edu.
+arXiv:2301.01847v1  [cond-mat.stat-mech]  4 Jan 2023
+
+2
+I.
+ENTROPY DISTRIBUTIONS FOR MAIN TEXT SYSTEMS
+In the main text, we presented robustness versus frequency plots for RNA folding, spin glass ground state, and
+quantum circuit PrGP maps. For the spin glass ground state and quantum circuit PrGP maps, data for a single
+representative realization were presented in the main text.
+In Figure S1, we plot the distribution of phenotype
+entropy S(g) across all genotypes g for each of these PrGP maps. For RNA folding and spin glasses, we observe that
+the entropy distributions shift rightward as the disorder parameter increases. For RNA, this corresponds to increasing
+temperature and for spin glasses, this corresponds to increasing external field variance σ2
+h. For the quantum circuits,
+we plot both exact and experimental results (from the 7-qubit IBM quantum computer); the experimental entropy
+distribution is shifted rightward relative to the exact result, due to measurement noise as well as a finite number of
+experimental trials.
+a
+b
+c
+FIG. S1.
+Phenotype entropy distributions for the (a) RNA folding, (b) spin glass ground state, and (c) quantum circuit
+PrGP maps whose robustness plots are presented in the main text. As disorder parameters increase in (a) and (b) due to
+increased temperature and increased external field variance, respectively, the entropies shift rightward. The same occurs due
+to measurement noise in (c).
+
+3
+II.
+EXTENDED DATA FOR MAIN TEXT RNA FOLDING PrGP MAP, GC ALPHABET, ℓ = 20, k = 2
+In the main text, we presented robustness versus frequency and robustness versus log10(frequency) plots for RNA
+folding PrGP and DGP maps at three temperatures.
+For clarity, we have included robustness versus frequency,
+robustness versus log10(frequency), and log10(robustness) versus log10(frequency) plots separately for PrGP and DGP
+maps in Figure S2. First, we see that the DGP map results reproduce the expected ρn ∝ log fn relationship for most
+phenotypes, with significant elevation above the random null model expectation. We also note that there is little
+temperature dependence in DGP robustness calculations, which suggests the effect of temperature does little to alter
+the exact ground state phenotype. In contrast, our PrGP map results showcase a different robustness behavior in
+which as simulation temperature increases, there is a gradual but clear suppression of the robustness versus frequency
+relationship; see main text for discussion of these features. In the PrGP map results we also note a biphasic behavior in
+which for high frequency phenotypes, the PrGP map robustness, similar to the DGP map robustness, is substantially
+elevated above the random null expectation and for lower frequencies, the robustness behaves more like the random
+model.
+In Table S1, we include the Pearson correlation coefficient r and Spearman rank correlation coefficient ρ for each
+map (PrGP, DGP), temperature (20 °C, 37 °C, 70 °C), and axis transformation presented in Figure 2(a-b) and
+Figure S2. The primary feature we point out is the relative decrease of the PrGP Pearson r coefficients in robustness
+versus log10(frequency) plots as compared to the DGP plots; this suggests a deviation from the empirical ρn ∝ log fn
+trend observed in DGP studies.
+In the GP map literature, phenotype bias, the finding that phenotype frequencies can vary over many orders of
+magnitude with a small number of phenotypes being the targets of a large number of genotypes, has been shown for
+many systems [1–4]. In Figure S3, we present plots of log10(frequency) versus normalized rank and log10(frequency)
+versus log10(normalized rank) for each temperature and map pairing which show phenotype bias for this RNA folding
+system. Notably, the log10(frequency) versus log10(normalized rank) plot suggests a deviation from Zipf’s law.
+Figure S4 presents transition probabilities φmn for the most frequently occurring phenotype n to the other pheno-
+types m due to a single nucleotide mutation for both PrGP and DGP maps at three different temperatures. For each
+respective map, a plot including and excluding the most robust transition (i.e. from phenotype n → n) is shown for
+added clarity. This figure demonstrates that the off-diagonal transition probabilities for PrGP maps maintained an
+approximate relationship φmn ∝ fm for m ̸= n in concordance with DGP maps, and in concordance with the random
+null expectation for PrGP maps (see main text). A proportionality constant not equal to 1 for φmn ∝ fm with m ̸= n
+is likely due to transition probability mass that is acquired by the diagonal element φnn. It is also apparent that the
+most robust transition is much more likely than the transition to any other phenotype, in support of our claim that
+PrGP maps, like DGP maps, exhibit enhanced robustness.
+System Alphabet, Length Map Temperature
+Axes
+Pearson r Spearman ρ
+RNA
+GC, 20
+PrGP
+20 °C
+Robust v. Freq
+0.807
+0.953
+RNA
+GC, 20
+PrGP
+20 °C
+Robust v. log10(Freq)
+0.794
+0.953
+RNA
+GC, 20
+PrGP
+20 °C
+log10(Robust) v. log10(Freq)
+0.946
+0.954
+RNA
+GC, 20
+PrGP
+37 °C
+Robust v. Freq
+0.837
+0.955
+RNA
+GC, 20
+PrGP
+37 °C
+Robust v. log10(Freq)
+0.798
+0.955
+RNA
+GC, 20
+PrGP
+37 °C
+log10(Robust) v. log10(Freq)
+0.946
+0.953
+RNA
+GC, 20
+PrGP
+70 °C
+Robust v. Freq
+0.849
+0.934
+RNA
+GC, 20
+PrGP
+70 °C
+Robust v. log10(Freq)
+0.767
+0.934
+RNA
+GC, 20
+PrGP
+70 °C
+log10(Robust) v. log10(Freq)
+0.940
+0.934
+RNA
+GC, 20
+DGP
+20 °C
+Robust v. Freq
+0.683
+0.887
+RNA
+GC, 20
+DGP
+20 °C
+Robust v. log10(Freq)
+0.897
+0.887
+RNA
+GC, 20
+DGP
+20 °C
+log10(Robust) v. log10(Freq)
+0.839
+0.860
+RNA
+GC, 20
+DGP
+37 °C
+Robust v. Freq
+0.689
+0.877
+RNA
+GC, 20
+DGP
+37 °C
+Robust v. log10(Freq)
+0.884
+0.877
+RNA
+GC, 20
+DGP
+37 °C
+log10(Robust) v. log10(Freq)
+0.836
+0.856
+RNA
+GC, 20
+DGP
+70 °C
+Robust v. Freq
+0.745
+0.917
+RNA
+GC, 20
+DGP
+70 °C
+Robust v. log10(Freq)
+0.909
+0.917
+RNA
+GC, 20
+DGP
+70 °C
+log10(Robust) v. log10(Freq)
+0.882
+0.913
+TABLE S1. Pearson and Spearman correlation coefficients for all robustness versus frequency plots in main text/Supplemental
+Material for RNA k = 2, ℓ = 20 simulations with reduced alphabet, for each simulation temperature.
+
+4
+PrGP
+DGP
+FIG. S2. Plots of (left) robustness versus frequency, (middle) robustness versus log10(frequency), and (right) log10(robustness)
+versus log10(frequency) for RNA folding (top row) PrGP maps and (bottom row) DGP maps for three temperatures. These
+data are the same results from main text Figure 2, with axis scaling adjusted and with PrGP and DGP data shown separately
+for clarity. The dashed line is the random null expectation for both PrGP and DGP maps given by φmn = fm for all m and n.
+FIG. S3. Plots of (left) log10(frequency) versus normalized rank and (right) log10(frequency) versus log10(normalized rank) for
+RNA folding PrGP and DGP maps for three temperatures. When computing ranks, ties were broken arbitrarily.
+
+5
+PrGP
+DGP
+20 °C
+37 °C
+70 °C
+All transitions included
+All transitions included
+Robust transition removed
+Robust transition removed
+FIG. S4.
+Plots of transition probabilities versus frequency for RNA folding (left) PrGP maps and (right) DGP maps for
+three temperatures, (top) 20◦ C, (middle) 37◦ C, and (bottom) 70◦ C. For each respective map, plots include either (left) all
+transitions or (right) have the most robust transition removed. The dashed line is the random null expectation for both PrGP
+and DGP maps given by φmn = fm for all m and n.
+
+6
+III.
+VALIDATION TRIAL FOR RNA FOLDING PrGP MAP, FULL ALPHABET, ℓ = 12, k = 4
+Here, we present results of a validation trial for RNA folding PrGP maps for sequences of length ℓ = 12 utilizing
+the full alphabet of size k = 4, {A, C, G, U}. In Figure S5, we present robustness versus frequency, robustness versus
+log10(frequency), and log10(robustness) versus log10(frequency) plots for RNA folding for PrGP and DGP maps. As
+with the reduced alphabet case, we see both PrGP and DGP map results show significant elevation above the random
+null model expectation, with PrGP map results demonstrating a gradual but clear suppression of the robustness versus
+frequency relationship compared to DGP map results. The expected ρn ∝ log fn relationship for phenotypes in the
+DGP map results as well as the biphasic behavior of the PrGP map results is present but less clear in this case, likely
+due to a small size effect from the limited number of phenotypes present in this complete alphabet (k = 4, ℓ = 12)
+system compared to the reduced alphabet system (K = 2), which contains sequences of longer length (ℓ = 20). Also
+in Figure S5, we plot the distribution of phenotype entropy S(g) across all genotypes g; most phenotype entropies are
+zero due to their being deterministic because for the RNA folding, k = 4, ℓ = 12 system most genotypes do not fold.
+In Table S2, we include the Pearson correlation coefficient r and Spearman rank correlation coefficient ρ for each
+map (PrGP, DGP) and axis transformation presented in Figure S5. In Figure S6, we present plots of log10(frequency)
+versus normalized rank and log10(frequency) versus log10(normalized rank) for each temperature and map pairing
+which show phenotype bias for this RNA folding system. Notably, the log10(frequency) versus log10(normalized rank)
+plot suggests a deviation from Zipf’s law.
+Figure S7 presents transition probabilities φmn for the most frequently occurring phenotype n to the other phe-
+notypes m due to a single nucleotide mutation for both PrGP and DGP maps. For each respective map, a plot
+including and excluding the most robust transition is shown for added clarity. This figure demonstrates that the
+off-diagonal transition probabilities for PrGP maps maintained an approximate relationship φmn ∝ fm for m ̸= n in
+concordance with DGP maps, and in concordance with the random null expectation for PrGP maps (see main text).
+A proportionality constant not equal to 1 for φmn ∝ fm with m ̸= n is likely due to transition probability mass that
+is acquired by the diagonal element φnn. It is also apparent that the most robust transition is much more likely than
+the transition to any other phenotype, in support of our claim that PrGP maps, like DGP maps, exhibit enhanced
+robustness.
+FIG. S5.
+Plots of (top left) robustness versus frequency, (top middle) robustness versus log10(frequency), and (top right)
+log10(robustness) versus log10(frequency) for RNA folding PrGP and DGP maps. Plots of (bottom left) density versus phenotype
+entropy and (bottom right) log10(density) versus phenotype entropy. The dashed line is the random null expectation for both
+PrGP and DGP maps given by φmn = fm for all m and n.
+
+7
+System Alphabet, Length Map
+Axes
+Pearson r Spearman ρ
+RNA
+AUCG, 12
+PrGP
+Robust v. Freq
+0.879
+0.684
+RNA
+AUCG, 12
+PrGP
+Robust v. log10(Freq)
+0.846
+0.684
+RNA
+AUCG, 12
+PrGP log10(Robust) v. log10(Freq)
+0.948
+0.684
+RNA
+AUCG, 12
+DGP
+Robust v. Freq
+0.788
+0.244
+RNA
+AUCG, 12
+DGP
+Robust v. log10(Freq)
+0.815
+0.244
+RNA
+AUCG, 12
+DGP log10(Robust) v. log10(Freq)
+0.948
+0.684
+TABLE S2. Pearson and Spearman correlation coefficients for all robustness versus frequency plots for RNA k = 4, ℓ = 12
+validation trial with reduced alphabet. Simulations were conducted at 37 ◦C.
+FIG. S6. Plots of (left) log10(frequency) versus normalized rank and (right) log10(frequency) versus log10(normalized frequency)
+for RNA folding PrGP and DGP maps. When computing ranks, ties were broken arbitrarily.
+PrGP
+DGP
+All transitions included
+Robust transition removed
+FIG. S7. Plots of transition probabilities versus frequency RNA folding (top) PrGP maps and (bottom) DGP maps. For each
+respective map, plots include either (left) all transitions or (right) have the most robust transition removed. The dashed line
+is the random null expectation for both PrGP and DGP maps given by φmn = fm for all m and n.
+
+8
+IV.
+EXTENDED DATA FOR MAIN TEXT SPIN GLASS PrGP MAP
+In the main text, we compared a spin glass DGP map with a fixed random external field {h0,i} with our spin glass
+PrGP map, which introduces a Gaussian distribution to the external field whose means are fixed at {h0,i} and whose
+variance σ2
+h is varied as an independent variable. Figure S8 shows the topology of the graph G(V, E) (with |V | = 9,
+|E| = 15) that corresponds to the spin glass PrGP map data presented in the main text.
+Main text Figure 2(c, d) presents robustness versus frequency and robustness versus log10(frequency) data; Fig-
+ure S9 additionally plots log10(robustness) versus log10(frequency) for these same data. These three plots collectively
+demonstrate that, as with RNA folding GP maps, as the disorder parameter increases the uncertainty in the genotype-
+phenotype pairing, the robustness versus frequency relationship in PrGP maps becomes suppressed relative to the
+DGP map limit. These spin glass results are highly suggestive of a biphasic robustness relationship where at high
+frequencies, ρn is substantially enhanced above the random null expectation and behavior close to the deterministic
+limit is observed. However, as is clear from Figure S9, nearly linear behavior is observed for the smallest frequencies
+with the empirical robustness nearly parallel to the random expectation, suggesting ρn ∝ fn. See main text for
+discussion of these features.
+In Table S3, we include the Pearson correlation coefficient r and Spearman rank correlation coefficient ρ for each
+map (PrGP, DGP), external field variance (σ2
+h = 0.001, σ2
+h = 0.01, σ2
+h = 0.1), and axis transformation presented
+in Figure 2(c, d) and Figure S9. The primary feature we point out is the relative decrease of the PrGP Pearson r
+coefficients in robustness versus log10(frequency) plots as compared to the DGP (deterministic) plot; this suggests a
+deviation from the empirical ρn ∝ log fn trend observed in the spin glass DGP study [5].
+In Figure S10, we present plots of log10(frequency) versus normalized rank and log10(frequency) versus log10(normalized
+rank) for each external field variance and the deterministic case. Notably, the log10(frequency) versus log10(normalized
+rank) plot suggests a deviation from Zipf’s law.
+Figure S11 presents transition probabilities φmn for the most frequently occurring ground state n to the other
+ground states m due to a single bond perturbation.
+For each setting of external random field variance, a plot
+including and excluding the most robust transition is shown for added clarity. This figure demonstrates that the
+off-diagonal transition probabilities for PrGP maps maintained an approximate relationship φmn ∝ fm for m ̸= n in
+concordance with DGP maps, and in concordance with the random null expectation for PrGP maps (see main text).
+A proportionality constant not equal to 1 for φmn ∝ fm with m ̸= n is likely due to transition probability mass that
+is acquired by the diagonal element φnn. It is also apparent that the most robust transition is much more likely than
+the transition to any other phenotype, in support of our claim that PrGP maps, like DGP maps, exhibit enhanced
+robustness.
+1
+2
+3
+4
+5
+6
+7
+8
+9
+G(V,E) for Main Text Data
+FIG. S8. Graph G(V, E) corresponding to the spin glass PrGP map data presented in the main text.
+
+9
+6
+5
+4
+3
+2
+1
+log10(Frequency)
+4
+3
+2
+1
+0
+log10(Robustness)
+2
+h = 0.001
+2
+h = 0.01
+2
+h = 0.1
+Deterministic
+FIG. S9. Plot of log10(robustness) versus log10(frequency) for each spin glass ground state at three different external field
+variances for the spin glass PrGP maps. The DGP map limiting case is also plotted for comparison. This is the same as the
+main text plots but with axes scaled as log-log. The dashed line is the random null expectation for both PrGP and DGP maps
+given by φmn = fm for all m and n.
+System
+Map
+σ2
+h
+Axes
+Pearson r Spearman ρ
+Spin glass PrGP
+0.001
+Robust v. Freq
+0.766
+0.962
+Spin glass PrGP
+0.001
+Robust v. log10(Freq)
+0.940
+0.962
+Spin glass PrGP
+0.001
+log10(Robust) v. log10(Freq)
+0.920
+0.962
+Spin glass PrGP
+0.01
+Robust v. Freq
+0.874
+0.985
+Spin glass PrGP
+0.01
+Robust v. log10(Freq)
+0.924
+0.985
+Spin glass PrGP
+0.01
+log10(Robust) v. log10(Freq)
+0.986
+0.985
+Spin glass PrGP
+0.1
+Robust v. Freq
+0.976
+0.987
+Spin glass PrGP
+0.1
+Robust v. log10(Freq)
+0.954
+0.987
+Spin glass PrGP
+0.1
+log10(Robust) v. log10(Freq)
+0.989
+0.987
+Spin glass DGP Deterministic
+Robust v. Freq
+0.930
+0.962
+Spin glass DGP Deterministic
+Robust v. log10(Freq)
+0.964
+0.962
+Spin glass DGP Deterministic log10(Robust) v. log10(Freq)
+0.962
+0.962
+TABLE S3. Pearson and Spearman correlation coefficients for all robustness versus frequency plots for the spin glass PrGP
+map with |V | = 9 and |E| = 15 whose data are shown in the main text and here in the Supplemental Material.
+FIG. S10. Plots of (left) log10(frequency) versus normalized rank and (right) log10(frequency) versus log10(normalized frequency)
+for spin glass ground states for PrGP maps at three external field variances and DGP maps the deterministic case. When
+computing ranks, ties were broken arbitrarily.
+
+10
+h
+2 = 0.001
+h
+2 = 0.01
+h
+2 = 0.1
+All transitions included
+Robust transition removed
+FIG. S11. Plots of transition probabilities versus frequency for spin glass ground states for PrGP maps at three external field
+variances, (top) σ2
+h = 0.001, (middle) σ2
+h = 0.01, and (bottom) σ2
+h = 0.1. For each, plots include either (left) all transitions
+or (right) have the most robust transition removed. The dashed line is the random null expectation for both PrGP and DGP
+maps given by φmn = fm for all m and n.
+
+11
+V.
+VALIDATION TRIAL FOR SPIN GLASS PrGP MAP
+We provide a second spin glass PrGP map trial here in the Supplemental Material to illustrate that the spin glass
+trends described above and in the main text hold across multiple random graph instances. We generate a new G(V, E),
+once again with |V | = 9, |E| = 15 with topology shown in Figure S12. Figure S13 presents robustness versus frequency,
+robustness versus log10(frequency), and log10(robustness) versus log10(frequency) for spin glass PrGP maps at three
+different external field variances and for the deterministic case for DGP maps.
+The results from this validation
+trial exhibit the same behavior as observed in the trial presented in the main text. In particular, we see that as
+the disorder parameter increases the uncertainty in the genotype-phenotype pairing, the robustness versus frequency
+relationship in PrGP maps becomes suppressed relative to the DGP map limit. Again, these spin glass results are
+highly suggestive of a biphasic robustness relationship where at high frequencies, ρn is substantially enhanced above
+the random null expectation and behavior close to the deterministic limit is observed. However, as is clear from
+Figure S9, nearly linear behavior is observed for the smallest frequencies with the empirical robustness nearly parallel
+to the random expectation, signaling ρn ∝ fn. See main text for discussion of these features. Additionally, Figure S13
+plots the distribution of phenotype entropy S(g) across all genotypes g for PrGP maps at each external field variance
+experimental value. As is the case in Figure S1, we observe that the entropy distributions shift rightward as the
+disorder parameter increases.
+In Table S4, we include the Pearson correlation coefficient r and Spearman rank correlation coefficient ρ for each
+map (PrGP, DGP), external field variance (σ2
+h = 0.001, σ2
+h = 0.00, σ2
+h = 0.1), and axis transformation presented
+in Figure S13.
+The primary feature we point out is the relative decrease of the PrGP Pearson r coefficients in
+robustness versus log10(frequency) plots as compared to the DGP (deterministic) plot; this suggests a deviation from
+the empirical ρn ∝ log fn trend observed in the spin glass DGP study [5].
+1
+2
+3
+4
+5
+6
+7
+8
+9
+G(V,E) for Validation Trial
+FIG. S12. Graph G(V, E) corresponding to the spin glass PrGP map validation trial data shown here in the Supplemental
+Material.
+
+12
+FIG. S13. Plots of (leftmost) robustness versus frequency, (middle left) robustness versus log10(frequency), and (middle right)
+log10(robustness) versus log10(frequency) for spin glass ground states for PrGP maps at three different external field variances
+and for the deterministic case for DGP maps. Additionally, the (rightmost) density versus phenotype entropy for the spin glass
+ground states at three difference external field variances is plotted. The dashed line is the random null expectation for both
+PrGP and DGP maps given by φmn = fm for all m and n.
+System
+Map
+σ2
+h
+Axes
+Pearson r Spearman ρ
+Spin glass PrGP
+0.001
+Robust v. Freq
+0.806
+0.994
+Spin glass PrGP
+0.001
+Robust v. log10(Freq)
+0.940
+0.994
+Spin glass PrGP
+0.001
+log10(Robust) v. log10(Freq)
+0.950
+0.994
+Spin glass PrGP
+0.01
+Robust v. Freq
+0.932
+0.996
+Spin glass PrGP
+0.01
+Robust v. log10(Freq)
+0.916
+0.996
+Spin glass PrGP
+0.01
+log10(Robust) v. log10(Freq)
+0.993
+0.996
+Spin glass PrGP
+0.1
+Robust v. Freq
+0.993
+0.997
+Spin glass PrGP
+0.1
+Robust v. log10(Freq)
+0.981
+0.997
+Spin glass PrGP
+0.1
+log10(Robust) v. log10(Freq)
+0.997
+0.997
+Spin glass DGP Deterministic
+Robust v. Freq
+0.962
+0.995
+Spin glass DGP Deterministic
+Robust v. log10(Freq)
+0.993
+0.995
+Spin glass DGP Deterministic log10(Robust) v. log10(Freq)
+0.990
+0.995
+TABLE S4. Pearson and Spearman correlation coefficients for all robustness versus frequency plots for the spin glass PrGP
+map validation trial with |V | = 9 and |E| = 15 whose data are shown above in this section.
+
+13
+VI.
+QUANTUM CIRCUIT GENERATION ALGORITHM
+In this study, we generated quantum circuits with 7 qubits and 4 layers. We take the genotype of the quantum
+circuit PrGP map to be a subset of single qubit gates (which are varied to reflect each genotype). We first start by
+seeding the circuit randomly with CNOT gates which cannot participate in the genotype gate list. Only certain pairs
+of qubits which are physically connected in the 7-qubit ibm lagos v1.2.0 quantum computer can participate in the
+same CNOT gate. The remaining open places are seeded with single qubit gates, and we choose ℓ = 4 of these gates
+to be the variable gates for the genotype. The alphabet chosen is of size k = 8: {Z, X, Y, H, S, S†, T, T †}. Circuit
+diagrams used in our experimental trials are shown in the subsequent sections.
+
+14
+VII.
+EXTENDED DATA FOR MAIN TEXT QUANTUM CIRCUIT PrGP MAP
+To our knowledge, this work is the first to analyze the structural properties of quantum circuit GP maps. We
+generate random quantum circuits as described in the main text and in the previous section with 7 qubits and 4 layers
+of gates. Figure S14 shows a schematic representation of the random quantum circuit generated for the quantum
+circuit PrGP map data presented in main text Figure 2(e, f) and in this Supplemental Material section.
+Main text Figure 2(e, f) presents robustness versus frequency and robustness versus log10(frequency) data using
+both exact and experimental phenotype probability vectors for robustness calculations; Figure S15 additionally plots
+log10(robustness) versus log10(frequency) for these same data. Collectively, we see that for the exact probabilities, the
+results strongly support the enhanced ρn ∝ log fn scaling. The spread of phenotypes observed in the frequency domain
+is due to superposition and/or entanglement and many of the phentoypes are degenerate with identical frequency and
+robustness. This degeneracy is broken in our experimental measurements, which also exhibit measurement noise.
+Moreover, the frequency and robustness of these logarithmically scaling phenotypes is suppressed relative to the exact
+case as probability density is drawn towards additional phenotypes which are observed experimentally which were not
+observed in the exact case. These quantum circuit PrGP map results are perhaps most illustrative of our suggested
+biphasic robustness scaling. The low frequency phenotypes which are introduced due to measurement noise in the
+experimental trials lie much closer to the random null expectation than the higher frequency phenotypes observed in
+the exact calculations, which rather scale with enhanced robustness similar to what is seen in standard DGP maps.
+In Table S5, we include the Pearson correlation coefficient r and Spearman rank correlation coefficient ρ for both
+exact and experimental quantum circuit PrGP results for each axis transformation presented in main text Figure 2(e,
+f) and Figure S15. The primary features we point out are the high Perason correlation r = 0.998 of the robustness
+versus log10(frequency) relationship for the exact phenotype probability vectors, and the relative decrease of the
+experimental Pearson r coefficients in robustness versus log10(frequency) plot as compared to the exact plot. This
+suggests that the exact relationship exhibits behavior similar to the empirical ρn ∝ log fn trend observed in DGP
+studies, and that the experimental trials introduce measurement noise which induces a deviation from the exact
+results.
+In Figure S16, we present plots of log10(frequency) versus normalized rank and of log10(frequency) versus
+log10(normalized rank) for experimental and exact quantum circuit PrGP map results.
+Notably, the plot show-
+ing log10(frequency) versus log10(normalized rank) suggests a deviation from Zipf’s law.
+Figure S17 presents transition probabilities φmn for the most frequently occurring circuit output state n to the
+other circuit output states m due to a single variable gate perturbation. For both experimental and exact phenotype
+probability vectors, a plot including and excluding the most robust transition is shown for added clarity. This figure
+demonstrates that the off-diagonal transition probabilities for quantum circuit PrGP maps are positively correlated
+with the frequency fm, though there appears to be some additional nonrandom relationship which is not predicted
+from standard DGP or PrGP theory. It is also apparent that the most robust transition is much more likely than
+the transition to any other phenotype, in support of our claim that PrGP maps, like DGP maps, exhibit enhanced
+robustness.
+System
+Map Trial Exact or Exp
+Axes
+Pearson r Spearman ρ
+Quantum circuit PrGP
+1
+Exact
+Robust v. Freq
+0.926
+0.996
+Quantum circuit PrGP
+1
+Exact
+Robust v. log10(Freq)
+0.998
+0.996
+Quantum circuit PrGP
+1
+Exact
+log10(Robust) v. log10(Freq)
+0.993
+0.996
+Quantum circuit PrGP
+1
+Experimental
+Robust v. Freq
+0.912
+0.987
+Quantum circuit PrGP
+1
+Experimental
+Robust v. log10(Freq)
+0.712
+0.987
+Quantum circuit PrGP
+1
+Experimental
+log10(Robust) v. log10(Freq)
+0.983
+0.987
+TABLE S5. Pearson and Spearman correlation coefficients for all robustness versus frequency plots quantum circuit PrGP map
+whose robustness data was presented in the main text and in the above log-log plot. This includes both exact results as well
+as experimental results for realization/Trial 1, whose circuit is also printed earlier in this section.
+
+15
+G3
+G2
+G1
+G0
+Trial 1 - Main Text
+FIG. S14. Random circuit generated for quantum circuit trial 1, whose robustness data is plotted in the main text and below
+in the remainder of this section. The genotype is the set of variable gates g = (G0, G1, G2, G3), so the length of the input
+sequence is ℓ = 4 drawn from an alphabet of k = 8 single qubit gates: {Z, X, Y, H, S, S†, T, T †}.
+6
+4
+2
+0
+log10Frequency
+6
+5
+4
+3
+2
+1
+0
+1
+log10Robustness
+Experimental
+Exact
+FIG. S15. Plot of log10(robustness) versus log10(frequency) for the quantum circuit in trial 1 for experimental and exact data.
+The dashed line is the random null expectation for both PrGP and DGP maps given by φmn = fm for all m and n.
+FIG. S16. Plots of (left) log10(frequency) versus normalized rank and (right) log10(frequency) versus log10(normalized frequency)
+for the quantum circuit trial 1 for experimental and exact data. The dashed line is the random null expectation for both PrGP
+and DGP maps given by φmn = fm for all m and n.
+
+16
+All transitions included
+Robust transition removed
+Experimental
+Exact
+FIG. S17. Plots of (top) transition probabilities versus frequency for quantum circuit trial 1 for experimental and (bottom)
+exact data. For each model framework, plots include either (left) all transitions or (right) have the most robust transition
+removed. The dashed line is the random null expectation for both PrGP and DGP maps given by φmn = fm for all m and n.
+The dashed line is the random null expectation for both PrGP and DGP maps given by φmn = fm for all m and n.
+
+17
+VIII.
+VALIDATION TRIALS FOR QUANTUM CIRCUIT PrGP MAP
+To validate the quantum circuit PrGP map results presented in the main text and Supplemental Material, six
+additional trials were conducted. A schematic of the random quantum circuit generated for the first of these validation
+trials is shown in Figure S18. Figure S19 presents robustness versus frequency, robustness versus log10(frequency),
+and log10(robustness) versus log10(frequency) for this quantum circuit PrGP map validation trial. As with the first
+quantum circuit PrGP map trial, these data strongly support the enhanced ρn ∝ log fn scaling.
+Again, we see
+the spread of phenotypes observed in the frequency domain due to superposition and/or entanglement and that
+many of the phentoypes are degenerate with identical frequency and robustness. This degeneracy is broken in our
+experimental measurements, which exhibit measurement noise. Once again, the frequency and robustness of these
+logarithmically scaling phenotypes is suppressed relative to the exact case as probability density is drawn towards
+additional phenotypes which are observed experimentally which were not observed in the exact case. These results
+illustrate our suggested biphasic robustness scaling in which the low frequency phenotypes, which are introduced
+due to measurement noise in the experimental trials, lie much closer to the random null expectation than the higher
+frequency phenotypes observed in the exact calculations, which rather scale with enhanced robustness similar to what
+is seen in standard DGP maps. Figure S19 also presents a plot of the distribution of phenotype entropy S(g) across all
+genotypes g for exact and experimental quantum circuit PrGP maps. Notably, the experimental entropy distribution
+is shifted rightward relative to the exact result due to measurement noise as well as a finite number of experimental
+trials.
+In Table S6, we include the Pearson correlation coefficient r and Spearman rank correlation coefficient ρ for both
+exact and experimental quantum circuit PrGP results for each axis transformation presented in Figure S19. The
+primary features we point out are the high Perason correlation r = 0.950 of the robustness versus log10(frequency)
+relationship for the exact phenotype probability vectors, and the relative decrease of the experimental Pearson r
+coefficients in robustness versus log10(frequency) plot as compared to the exact plot. This suggests that the exact
+relationship exhibits behavior similar to the empirical ρn ∝ log fn trend observed in DGP studies, and that the
+experimental trials introduce measurement noise which induces a deviation from the exact results.
+Figure S20 presents robustness versus frequency and robustness versus log10(frequency) plots as well as schematics
+of the corresponding random quantum circuits for validation trials 3-7. In each trial, the suggested biphasic robustness
+scaling is clear. Additionally, these trials support the enhanced ρn ∝ log fn scaling.
+Trial 2 (Validation)
+G3
+G2
+G1
+G0
+FIG. S18. Random circuit generated for quantum circuit trial 2, whose robustness and entropy data are plotted below as a
+validation trial. The genotype is the set of variable gates g = (G0, G1, G2, G3), so the length of the input sequence is ℓ = 4
+drawn from an alphabet of k = 8 single qubit gates: {Z, X, Y, H, S, S†, T, T †}.
+
+18
+FIG. S19. Plot of (leftmost) robustness versus frequency, (left middle) robustness versus log10(frequency), and (middle right)
+log10(robustness) versus log10(frequency) for quantum circuit trial 2 for experimental and exact data. Additionally, the (right-
+most) density versus phenotype entropy for quantum circuit trial 2 is plotted. The dashed line is the random null expectation
+for both PrGP and DGP maps given by φmn = fm for all m and n.
+System
+Map Trial Exact or Exp
+Axes
+Pearson r Spearman ρ
+Quantum circuit PrGP
+2
+Exact
+Robust v. Freq
+0.910
+0.973
+Quantum circuit PrGP
+2
+Exact
+Robust v. log10(Freq)
+0.950
+0.973
+Quantum circuit PrGP
+2
+Exact
+log10(Robust) v. log10(Freq)
+0.954
+0.973
+Quantum circuit PrGP
+2
+Experimental
+Robust v. Freq
+0.916
+0.979
+Quantum circuit PrGP
+2
+Experimental
+Robust v. log10(Freq)
+0.837
+0.979
+Quantum circuit PrGP
+2
+Experimental
+log10(Robust) v. log10(Freq)
+0.989
+0.979
+TABLE S6. Pearson and Spearman correlation coefficients for all robustness versus frequency plots quantum circuit PrGP
+map whose robustness data is shown above as a Validation trial (i.e. Trial 2). This includes both exact results as well as
+experimental results for Trial 2, whose circuit is also printed earlier in this section.
+
+19
+Experimental trial 3
+Experimental trial 4
+Experimental trial 5
+Experimental trial 6
+Experimental trial 7
+Trial 3
+G3
+G2
+G1
+G0
+Trial 4
+G3
+G2
+G1
+G0
+Trial 5
+G3
+G2
+G1
+G0
+Trial 6
+G3
+G2
+G1
+G0
+Trial 7
+G3
+G2
+G1
+G0
+FIG. S20. Plots of (left) robustness versus frequency and (middle) robustness versus log10(frequency) for quantum circuit
+trials 3-7, as well as (right) corresponding random quantum circuits for Trials 3-7 (validation trials). The genotype is the set
+of variable gates g = (G0, G1, G2, G3), so the length of the input sequence is ℓ = 4 drawn from an alphabet of k = 8 single
+qubit gates: {Z, X, Y, H, S, S†, T, T †}. The dashed line is the random null expectation for both PrGP and DGP maps given by
+φmn = fm for all m and n.
+
+20
+.
+[1] T. J¨org, O. C. Martin, and A. Wagner, Neutral network sizes of biological RNA molecules can be computed and are not
+atypically small, BMC Bioinformatics 9, 464 (2008).
+[2] S. E. Ahnert, Structural properties of genotype–phenotype maps, Journal of The Royal Society Interface 14, 20170275
+(2017).
+[3] S. Manrubia, J. A. Cuesta, J. Aguirre, S. E. Ahnert, L. Altenberg, A. V. Cano, P. Catal´an, R. Diaz-Uriarte, S. F. Elena,
+J. A. Garc´ıa-Mart´ın, P. Hogeweg, B. S. Khatri, J. Krug, A. A. Louis, N. S. Martin, J. L. Payne, M. J. Tarnowski, and
+M. Weiß, From genotypes to organisms: State-of-the-art and perspectives of a cornerstone in evolutionary dynamics, Physics
+of Life Reviews 38, 55 (2021).
+[4] S. F. Greenbury, S. Schaper, S. E. Ahnert, and A. A. Louis, Genetic Correlations Greatly Increase Mutational Robustness
+and Can Both Reduce and Enhance Evolvability, PLOS Computational Biology 12, e1004773 (2016).
+[5] V. Mohanty and A. A. Louis, Robustness and Stability of Spin Glass Ground States to Perturbed Interactions, Phys. Rev.
+E In press (2022), arXiv: 2012.05437 [cond-mat.dis-nn].
+

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+Compiled using MNRAS LATEX style file v3.0
+Data-Driven Selection and Spectral Classification of White Dwarf Stars
+Olivier Vincent★, P. Bergeron and P. Dufour
+Département de Physique, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, Québec H3C 3J7, Canada
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+The next generation of spectroscopic surveys is expected to provide spectra for hundreds of thousands of white dwarf (WD)
+candidates in the upcoming years. Currently, spectroscopic classification of white dwarfs is mostly done by visual inspection,
+requiring substantial amounts of expert attention. We propose a data-driven pipeline for fast, automatic selection and spectroscopic
+classification of WD candidates, trained using spectroscopically confirmed objects with available Gaia astrometry, photometry,
+and Sloan Digital Sky Survey (SDSS) spectra with signal-to-noise ratios ≥ 9. The pipeline selects WD candidates with improved
+accuracy and completeness over existing algorithms, classifies their primary spectroscopic type with ≳ 90% accuracy, and
+spectroscopically detects main sequence companions with similar performance. We apply our pipeline to the Gaia Data Release
+3 cross-matched with the SDSS Data Release 17 (DR17), identifying 424 096 high-confidence WD candidates and providing
+the first catalogue of automated and quantifiable classification for 36 523 WD spectra. Both the catalogue and pipeline are made
+available online. Such a tool will prove particularly useful for the undergoing SDSS-V survey, allowing for rapid classification
+of thousands of spectra at every data release.
+Key words: White Dwarfs – Methods: Data Analysis – Catalogues – Surveys
+1 INTRODUCTION
+The field of astronomy is entering an era of Big Data, with the volume
+of astronomical data doubling every 16 months and is predicted to
+keep doing so for the next few years (Smith & Geach 2022). Classical
+methods that rely on human supervision and specialist expertise are
+rapidly becoming insufficient to handle this stream of opportunities,
+and concerns are arising that they may strongly delay the important
+discoveries, or worse, completely miss them. Machine learning meth-
+ods have emerged as a natural response to these concerns and have
+already become commonplace in many physical sciences (see Carleo
+et al. 2019; Smith & Geach 2022, for recent reviews). Stellar science
+has also recently seen interesting applications of machine learning,
+including main sequence star spectral classification (Sharma et al.
+2020), stellar parameter inference (Ting et al. 2019; Chandra et al.
+2020), and trigonometric parallax calibration (Leung & Bovy 2019).
+Until now, the quantity of astronomical data available for the study
+of white dwarfs has remained small enough to be manageable using
+classical methods. Most of the spectroscopic data come from SDSS
+(Gunn et al. 2006), which gradually provided optical spectroscopy
+along with broadband photometry for the majority of the ∼33 000
+currently known white dwarfs (Harris et al. 2003; Kleinman et al.
+2004; Eisenstein et al. 2006; Kleinman et al. 2013; Kepler et al.
+2015, 2016; Fusillo et al. 2021). Newer surveys dramatically contrast
+with this figure, such as the Gaia survey (Gaia Collaboration et al.
+2016) that measured astrometry for over a billion objects, resulting
+not only in increasing the number of white dwarfs with parallax mea-
+surements by about 3 orders of magnitude (Bédard et al. 2017, and
+★ E-mail: o.vincent@umontreal.ca
+references therein), but also in the identification of ∼260 000 high-
+confidence white dwarf candidates (Fusillo et al. 2021). Available
+data sources also worth mentioning include the Survey Telescope
+And Rapid Response System photometry (Pan-STARRS; Chambers
+et al. 2016), the Two Micron Sky Survey near-infrared photometry
+(2MASS; Skrutskie et al. 2006), and the spectroscopic data from the
+Large Sky Area Multi-Object Fiber Spectroscopic Telescope (LAM-
+OST; Cui et al. 2012).
+The next generation of observatories and surveys (e.g. SDSS-V,
+DESI, 4MOST; Kollmeier et al. 2017; DESI Collaboration et al.
+2016; de Jong et al. 2014) will be providing spectroscopic data
+for a large number of white dwarf candidates, unlocking both un-
+precedented statistical analyses and detailed studies of white dwarfs.
+However, extracting the white dwarf observations from the billions
+of expected spectra will be an impossible task without the aid of
+automated tools. While machine learning methods for the selection
+of white dwarf candidates as well as the classification of DA ver-
+sus non-DA have started to surface (Gaia Collaboration et al. 2021;
+López-Sanjuan et al. 2022), the most recent spectroscopic catalogues
+are still built using visual inspection (Fusillo et al. 2021; Kepler et al.
+2021). More generally, studies of large white dwarf samples usually
+involve a number of manual steps that require substantial amounts of
+time that experts could and should be spending on more important
+aspects of the data analysis (Caron et al. 2022).
+As a first step towards addressing this lack of tools, we present a
+neural network-based pipeline for rapid and automated selection and
+spectroscopic classification for white dwarf candidates. The pipeline
+is comprised of three independent modules that identify white dwarf
+candidates based on Gaia astrometry and photometry, classify the
+main spectroscopic signature of white dwarf spectra, and detect the
+© 2022 The Authors
+arXiv:2301.05209v1  [astro-ph.SR]  12 Jan 2023
+
+2
+Vincent et al.
+presence of spectroscopic contamination by a main sequence star.
+The entire pipeline is trained using human-labelled spectroscopically
+confirmed objects, and spectroscopic modules are trained and tested
+using SDSS Data Release 16 spectra (DR16; Ahumada et al. 2020).
+In the future, each module of the pipeline will be able to serve as a
+base classification model for various tasks and surveys. As a proof-
+of-concept, we produce a catalogue containing 1.3M Gaia objects
+cross-matched with the SDSS Data Release 17 (DR17; Abdurro’uf
+et al. 2022), the latest and last data release of SDSS-IV.
+This paper is structured as follows. In Section 2, we describe
+the different components of the pipeline, as well as the methods
+and data used to train them. The performance of the pipeline on
+test data is presented in Section 3. We also test the spectroscopic
+modules in different data quality regimes, and on white dwarf spectra
+with multiple spectral signatures. We then showcase the pipeline
+by creating a catalogue using 1.3M white dwarf candidates from
+the Gaia survey cross-matched with the SDSS DR17 in Section 4.
+Finally, we summarize our work and provide concluding remarks in
+Section 6.
+2 METHODOLOGY
+2.1 Pipeline description
+The white dwarf selection and spectral classification pipeline con-
+sists of three different modules: one for white dwarf candidate se-
+lection among Gaia objects, one for classification of the primary
+spectroscopic type1, and one for the detection of spectral contami-
+nation from a main sequence companion. The pipeline is illustrated
+in Figure 1. Each module consists of 10 neural networks grouped to-
+gether to form a deep ensemble, which are known to offer improved
+generalization and performance over single neural networks (Lee
+et al. 2015; Lakshminarayanan et al. 2016). Since neural networks
+typically have millions of parameters, there exists a large number
+of different parameter combinations that might sufficiently approxi-
+mate the function the networks are trying to model. By ensembling
+them, we create a distribution of diverse functions from which we
+can compute statistics for our classifications (Fort et al. 2019). For
+each pipeline module, we take the mean and standard deviation of the
+predictions of their respective 10 networks as the final prediction and
+uncertainty, respectively. Within each module, the neural networks
+are built using an unique architecture (see Appendix A for more de-
+tails), but are trained using different initialization and portions of the
+training data sets.
+The candidate selection module is an ensemble of binary classi-
+fiers that take the three Gaia magnitudes (𝐺, 𝐺BP, 𝐺RP), their flux
+errors, both proper motion components (𝜇𝛼 and 𝜇𝛿), and the paral-
+lax measurement, along with all their respective uncertainties. Our
+choice of input data is based on the white dwarf candidate Random
+Forest classifier by Gaia Collaboration et al. (2021), and differences
+between our classifiers are explored in Section 4. Additionally, we
+include phot_bp_rp_excess_factor as an input parameter, as we
+found it helpful for identifying binaries composed of a white dwarf
+and a main sequence star (WD+MS). We also tested the corrected
+excess factor proposed by Riello et al. (2021) and found no notable
+differences in performance when compared to the uncorrected factor.
+1 In this paper, we follow the terminology described in Sion et al. (1983)
+where the upper case letter following the D (for degenerate) indicates the
+primary spectroscopic type in the optical spectrum, and where the following
+upper case letters indicate the secondary spectroscopic features.
+In total, we use 13 Gaia input parameters to predict the probability
+𝑃WD of an object being a white dwarf.
+Spectral classification by the two other modules requires a spectral
+coverage of at least 3840 to 7000 Å for the determination of the
+primary spectroscopic type, which we increase to 9000 Å for the
+detection of contamination from a main sequence companion. The
+first spectroscopic module outputs the probability 𝑃class that the
+object belongs to one of the 13 following classes: DA, DB, DC,
+DO, DQ, hotDQ, DZ, DAH, PG1159, cataclysmic variable (CV),
+sdB, sdO, or sdBO. The white dwarf classes follow the classification
+system proposed by Sion et al. (1983) and Fusillo et al. (2021),
+whereas subdwarfs follow the system proposed by Geier et al. (2017).
+The neural networks of this module are trained assuming a multiclass
+problem, meaning the classes are assumed to be mutually exclusive,
+and the sum of probabilities must equal unity. We emphasize that
+this approach does not attempt to classify secondary spectroscopic
+features, but does, however, provide a consistent primary type for
+hybrid white dwarfs (see Section 3.3). The main sequence companion
+module, similar to the candidate selection module, is made of binary
+classifier networks that output a probability 𝑃MS that the spectrum
+is contaminated by a MS star.
+2.2 Data selection and processing
+To train and test our networks, we made use of all objects with
+a confirmed spectral type in the Montreal White Dwarf Database
+(MWDD; Dufour et al. 2017), in the Gaia-SDSS catalogue of Fusillo
+et al. (2021, henceforth GF21), as well as the subdwarf star catalogues
+of Geier et al. (2017) and Geier (2020). We downloaded all SDSS
+DR16 spectra and Gaia DR3 data associated with these objects from
+the Science Archive Server2 and Gaia Archive3. In order to minimize
+human error in the spectral classifications, we discarded spectra with
+a signal-to-noise ratio (SNR) lower than 9 between 4500 and 5500
+Å, which are mostly DA and DB white dwarfs, as well as subdwarfs.
+Each module of the pipeline has further restrictions that are described
+in Section 3.
+All SDSS spectra go through the following preprocessing. We
+remove the sky emission lines around 5577 6300, and 6363 Å by re-
+placing a region of 9 pixels centered on those lines with interpolated
+values of the nearest neighbouring pixel. We then pseudo-continuum
+normalize the spectra using a running window of 50 pixels width
+and selecting pixels in the 85th percentile or higher, on which we fit
+a fourth order Chebyshev polynomial. The pseudo-continuum pix-
+els are restricted to the 3842-7000Årange for the main spectroscopic
+normalization module and to 3842-9000Åfor the main sequence com-
+panion detection module. Telluric pixels are also excluded.
+As a final preprocessing step for the spectroscopic module inputs,
+we compute the average and standard deviations for all pixels of all
+continuum-normalized spectra within their respective training sets,
+and zero-center the spectra by subtracting the average, and then by
+dividing the standard deviation. The same preprocessing procedure is
+applied to the Gaia parameters of the candidate selection module by
+using the mean and standard deviations of each parameter computed
+over the entire training set.
+2 https://www.sdss.org/
+3 https://gea.esac.esa.int/archive/
+MNRAS 000, 1–12 (2022)
+
+Data-Driven Classification of WD Stars
+3
+Gaia
+parameters
+Spectrum
+Module 1:
+Candidate Selection
+Module 2:
+Main Spec. Type
+Spectroscopically
+classified WD
+Module 3:
+WD+MS Detection
+𝑃WD
+𝑃class
+𝑃MS
+DA, DB, DC
+Figure 1. Schema of the pipeline. For a given object, the Gaia parameters are sent to the candidate selection module to determine the probability of being a white
+dwarf (𝑃WD). If this probability meets certain criteria, the spectrum of the object is sent to the first spectroscopic classification module, where the probability
+that its primary spectroscopic type belongs to one of the 13 possible classes (𝑃class) is calculated. If the most probable class is either a DA, DB or DC, the
+spectrum is also sent to the WD+MS module to determine the probability of a main sequence companion (𝑃MS).
+3 PIPELINE TRAINING AND VALIDATION
+3.1 Module 1: Identification of white dwarf candidates
+We begin by looking at the candidate selection module, which takes
+13 Gaia parameters as input, and outputs a probability 𝑃WD for the
+object to be a white dwarf (see Section 2). Our sample includes all
+objects in the MWDD as well as the GF21 catalogue with available
+spectral classification. These objects are split into two categories:
+white dwarfs and non-white dwarfs; this last category also includes
+subdwarfs and main sequence stars. We apply the following cuts:
+𝐺abs > 6 + 5(𝐺BP − 𝐺RP) to remove most main sequence stars, and
+parallax_over_error > 10−3 to remove any object with excessive
+parallax measurement error.
+Any object with missing Gaia parameters or any known MS+WD
+binary is also removed, leaving a total of 35 930 stars among which
+33 416 are spectroscopically confirmed white dwarfs, and 2514 are
+confirmed non-white dwarfs. We train the neural networks using a
+different random selection of 25 000 objects for each of them, validate
+each one using a different random selection of 5500 objects, and test
+them with their respective 5430 remaining objects.
+Using a probability threshold of 0.5, the networks correctly iden-
+tify, on average, 99.2% of white dwarfs (𝑃WD > 0.5) and 83.5%
+of non-white dwarfs (𝑃WD ≤ 0.5) in their respective test sets. This
+translates into about 1.4% contamination in objects classified as
+white dwarfs and 0.9% white dwarfs being missed by the networks.
+In order to minimize biases learned by individual networks, we as-
+semble them and use the average probability as the final prediction.
+We run the entire sample through the ensemble and show the re-
+sulting distribution of 𝑃WD over the Hertzsprung-Russell diagram
+(HRD) in the right panel of Figure 2. Objects located within the white
+dwarf locus show high probabilities of being a white dwarf, rapidly
+dropping as we move towards the main sequence.
+We also show the HRD of misclassified objects in the left panel
+of Figure 2 when applying the 𝑃WD = 0.5 threshold, highlighting
+that most misclassified white dwarfs reside within the edge of the
+main sequence locus. While it is unsurprising that the ensemble
+shows confusion in such regions, we note that these misclassified
+white dwarfs are mostly of hot spectral types (DOA, PG1159) or
+have photometric effective temperatures above ≳ 20000 K (Fusillo
+et al. 2021; Dufour et al. 2017). Consequently, the ensemble may be
+less sensitive to very hot white dwarfs found outside the white dwarf
+locus.
+We compare the ensemble probabilities with those of GF21 in
+1
+0
+1
+4
+6
+8
+10
+12
+14
+16
+PWD
+0.5, is WD
+PWD > 0.5, not WD
+1
+0
+1
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+PWD
+GBP
+GRP
+MG
+Figure 2. Gaia HRD of the sample used to test the candidate selection ensem-
+ble. Left: Red dots represent confirmed white dwarfs for which the ensemble
+predicts a low 𝑃WD, while teal dots are non-WD objects predicted to have
+a high 𝑃WD. Objects with correct classifications are colored in grey. Right:
+Distribution of 𝑃WD over the entire HRD.
+Figure 3 by plotting the differences between our 𝑃WD and theirs
+over the HRD. We find large differences (over 0.5) between the
+probabilities for 532 objects, primarily located at the edges of the
+white dwarf locus and main sequence tail. 471 of these objects are
+spectroscopically confirmed white dwarfs for which our ensemble
+predicts high probabilities (𝑃WD ≳ 0.85), while the GF21 catalogue
+indicates very low probabilities (𝑃WD ≲ 0.15). The remaining 61
+objects have subdwarf spectroscopic classifications and show a mix
+of very high or low 𝑃WD for both our predictions and those of GF21.
+Our ensemble shows excellent performance and appears more robust
+for objects located in ambiguous regions of the HRD. We attribute
+this to the fact that the neural networks can learn highly non-linear
+features in a larger parameter space to make their predictions, rather
+than providing a simple density estimation of the HRD.
+3.2 Module 2: Primary spectroscopic type
+This section focuses on the primary spectroscopic type classification
+module of our pipeline. We train the networks using spectra of non-
+MNRAS 000, 1–12 (2022)
+
+4
+Vincent et al.
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+GBP
+GRP
+4
+6
+8
+10
+12
+14
+16
+MG
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+PWD, ours
+PWD, GF21
+Figure 3. Comparison of the predicted 𝑃WD by our ensemble and those of
+Fusillo et al. (2021) over the HRD. Red points indicate that the network pre-
+dicts higher probabilities than GF21, while blue points indicate the opposite.
+Table 1. Sample of spectra used to train the networks and summary of the
+ensemble predictions
+Label
+𝑁
+𝑁agree
+𝑁uncert
+𝑁disagree
+DA
+17485
+17281
+93
+111
+DC
+1718
+1548
+65
+105
+DB
+1649
+1625
+7
+17
+DZ
+439
+424
+5
+10
+DQ/DQpec
+296
+260
+14
+22
+DAH
+179
+137
+27
+15
+DO/DAO
+142
+104
+12
+26
+hotDQ
+73
+70
+2
+1
+PG1159
+23
+15
+3
+5
+sdB
+759
+617
+51
+91
+sdOB
+389
+118
+35
+102
+sdO
+255
+203
+67
+119
+CV
+221
+208
+5
+8
+Total
+23628
+22610
+386
+632
+hybrid objects from the GF21 Gaia-SDSS catalogue and subdwarf
+catalogues from (Geier et al. 2017; Geier 2020). The number of
+spectra for each class is listed in Table 1. In order to increase the size
+of the DO and DQ classes, we include DAO white dwarfs in the DO
+class, and DQpec white dwarfs in the DQ class. We do not include
+objects with spectral classification found only in the MWDD as their
+types may have been assigned based on observations obtained by
+other means than the SDSS. We split the 23 628 spectra in Table
+1 into 21 265 for the training set, and 2363 for the validation set,
+ensuring the proportion of each spectral type remains the same. We
+do not use a testing set due to the very small number of PG1159,
+hotDQ, and DO stars, and instead cross-validate the networks by
+using a completely different validation set for each network, i.e. a
+different 10% of the dataset for each one.
+In order to determine whether a spectral classification is to be con-
+sidered reliable or flagged as uncertain, requiring a visual inspection,
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Threshold
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+F
+DA
+DB
+DC
+DO
+DQ
+hotDQ
+DZ
+PG1159
+CV
+DAH
+sdB
+sdO
+sdOB
+Figure 4. 𝐹𝛽 score as a function of classification threshold for each class
+label. The 𝛽 factor is set to 0.5, making precision twice as important as recall
+(see text).
+we rely on a prediction probability threshold based on the generalized
+𝐹𝛽 score:
+𝐹𝛽 = (1 + 𝛽2)
+Precision × Recall
+(𝛽2 × Precision) + Recall ,
+(1)
+where
+Precision = TP/(TP + FP) ,
+(2)
+Recall = TP/(TP + FN) .
+(3)
+Precision is a measure of purity for a given class, while recall is a
+measure of completeness, and both are calculated using the count
+of true positives (TP), false positives (FP), or false negatives (FN).
+Their importance can be weighted in the 𝐹𝛽 score using the 𝛽 pa-
+rameter, where 𝛽 > 1 places more weight on recall, and 𝛽 < 1 places
+more importance on precision. As we aim to minimize the need for
+visual inspection of spectra, we consider precision twice as important
+as recall and set 𝛽 = 0.5. We calculate 𝐹𝛽 for each class for every
+individual network using their respective validation set, and plot the
+average 𝐹𝛽 curve for each class in Figure 4. We find that a threshold
+of 0.6 provides the highest score for most classes. Depending on the
+case studied, different thresholds may prove optimal. For example, if
+the subdwarf classification is not of interest, a threshold of 0.5 may
+be more appropriate, or if DQ white dwarfs are the only objects of
+interest, a threshold of 0.75 may be more useful. In light of these re-
+sults, we consider a spectrum to be classified if the highest prediction
+probability for a class is above the threshold 𝑃class ≥ 0.6. Objects
+with no class probabilities above this threshold are tagged for visual
+inspection. These are discussed at the end of this section.
+Having defined what constitutes a confirmed classification, we
+now perform a cross-validation to verify the performance of the
+networks for objects with a non-hybrid spectral type (i.e., DA, DB,
+DQ, DZ, etc.). We calculate the confusion matrix for each of the
+ten networks with their respective validation set, normalize them
+row-wise and take their average, producing the confusion matrix
+MNRAS 000, 1–12 (2022)
+
+Data-Driven Classification of WD Stars
+5
+DA
+DB
+DC
+DO
+DQ
+hotDQ
+DZ
+PG1159
+CV
+DAH
+sdB
+sdO
+sdOB
+Prediction
+DA
+DB
+DC
+DO
+DQ
+hotDQ
+DZ
+PG1159
+CV
+DAH
+sdB
+sdO
+sdOB
+Label
+0.995
+0
+0.003
+0
+0
+0
+0
+0
+0
+0
+0.001
+0
+0
+0
+0.996 0.003 0.001
+0
+0
+0
+0
+0
+0
+0
+0
+0.001
+0.007 0.012 0.96 0.001 0.016
+0
+0.002
+0
+0
+0.001
+0
+0.001 0.001
+0.06
+0
+0
+0.897
+0
+0
+0
+0.009
+0
+0
+0
+0.034
+0
+0
+0
+0.044
+0
+0.949 0.007
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0.014
+0
+0
+0.972
+0
+0
+0
+0.014
+0
+0
+0
+0
+0
+0.012
+0
+0
+0
+0.988
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0.167
+0
+0
+0
+0.833
+0
+0
+0
+0
+0
+0.009
+0
+0.009
+0
+0
+0
+0
+0
+0.977 0.005
+0
+0
+0
+0.152
+0
+0.006
+0
+0
+0.006
+0
+0
+0
+0.835
+0
+0
+0
+0.01
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0.924 0.021 0.045
+0.026
+0
+0
+0.007
+0
+0
+0
+0
+0
+0
+0.131 0.771 0.065
+0.007
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0.207 0.033 0.752
+0.0
+0.2
+0.4
+0.6
+0.8
+Figure 5. Average confusion matrix of the primary spectroscopic type con-
+fidently predicted (𝑃class ≥ 0.6) by the networks for objects with a single
+known spectroscopic signature. The values are normalized row-wise.
+shown in Figure 5. The majority of white dwarf classes show ≳90%
+agreement between network predictions and human labels, while
+PG1159 and DAH show the lowest agreement at ∼83%. The networks
+predict a PG1159 class for ∼13% of human-labelled DO, and ∼17%
+of the human-labelled DAH as being DA, lowering the score of
+their respective types. Subdwarfs appear to be the most confused
+classes as many sdO and sdBO are predicted to be sdB. Even so,
+all three classes show good agreement with human labels. In what
+follows, we ensemble the ten neural networks, predict classes for
+the entire spectroscopic sample, and review spectra belonging to
+classes displaying the highest percentage of disagreeing predictions
+and labels. We list the number of disagreements for each class in Table
+1 as 𝑁disagree, along with the number of agreeing predictions/labels
+(𝑁agree) and number of spectra flagged for visual inspection (𝑁uncert).
+Visual inspection of 15 objects classified as DAH by GF21 but
+with different predictions reveals that the ensemble tends to confuse
+DAH with low SNR DA, especially if the blue part of the spectrum is
+of low quality. In low SNR spectra, magnetic line splitting looks very
+similar to wide, but noisy, Balmer lines. Such confusion has been
+shown to be frequent in the study of magnetic white dwarfs by Hardy
+et al. (in prep.), who have found 400 out of 651 white dwarf spectra to
+have erroneously been classified as magnetic in previous literature.
+In our case, however, human labels appear to be correct, and the
+ensemble indicates a 5-30% probability of the objects being a DAH,
+which can be used as an indicator of weakly detectable magnetic
+splitting. There is one notable object with differing classifications,
+Gaia DR3 2849930668862492544, predicted to be a hotDQ, which
+was classified as a DQA in the in-depth analysis of DQ/DZ white
+dwarfs by Coutu et al. (2019).
+A similar investigation of the 3 objects classified as PG1159 by
+GF21, but with different predictions, points to the presence of ionized
+helium in the spectra of these objects, a feature that the ensemble
+strongly associates with the DO class. We emphasize the very small
+number of known PG1159 stars, and the fact that neural networks
+Table 2. Sample of white dwarfs with known secondary spectroscopic features
+Label
+𝑁
+𝑁PIA
+𝑁uncert
+𝑁disagree
+DAB/DBA
+326
+271
+20
+35
+DAZ/DZA
+105
+92
+10
+3
+DBZ/DZB
+97
+86
+11
+0
+DBAZ
+59
+59
+0
+0
+DABZ
+20
+17
+3
+0
+DZBA
+14
+13
+1
+0
+DZAB
+5
+4
+1
+0
+DOZ
+2
+1
+1
+0
+DQZ
+1
+0
+1
+0
+DA+MS
+783
+750
+22
+11
+DB+MS
+37
+32
+1
+4
+DC+MS
+23
+16
+4
+3
+Total
+1472
+1341
+75
+56
+are data-driven algorithms that strongly depend on the number of
+available examples to learn from (He & Garcia 2009; Zhu et al.
+2015). It is thus not surprising to see a lower performance for the
+class with the least number of spectra. Furthermore, PG1159 and
+DO are suspected to share a common spectral evolution pathway
+(Bédard et al. 2022), and so the decision boundary between the two
+is ill-defined by nature. Objects found on the fine line between DO and
+PG1159 can usually be identified with high prediction probabilities
+for both classes, typically ≳ 0.6 for the first, and ≳ 0.2 for the second.
+As for the apparent confusion between subdwarf types, we strongly
+suspect the culprit to be common features among the various types.
+Indeed, according to the classification scheme proposed by Geier
+et al. (2017), both sdOB and sdO subdwarfs may show a mix of
+hydrogen, neutral helium, and ionized helium lines. The consider-
+able feature overlap, along with their occasional presence, of these
+classes could easily induce confusion to both our ensemble and a
+human classifier, resulting in errors in not only the predictions but
+labels as well. While a subtype classification has been proposed by
+Geier (2020) to separate hydrogen-rich from helium-rich spectra, the
+number of objects for each class would be too small for the networks
+to learn meaningful features.
+The results discussed so far have been restricted to high-confidence
+predictions, i.e. those with a class probability above the threshold
+𝑃class > 0.6. Among the 23 628 spectra used to train and vali-
+date the networks, 2.7% have their highest predictions falling be-
+low this threshold. We list the number of uncertain spectra per
+label in Table 1. The majority of cases are subdwarfs, consistent
+with the fact that their classes are the most confusing to the en-
+semble. Uncertain white dwarf classifications can be grouped into
+three broad categories: (1) their spectra is close to an ill-defined
+boundary between multiple classes, (2) the spectra have low SNR
+in regions where important class-specific features are found, and
+(3) the spectra may possess an unusual feature among its class.
+Categories (1) and (2) are self-explanatory and the most affected
+classes share the same explanations as the misclassifications dis-
+cussed above. Spectra belonging to category (3) include rare objects
+such as Gaia DR33985469616188225152, a DQ with oxygen lines
+(Gänsicke et al. 2010), or Gaia DR3 3731667388643923840, also
+a DQ but with metal traces (Coutu et al. 2019; Farihi et al. 2022).
+A simple approach to filter out the majority of uninteresting spectra
+with uncertain classifications would be to apply a SNR cut, keeping
+most spectra in categories (1) and (3).
+MNRAS 000, 1–12 (2022)
+
+6
+Vincent et al.
+3.3 White dwarfs with secondary spectroscopic features
+We now turn to white dwarfs with secondary spectroscopic features,
+such as DAZ, DBA, etc. Since the networks are trained on objects
+with only primary spectroscopic types, it is important to asses their
+predictions when confronted with ambiguous data. Our white dwarf
+sample with known secondary spectroscopic features is listed in Ta-
+ble 2 and totals 1472 spectra, most of which are labelled DBA/DAB
+and WD+MS binaries. The second column of Table 2 lists the number
+of spectra with a predicted primary spectroscopic type matching one
+of the human labels of their respective class (Prediction In Any class;
+𝑁PIA), while the third and fourth columns list the number of spectra
+flagged for visual inspection and disagreeing predictions/labels, re-
+spectively. The ensemble predictions show excellent agreement with
+human labels, with ∼91% of all spectra matching one of the possible
+types and only ∼3% disagreement. Upon visual inspection, nearly all
+spectra with predictions inconsistent with their labels were found to
+be misclassified in the GF21 catalogue.
+Out of 38 spectra labelled as having secondary spectroscopic
+features for which the ensemble predicts no matching primary
+type, only two appear to have been erroneously predicted by
+the networks. The ensemble prediction for the spectra of Gaia
+DR3 3781616827503753088 and Gaia DR3 1028779636740111872
+points to a DC type instead of either a DA or DB that would match
+their proposed DBA label, probably due to the weak strength of the
+lines. As a matter of fact, Gaia DR3 1028779636740111872 was
+classified as a DB+DC binary by Kleinman et al. (2013), and the
+helium lines are most likely diluted by the DC companion. 11 other
+spectra are labelled as DBA/DAB, although they clearly show ionized
+helium lines, and the ensemble predicts a DO primary class for these
+spectra, which we confirm to be correct by cross-checking the predic-
+tions with the spectral types in the MWDD. 22 spectra are labelled as
+DBA/DAB/DAZ but are predicted to be subdwarfs by the ensemble,
+which we also confirm using the MWDD. One spectrum of Gaia
+DR3 1884548739436672640 shows carbon in its spectrum and is
+correctly predicted to be a DQ, although it is labelled as a DBA. The
+spectrum for Gaia DR3 1587462866571138048 is labelled as DZA,
+while the ensemble predicted a DB type, consistent with the DBZA
+spectral type in the MWDD. The predictions made by the ensemble
+for the last two objects are also consistent with the spectral types
+DQA and DBAZ given by Coutu et al. (2019). The ensemble seems
+to provide a primary spectroscopic type reliably when classifying
+white dwarfs with secondary spectroscopic features.
+The situation is quite similar when classifying the 1472 spec-
+tra labelled as having a main sequence companion, as most spec-
+tra with predictions inconsistent with their labels were found to
+have wrong labels. We note that although there are differences be-
+tween the predicted and labelled primary spectroscopic types for
+the cases discussed below, a main sequence companion is indeed
+always present. Out of the 18 spectra with disagreeing predictions
+and labels, the only spectrum with a possible erroneous ensemble
+prediction belongs to the object Gaia DR3 2464385576553809792,
+which is predicted to be a DZ but labelled as a DC+MS. The spec-
+trum is dominated by the MS companion, showing unusual features
+to the ensemble, which likely confuse it. Prediction uncertainties for
+this spectrum are very high, with the DZ type being predicted with
+61% probability, but with 37% uncertainty. Interestingly, the DC
+type is predicted with 24% probability, but with 34% uncertainty.
+Such uncertainties may be used to discern spurious high-confidence
+classifications, and even give a hint as to which class is the cor-
+rect one. 9 of 18 spectra labelled as DA+MS, but predicted to be
+CV, all show emission in at least one of their hydrogen lines. Two
+spectra for the object Gaia DR3 922604914151538816 labelled as
+DB+MS show ionized helium lines and are predicted to be DO, con-
+sistent with the DO+MS classification in the MWDD. The spectrum
+for Gaia DR3 1219552974402511104 is labelled as DA+MS but is
+predicted to be a subdwarf, probably due to its narrow hydrogen
+lines. A spectrum for Gaia DR3 2836746940329352448 labelled as
+DC+MS is predicted to be a DB, consistent with the MWDD, and
+one available spectrum for Gaia DR3 733784442283662464 labelled
+DB+MS does not show any obvious features, consistent with the en-
+semble prediction suggesting a DC. Finally, a spectrum for Gaia
+DR3 2536561952205972608 labelled as a DA+MS but predicted to
+be a DC also does not seem to show any obvious features. However,
+higher SNR spectra of the same object reveal it is indeed a DA+MS.
+These verifications lend support to the ensemble providing primary
+spectroscopic types that are more robust than visual inspection for
+WD+MS spectra.
+3.4 Module 3: Main sequence companionship
+The third and final module of the pipeline makes use of SDSS spec-
+tra to predict the probability of contamination by a main sequence
+companion. Since the redder part of white dwarf spectra often con-
+tains the most obvious signature of the presence of a main sequence
+companion, we rely on a wider wavelength coverage than the main
+classification module, extending to 9000 Å. We use the GF21 cata-
+logue to form two groups for our training sample: all white dwarfs
+labelled as having a main sequence companion form the positive
+group, while those with a DA, DB or DC type form the negative
+group, totalling 832 and 20 486 spectra for each group, respectively.
+We then train each neural network using the same approach as
+for the candidate selection module by randomly selecting, for each
+network, 16 000 and 2700 spectra for training and validation, re-
+spectively, keeping the remaining 2618 spectra for testing. Using a
+probability threshold of 0.5, the networks correctly identify, on aver-
+age, 91.6% of WD+MS spectra and 99.8% of uncontaminated white
+dwarfs. About 0.4% of spectra labelled as single white dwarfs are
+predicted to have a companion, and about 4.9% of spectra labelled as
+having a companion are predicted to be single white dwarfs. We vi-
+sually inspected the spectra with disagreeing predictions and labels,
+and found that 36 out of 76 spectra labelled as having no companion
+turn out to obviously have one, consistent with the very high prob-
+abilities (≳0.8) given by the network. Out of 41 spectra predicted
+as having no companion but labelled as having one, 9 show obvious
+companion contamination. Among these, 4 are DB+MS and 1 is a
+DC+MS. We find no obvious clues as to why these objects were
+erroneously classified, but we suspect the type of white dwarf may
+be the cause, as DB and DC binaries are much less numerous and
+may be harder for the ensemble to recognize. We also notice 11 CVs
+and 5 subdwarfs that were mislabelled as WD+MS in the GF21 cata-
+logue, for which the networks give very small probabilities of being
+a WD+MS, consistent with their real classification. We find the rest
+of the spectra with disagreeing predictions and labels too visually
+ambiguous to determine companionship by visual inspection alone.
+3.5 Performance vs SNR
+The SNR of a spectrum is perhaps one of the most important factors
+affecting the visibility of spectroscopic features, as well as the overall
+spectral shape, making the same object look radically different when
+observed at low and high SNR. Moreover, large surveys generally do
+not provide uniform SNR distributions of spectra, and so classifica-
+tion algorithms may struggle to classify correctly spectra of objects
+MNRAS 000, 1–12 (2022)
+
+Data-Driven Classification of WD Stars
+7
+]0-9]
+]9-18] ]18-27] ]27-36] ]36-45] ]45-54]
+>54
+Signal-to-noise
+0
+2000
+4000
+6000
+8000
+10000
+12000
+Number of spectra
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Weighted F
+]0-9]
+]9-18] ]18-27] ]27-36] ]36-45] ]45-54]
+>54
+Signal-to-noise
+0
+2000
+4000
+6000
+8000
+10000
+12000
+Number of spectra
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Weighted F
+Figure 6. SNR histograms of spectra used to train and to test the spectro-
+scopic classification modules of our pipeline, overlaid with the class-weighted
+average 𝐹𝛽-score. The top panel is for the primary spectroscopic type clas-
+sification module, while the bottom is for the main sequence companion
+module.
+that have much higher (or lower) SNR than the bulk of spectra of its
+class. Below, we demonstrate that the two spectroscopic classifica-
+tion modules of our pipeline perform well over the entire SNR range
+of the SDSS spectra.
+To evaluate the performance of the primary spectroscopic type
+module in different SNR bins, we plot the SNR histogram of all
+spectra in the upper panel of Figure 6, including white dwarfs with
+secondary types and main sequence binaries discussed previously,
+and overlay the class-weighted average of the 𝐹𝛽-score. The primary
+spectroscopic type module shows an excellent score at 𝐹𝛽 ∼ 0.95 for
+spectra under SNR ∼45, with a slight drop to ∼0.91 for higher SNR.
+Though it may seem counter-intuitive that higher SNR spectra are
+slightly more difficult to classify for the neural networks, one must
+keep in mind the small number of spectra relative to the bulk at SNR
+9-18, along with the fact that classes with few known objects – thus
+more difficult to classify – tend to have a higher SNR.
+We find similar results for the main sequence companion module,
+for which we plot the histogram and class-weighted 𝐹𝛽-score in the
+lower panel of Figure 6 using the sample of spectra described in
+Section 3.4. The module also shows an excellent score of 𝐹𝛽 ≳ 0.9
+over the entire SNR > 9 range, generally increasing along with higher
+SNR. This trend differs from that found for the primary spectroscopic
+Table 3. Sample of low SNR spectra used to test the main spectroscopic type
+classification module
+Label
+𝑁
+𝑁agree
+𝑁uncert
+𝑁disagree
+DA
+7531
+7438
+45
+48
+DC
+1241
+995
+97
+149
+DB
+331
+321
+5
+5
+DZ
+362
+315
+20
+27
+DQ
+47
+41
+4
+2
+DAH
+15
+10
+0
+5
+DO
+1
+0
+0
+1
+hotDQ
+7
+5
+2
+0
+PG1159
+0
+0
+0
+0
+sdB
+20
+2
+6
+12
+sdOB
+3
+0
+1
+2
+sdO
+8
+0
+4
+4
+CV
+75
+72
+1
+2
+Total
+9734
+9292
+185
+257
+type module because the detection of MS companions becomes easier
+with better signal, but also because the classification is set as a binary
+problem (WD vs WD+MS), removing the effect that might have been
+caused by rare classes whose spectra are mostly found at high SNR.
+Even though we do not use low-SNR (≤ 9) spectra to train and to
+test the neural networks of our pipeline, they constitute ∼30% of all
+spectra in the GF21 catalogue, warranting at the very least a short
+performance assessment of the spectroscopic classification modules
+in this SNR regime. We verify the global performance by including a
+bin containing SNR ≤ 9 spectra in the histograms displayed in Figure
+6. Both modules display very high scores on par with higher SNR
+spectra, with the primary spectroscopic type module showing a ∼0.96
+score, and the main sequence companion module showing a 0.86
+score. We further study the high score of the primary spectroscopic
+type module with its confusion matrix in Figure 7, showing a ≳90%
+agreement for most classes when applying the 𝑃class > 0.6 threshold.
+The only exception is for subdwarfs, for which the ensemble predicts
+a DA type for 27 out of the 31 spectra. These classes are known to
+be notoriously difficult to distinguish in the low SNR regime. The
+confusion between DA and DAH also remains present, with 4 of
+the 15 labelled DAH being predicted as DA. A more detailed list of
+low-SNR spectra used to test the main spectroscopic type module
+and their resulting classification is provided in Table 3.
+We conclude that the spectral classification modules of our pipeline
+are reliable over the entire SNR range, including noisier spectra
+(SNR< 9) on which the networks were not trained. We caution, how-
+ever, that neural networks are known for producing overconfident
+predictions on out-of-distribution data (Nguyen et al. 2014; Good-
+fellow et al. 2014), and recommend that extra care be taken when
+interpreting results in low-SNR regimes.
+4 GAIA-SDSS WHITE DWARF CATALOGUE
+In this section, we use our pipeline to identify white dwarf candidates
+from a large sample of Gaia objects and classify them spectroscop-
+ically using SDSS DR17 spectra. The results are made available as
+an online catalogue, along with recommendations on how to use it,
+as well as the pipeline itself.
+MNRAS 000, 1–12 (2022)
+
+8
+Vincent et al.
+A
+B
+C
+O
+Q
+hot
+Z
+P
+V
+DAH
+sdB
+sdO
+sdOB
+Prediction
+A
+B
+C
+O
+Q
+hot
+Z
+P
+V
+DAH
+sdB
+sdO
+sdOB
+Label
+0.994
+0
+0.004
+0
+0
+0
+0
+0
+0
+0.001
+0
+0
+0
+0
+0.985 0.012
+0
+0
+0.003
+0
+0
+0
+0
+0
+0
+0
+0.084 0.031 0.871
+0
+0.004 0.004 0.005
+0
+0
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0.953
+0
+0.023
+0
+0.023
+0
+0
+0
+0
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+0
+0
+0.012
+0
+0.067
+0
+0
+0
+0.921
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0.014
+0
+0.014
+0
+0
+0
+0
+0
+0.973
+0
+0
+0
+0
+0.267
+0
+0.067
+0
+0
+0
+0
+0
+0
+0.667
+0
+0
+0
+0.857
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0.143
+0
+0
+1
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Figure 7. Primary spectroscopic type confusion matrix of confident predic-
+tions (𝑃class > 0.6) for SNR≤ 9 spectra.
+4.1 Candidate selection
+As our starting point, we calculate the probability of being a white
+dwarf for the 1.3M objects found in the GF21 Gaia main catalogue.
+We apply a probability threshold of 𝑃WD > 0.75 and an uncertainty
+limit of 0.02, resulting in 424 096 white dwarf candidates. Of these
+candidates, 25 205 are spectroscopically confirmed white dwarfs,
+and 50 are non-WD according to the MWDD and/or the GF21 SDSS
+catalogue. We reclassify as candidates 131 non-WD objects found
+within the white dwarf locus in the Gaia HRD since their spectra have
+very low SNR (≲ 5) and are too noisy for reliable visual classification.
+We show in Figure 8 the Gaia HRD of the candidates as well as
+spectroscopically confirmed white dwarfs and non-WD objects. For
+visibility purposes, a random selection of 15% of the candidates
+and a quarter of the confirmed white dwarfs is displayed, while all
+non-WD objects are kept.
+A large number of white dwarf candidates can be seen above
+the faint end of the white dwarf sequence, as delimited by the or-
+ange dashed line in Figure 8. This region is typically populated
+by WD+MS binaries (Rebassa-Mansergas et al. 2016, 2021) and
+here we look for clues to confirm whether this is the case. We find
+that 2583 out of the 4311 candidates above the line have renormal-
+ized unit weight error (ruwe) greater than 1.1, a quantity repre-
+senting the quality of the astrometric solutions, for which a value
+between 1.1 ≲ ruwe ≲ 1.4 has been found to indicate possible
+movement perturbations caused by an unresolved companion (Be-
+lokurov et al. 2020). An additional 583 candidates show red flux
+excess (phot_bp_rp_excess_factor ≳ 1.3) that is at least 0.2
+larger than those of other objects with a similar absolute 𝐺 magni-
+tude. Riello et al. (2021) found that many objects with large excess
+factors tend to either have emission lines in the wavelength range
+where the RP passband has a larger transmissivity with respect to
+the 𝐺 passband, or to be blended sources. High values of ruwe and
+phot_bp_rp_excess_factor may thus imply that the white dwarf
+candidates inside the region delimited by orange dashed lines in Fig-
+ure 8 are genuine white dwarf binary systems. For ease of selection
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+GBP
+GRP
+6
+8
+10
+12
+14
+16
+MG
+Confirmed WD
+Confirmed non-WD
+Figure 8. Gaia HRD of white dwarf candidates selected among 1.3M Gaia
+objects. Candidates are shown in grey, while spectroscopically confirmed
+white dwarfs and non-white dwarfs are shown in blue and red, respectively.
+Objects found to the right of the region delimited by the orange dashed lines
+are likely WD+MS objects (see Section 4.1).
+(or removal), we include a binary flag above_locus in our catalogue
+for the 4317 objects located within the region.
+We compare our selection of candidates to the GF21 main cata-
+logue by extracting all objects with 𝑃WD,GF21 > 0.75. We find our
+selection to contain 25 816 candidates not in the GF21 selection,
+including 869 spectroscopically confirmed white dwarfs and 28 non-
+WD objects, the former set mainly consisting of DA and DA+MS
+white dwarfs. There are 48 candidates present in the GF21 selection,
+but not in ours, including 28 confirmed non-WD and no confirmed
+white dwarfs.
+We also compare our candidate selection within 100 parsecs of
+the Sun to the Gaia Catalogue of Nearby Stars (GCNS, Gaia Col-
+laboration et al. 2021), which provides a probability for objects to
+be white dwarfs using a Random Forest algorithm. Apart from the
+choice of algorithm, the main difference between our two methods
+lies in the choice of training data. Our training set is restricted to the
+white dwarf region in the Gaia HRD through color cuts, while Gaia
+Collaboration et al. (2021) use the entire HRD space. Moreover, our
+non-WD training examples are solely spectroscopically confirmed
+objects, whereas the GCNS uses any object that is not a spectroscop-
+ically confirmed white dwarf as part of their non-WD training set,
+which may include yet-to-be confirmed white dwarfs. We compare
+objects in the GCNS with a probability of being a white dwarf above
+0.75 and find a single candidate not present in our own selection,
+but we find 366 candidates present in our selection but missing in
+the GCNS, among which 107 are spectroscopically confirmed white
+dwarfs and no confirmed non-WD objects. Overall, our candidate
+selection appears to be more complete while also being less contam-
+inated than other catalogues.
+4.2 Spectroscopic classification
+Optical spectra for the white dwarf candidates are obtained by calcu-
+lating the position of the Gaia objects at the J2000 epoch using their
+proper motions, then by cross-matching them with the SDSS DR17
+MNRAS 000, 1–12 (2022)
+
+Data-Driven Classification of WD Stars
+9
+Table 4. Primary spectral type classification predicted by our pipeline for
+white dwarf candidates in the Gaia-SDSS DR17 sample.
+Class
+𝑁 conf
+Gaia
+𝑁 uncert
+Gaia
+𝑁 conf
+spec
+𝑁 uncert
+spec
+DA
+21434
+178
+28093
+211
+DB
+1887
+35
+2758
+43
+DC
+2007
+341
+2518
+416
+DO
+68
+5
+102
+5
+DQ
+265
+21
+362
+26
+hotDQ
+131
+22
+181
+26
+DZ
+896
+80
+1120
+96
+PG1159
+12
+0
+17
+1
+CV
+157
+9
+225
+12
+DAH
+174
+17
+237
+21
+sdB
+13
+5
+18
+6
+sdO
+6
+0
+9
+1
+sdOB
+9
+7
+11
+8
+the latest and last data release of SDSS-IV (Blanton et al. 2017). This
+data release includes new observations through January 2021, as well
+as updates to some calibration files affecting all eBOSS spectra taken
+after the summer of 2017. Therefore, all spectra after MJD 58000 are
+different from their equivalent DR16 version and can be considered
+as unseen data for the networks. We also supplement our sample
+with 3591 spectra from the MWDD and the GF21 SDSS catalogue
+missed by our cross-match procedure. We remove all spectra that do
+not have the full coverage of 3842-7000 Å required by our networks,
+leaving a total of 36 523 spectra belonging to 27 866 unique Gaia
+white dwarf candidates.
+We pass all the spectra through the primary spectroscopic type
+classification module and present the results in Table 4, where 𝑁conf
+Gaia
+is the number of unique Gaia objects for which their highest SNR
+spectrum has a confident prediction above the 𝑃class > 0.6 threshold,
+𝑁uncert
+Gaia
+is the number of unique Gaia objects whose predictions for
+their highest SNR spectrum fall below or equal to the threshold.
+Also included are the number of spectra predicted to belong to each
+class, 𝑁conf
+spec and 𝑁uncert
+spec , following the same notation as for the
+Gaia columns. About 97.5% of both spectra and unique Gaia objects
+were assigned a high probability primary spectral type, reducing the
+number of spectra requiring visual inspection from 36 523 to 872 in
+less than a minute.
+A surprising result from the automated classification is the iden-
+tification of 131 hotDQ Gaia objects, a much larger number than
+what is currently known in the literature (Dufour et al. 2008; Koester
+& Kepler 2019; Fusillo et al. 2021). To our knowledge, 66 of these
+have already been identified as hotDQ by at least one study, 49 have
+previously been classified as another type than hotDQ (usually DAH
+or DQ), and the remaining 16 appear to be new discoveries. The
+large number of hotDQ may also be due to differing definitions of
+this class. Indeed, GF21 do not state what features were used to dis-
+tinguish hotDQ from other DQ white dwarfs, and seem to include
+warmDQ (Dufour et al. 2013) in the hotDQ class. In their analysis of
+carbon-atmosphere white dwarfs, Coutu et al. (2019) classify white
+dwarfs showing molecular carbon bands as DQ, neutral atomic car-
+bon lines as warmDQ, and ionizedcarbon linesas hotDQ. Given these
+definitions, visual inspection of the spectra labelled and predicted as
+hotDQ reveals some of the objects may actually be warmDQ. A more
+detailed analysis of these objects would allow to confirm their true
+classes.
+As a final step, all candidates classified as DA, DB, and DC white
+dwarfs by our neural networks are passed through the third classi-
+fication module to determine whether they have a MS companion.
+Assuming a probability threshold of 0.5, we find 1380 spectra show-
+ing signs of companionship, a number comparable to those identified
+by GF21. To verify how the pipeline deals with WD+MS systems, we
+compare our results with the spectroscopic catalogue of WD+MS bi-
+naries in SDSS DR12 published by Rebassa-Mansergas et al. (2016).
+We first cross-match the 979 SDSS objects in their catalogue with
+the Gaia DR3 and find 258 within the 1.3M objects in the GF21
+main catalogue. The list of white dwarf candidates produced by our
+candidate selection module contains 211 of these, all of which were
+correctly classified as WD+MS by our pipeline.
+Binary systems were excluded from the training sets of both the
+candidate selection and the main spectroscopic type modules. Con-
+sequently, our pipeline may not be the optimal tool for discovering
+such objects. However, the MS companion module could be used as
+a stand-alone classifier to identify new WD+MS binaries in samples
+with appropriate selection criteria. The module could also bene-
+fit from training on synthetic data, as current WD+MS samples are
+strongly biased towards relatively equal contribution from both mem-
+bers (Rebassa-Mansergas et al. 2021), and may prove to be able to
+classify objects for which visual inspection is too ambiguous.
+5 DATA AVAILABILITY
+The results of Section 4 are available on the MWDD website4 and
+in the VizieR catalogue access tool as two catalogues. The first
+catalogue includes the probability of being a white dwarf for the
+1.3M Gaia objects in Section 5, along with all Gaia parameters used
+for our analysis and discussion. See Table 5 for a list of column names
+and description. The second catalogue contains the list of objects
+for which SDSS spectra were found, as well as their spectroscopic
+classification results, in addition to the same columns as in the Gaia
+candidate catalogue. We provide the fiberid, mjd and plateid as
+a way to identify the spectra in the SDSS database. Note that Gaia
+objects may have multiple spectra, or vice versa. A list of the columns
+unique to the results of the Gaia-SDSS catalogue is shown in Table
+6.
+Granular control over the completeness and contamination rate
+for both the white dwarf candidate selection and the spectroscopic
+classifications can be achieved by imposing different prediction prob-
+ability thresholds or prediction uncertainty limits. In principle, the
+prediction probability and uncertainty could be combined to look for
+spectra that show secondary spectroscopic signatures, though this
+has not been tested here. For a good balance between complete-
+ness and contamination, we recommend the following thresholds:
+𝑃WD > 0.75 and an uncertainty limit of 0.02 for candidate selec-
+tion, 𝑃class > 0.6 for the primary spectroscopic type with the highest
+prediction probability, and 𝑃MS > 0.5 for MS companion detection.
+We also remind the reader to be cautious when interpreting predic-
+tions for spectra with low SNR (≲ 9), as the neural networks may be
+overconfident in their predictions (see Section 3).
+The modules of the pipeline are made available to try and to
+use on the MWDD website5. A description of the required inputs,
+file formats can be found there. Future versions of the pipeline and
+machine learning tools will also be uploaded on this page. By making
+these publicly available, we aim to facilitate the transition from the
+traditionally manual approaches of white dwarf analysis to rapid,
+4 montrealwhitedwarfdatabase.org
+5 montrealwhitedwarfdatabase.org/MLTools
+MNRAS 000, 1–12 (2022)
+
+10
+Vincent et al.
+Table 5. Columns unique to our Gaia candidate catalogue
+Column Header
+Description
+source_id
+Unique source identifier in Gaia DR3
+P_wd
+Probability of being a white dwarf (see Section 2)
+P_wd_u
+Uncertainty on the probability of being a white dwarf
+above_locus
+Binary flag indicating whether the object is above the white dwarf locus and may be a WD+MS binary (see Section 4.1)
+Table 6. Columns unique to the Gaia-SDSS DR17 catalogue. All columns in the Gaia candidate catalogue are also included, but not shown here.
+Column Header
+Description
+P_{class}
+Probability of the spectrum being of primary spectroscopic type {class} (see Section 2 for the 13 possible classes)
+P_{class}_u
+Uncertainty on the probability of the spectrum being of primary spectroscopic type {class}
+P_ms
+Probability of the presence of a MS companion in the spectrum (see Section 2)
+P_ms_u
+Uncertainty on the probability of the presence of a MS companion in the spectrum
+automated statistical tools for the entire community, and encourage
+other teams to do the same. As the field of astronomy enters a new era
+of Big Data, collaborative efforts will be more important than ever
+to ensure science is not hampered by sheer quantity of observations,
+and to allow astronomers to focus on interesting cases waiting to be
+discovered.
+6 CONCLUSION
+In this paper, we presented a fully automated, data-driven pipeline
+for white dwarf candidate selection and spectroscopic classification
+based on neural network ensembles. The pipeline is composed of
+three modules that can be used independently for a variety of pur-
+poses. The first module calculates the probability of being a white
+dwarf given Gaia photometric and astrometric data, and correctly
+identified > 99% of white dwarfs in the test set, with a contamina-
+tion rate of ∼1.4%. The second module predicts the primary spectro-
+scopic type of a white dwarf given an optical spectrum with > 90%
+precision for most classes according to cross-validation tests. The
+last module calculates the probability of main sequence star contam-
+ination being present in the spectrum with > 91% precision on its
+test set. The two spectroscopic modules were trained with SDSS DR
+16 spectra.
+We applied our pipeline to 1.3M Gaia objects located in, or near
+the white dwarf locus in the Gaia HRD and found 424 096 high
+probability white dwarf candidates for which we cross-matched 36
+523 SDSS DR17 spectra, creating the first white dwarf catalogue
+with quantifiable spectroscopic classifications. The entire process
+is orders of magnitude faster than the current manual inspection
+approach, taking about 10 minutes on a Mac M1 laptop, with about
+9 minutes taken by the candidate selection module. In addition to the
+benefits of quantifiable classifications and speed, neural networks
+remove the need to select manually the relevant features by learning
+which ones best distinguish one class from the others.
+The pipeline presented here will be particularly useful for the
+SDSS-V, as the spectroscopic classification modules are already
+trained on data observed by the same instruments. The pipeline can
+also serve as a base model for other surveys (e.g., DESI, 4MOST,
+LAMOST), where fine-tuning and transfer learning methods can be
+applied to adapt the spectroscopic modules to the new data distri-
+bution. Deep ensembles seem to offer benefits in out-of-distribution
+settings (Ovadia et al. 2019; Gustafsson et al. 2020), although with
+some limitations (Rahaman & Thiery 2021), and may require only
+small adjustments to perform well on data from surveys other than
+SDSS. Future work will aim to provide secondary spectroscopic fea-
+ture identification and improved classification for rare classes such as
+PG1159 and hotDQ. In particular, the addition of synthetic spectra
+to augment the training data offers a promising avenue that will be
+investigated. Such machine learning tools will soon become indis-
+pensable as observations from the next generation of spectroscopic
+surveys will provide millions of spectra for hundreds of thousands of
+white dwarf candidates.
+ACKNOWLEDGEMENTS
+This work was supported in part by the NSERC Canada and by
+the Fund FRQ-NT (Québec). This work has made use of data
+from the European Space Agency (ESA) mission Gaia (https:
+//www.cosmos.esa.int/gaia), processed by the Gaia Data Pro-
+cessing and Analysis Consortium (DPAC, https://www.cosmos.
+esa.int/web/gaia/dpac/consortium). Funding for the DPAC
+has been provided by national institutions, in particular the insti-
+tutions participating in the Gaia Multilateral Agreement. Funding
+for the Sloan Digital Sky Survey (https://www.sdss.org) has
+been provided by the Alfred P. Sloan Foundation, the U.S. Depart-
+ment of Energy Office of Science, and the Participating Institutions.
+SDSS-IV acknowledges support and resources from the Center for
+High-Performance Computing at the University of Utah, and is man-
+aged by the Astrophysical Research Consortium for the Participating
+Institutions of the SDSS Collaboration. This research has also made
+use of the NASA Astrophysics Data System Bibliographic Services;
+the Montreal White Dwarf Database (Dufour et al. 2017); the SIM-
+BAD database, operated at the Centre de Données astronomiques de
+Strasbourg (Wenger et al. 2000)
+DATA AVAILABILITY
+A full description of the data and software availability is provided in
+Section 5.
+MNRAS 000, 1–12 (2022)
+
+Data-Driven Classification of WD Stars
+11
+REFERENCES
+Abadi M., et al., 2015, TensorFlow: Large-Scale Machine Learning on Het-
+erogeneous Systems, https://www.tensorflow.org/
+Abdurro’uf et al., 2022, ApJS, 259, 35
+Ahumada R., et al., 2020, ApJS, 249, 3
+Bédard A., Bergeron P., Fontaine G., 2017, ApJ, 848, 11
+Bédard A., Bergeron P., Brassard P., 2022, ApJ, 930, 8
+Belokurov V., et al., 2020, MNRAS, 496, 1922
+Blanton M. R., et al., 2017, AJ, 154, 28
+Carleo G., Cirac I., Cranmer K., Daudet L., Schuld M., Tishby N., Vogt-
+Maranto L., Zdeborová L., 2019, Reviews of Modern Physics, 91, 045002
+Caron A., Bergeron P., Blouin S., Leggett S. K., 2022, arXiv e-prints, p.
+arXiv:2212.08014
+Chambers K. C., et al., 2016, arXiv e-prints, p. arXiv:1612.05560
+Chandra V., Hwang H.-C., Zakamska N. L., Budavári T., 2020, MNRAS,
+497, 2688
+Coutu S., Dufour P., Bergeron P., Blouin S., Loranger E., Allard N. F., Dunlap
+B. H., 2019, ApJ, 885, 74
+Cui X.-Q., et al., 2012, Research in Astronomy and Astrophysics, 12, 1197
+DESI Collaboration et al., 2016, arXiv e-prints, p. arXiv:1611.00036
+Dufour P., Fontaine G., Liebert J., Schmidt G. D., Behara N., 2008, ApJ, 683,
+978
+Dufour P., Vornanen T., Bergeron P., Fontaine Berdyugin A., 2013, in Krze-
+siński J., Stachowski G., Moskalik P., Bajan K., eds, Astronomical Society
+of the Pacific Conference Series Vol. 469, 18th European White Dwarf
+Workshop.. p. 167
+Dufour P., Blouin S., Coutu S., Fortin-Archambault M., Thibeault C., Berg-
+eron P., Fontaine G., 2017, in Tremblay P. E., Gaensicke B., Marsh T.,
+eds, Astronomical Society of the Pacific Conference Series Vol. 509, 20th
+European White Dwarf Workshop. p. 3 (arXiv:1610.00986)
+Eisenstein D. J., et al., 2006, ApJS, 167, 40
+Farihi J., Dufour P., Wilson T. G., 2022, arXiv e-prints, p. arXiv:2208.05990
+Fort
+S.,
+Hu
+H.,
+Lakshminarayanan
+B.,
+2019,
+arXiv
+e-prints,
+p.
+arXiv:1912.02757
+Fusillo N. P. G., et al., 2021, arXiv
+Gaia Collaboration et al., 2016, A&A, 595, A1
+Gaia Collaboration et al., 2021, A&A, 649, A6
+Gänsicke B. T., Koester D., Girven J., Marsh T. R., Steeghs D., 2010, Science,
+327, 188
+Geier S., 2020, A&A, 635, A193
+Geier S., Østensen R. H., Nemeth P., Gentile Fusillo N. P., Gänsicke B. T.,
+Telting J. H., Green E. M., Schaffenroth J., 2017, A&A, 600, A50
+Goodfellow I. J., Shlens J., Szegedy C., 2014, arXiv e-prints, p.
+arXiv:1412.6572
+Gunn J. E., et al., 2006, AJ, 131, 2332
+Gustafsson F. K., Danelljan M., Schön T. B., 2020, 2020 IEEE/CVF Confer-
+ence on Computer Vision and Pattern Recognition Workshops (CVPRW),
+pp 1289–1298
+Harris H. C., et al., 2003, AJ, 126, 1023
+He H., Garcia E. A., 2009, IEEE Transactions on Knowledge and Data Engi-
+neering, 21, 1263
+Kepler S. O., et al., 2015, MNRAS, 446, 4078
+Kepler S. O., et al., 2016, MNRAS, 455, 3413
+Kepler S. O., Koester D., Pelisoli I., Romero A. D., Ourique G., 2021, MN-
+RAS, 507, 4646
+Kingma D. P., Ba J., 2014, arXiv e-prints, p. arXiv:1412.6980
+Kleinman S. J., et al., 2004, ApJ, 607, 426
+Kleinman S. J., et al., 2013, ApJS, 204, 5
+Koester D., Kepler S. O., 2019, A&A, 628, A102
+Kollmeier J. A., et al., 2017, arXiv e-prints, p. arXiv:1711.03234
+Krizhevsky A., Sutskever I., Hinton G. E., 2012, Communications of the
+ACM, 60, 84
+Lakshminarayanan B., Pritzel A., Blundell C., 2016, arXiv e-prints, p.
+arXiv:1612.01474
+Lee S., Purushwalkam S., Cogswell M., Crandall D., Batra D., 2015, arXiv
+e-prints, p. arXiv:1511.06314
+Leung H. W., Bovy J., 2019, MNRAS, 489, 2079
+López-Sanjuan C., et al., 2022, A&A, 658, A79
+Maas A. L., 2013.
+Nguyen A., Yosinski J., Clune J., 2014, arXiv e-prints, p. arXiv:1412.1897
+Ovadia Y., et al., 2019, arXiv e-prints, p. arXiv:1906.02530
+Rahaman R., Thiery A. H., 2021, in NeurIPS.
+Rebassa-Mansergas A., Ren J. J., Parsons S. G., Gänsicke B. T., Schreiber
+M. R., García-Berro E., Liu X. W., Koester D., 2016, MNRAS, 458, 3808
+Rebassa-Mansergas A., et al., 2021, MNRAS, 506, 5201
+Riello M., et al., 2021, A&A, 649, A3
+Sharma K., Kembhavi A., Kembhavi A., Sivarani T., Abraham S., Vaghmare
+K., 2020, MNRAS, 491, 2280
+Sion E. M., Greenstein J. L., Landstreet J. D., Liebert J., Shipman H. L.,
+Wegner G. A., 1983, ApJ, 269, 253
+Skrutskie M. F., et al., 2006, AJ, 131, 1163
+Smith M. J., Geach J. E., 2022, arXiv e-prints, p. arXiv:2211.03796
+Srivastava N., Hinton G. E., Krizhevsky A., Sutskever I., Salakhutdinov R.,
+2014, J. Mach. Learn. Res., 15, 1929
+Ting Y.-S., Conroy C., Rix H.-W., Cargile P., 2019, ApJ, 879, 69
+Zhu X., Vondrick C., Fowlkes C., Ramanan D., 2015, arXiv e-prints, p.
+arXiv:1503.01508
+de Jong R. S., et al., 2014, in Ramsay S. K., McLean I. S., Takami H.,
+eds, Society of Photo-Optical Instrumentation Engineers (SPIE) Confer-
+ence Series Vol. 9147, Ground-based and Airborne Instrumentation for
+Astronomy V. p. 91470M, doi:10.1117/12.2055826
+APPENDIX A: NEURAL NETWORK ARCHITECTURES
+The architecture details for our three pipeline modules are briefly de-
+scribed here. All modules are implemented using the tensorflow
+(version 2.10) Python library (Abadi et al. 2015). For every net-
+work, regardless of the module, we use an initial learning rate of
+0.01 and the ReduceLROnPlateau callback with a factor of 0.33,
+min_delta of 10−3, and patience of 3. We use the tensorflow
+implementation of the Adam optimizer (Kingma & Ba 2014).
+Starting with the candidate selection module, each network is
+composed of 5 fully-connected layers with 56 hidden units in the first
+layer, doubling up for each successive layer, followed by a single-unit
+fully-connected layer that outputs the probability of an object being
+a white dwarf. The output layer has a sigmoid activation function,
+while all other layers use a LeakyReLU activation (Maas 2013) with
+the leak parameter set to 0.1, along with dropout (Srivastava et al.
+2014) set to 0.5. We train the networks for 100 epochs using a binary
+cross-entropy loss.
+The primary spectroscopic type module neural networks use a
+mix of fully-connected and convolutional (Krizhevsky et al. 2012)
+layers. Each network has 4 convolutional layers with a kernel size of
+5, 14 feature maps and a stride of 2, followed by a fully-connected
+layer with 100 hidden units, and a final fully-connected layer with 13
+hidden units for the output. The output layer has a sigmoid activation
+function, while all other layers use a LeakyReLU activation with the
+leak parameter set to 0.1, along with dropout set to 0.4. We also
+apply dropout to the input with a 0.2 probability. Each network is
+trained for 30 epochs using a categorical cross-entropy loss.
+The main sequence companion detection module neural networks
+have 3 fully-connected layers with 512 hidden units in the first layer,
+doubling down for every successive layer, followed by a single-unit
+fully-connected output layer. The output layer has a sigmoid activa-
+tion function, while all other layers use a LeakyReLU activation with
+the leak parameter set to 0.1, a dropout set to 0.5, and L1L2 kernel
+regularization with default parameters. We also apply dropout to
+the input with a 0.2 probability. We train the networks for 100 epochs
+using a binary cross-entropy loss.
+MNRAS 000, 1–12 (2022)
+
+12
+Vincent et al.
+This paper has been typeset from a TEX/LATEX file prepared by the author.
+MNRAS 000, 1–12 (2022)
+