Faster Wasserstein Distance Estimation with the Sinkhorn Divergence

Chizat, Lenaic; Roussillon, Pierre; Léger, Flavien; Vialard, François-Xavier; Peyré, Gabriel

doi:10.48550/arxiv.2006.08172

Cited by 7 publications

(12 citation statements)

References 30 publications

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“…The performance of other algorithms can be further investigated. Finally, it is possible to reduce the computational time for our method by parallel computing, faster Monte Carlo sampling methods (Rubinstein and Kroese 2016) and more efficient computation algorithms of Wasserstein distances (Chizat et al 2020), which we leave for future research. ).…”

Section: Discussionmentioning

confidence: 99%

A nonparametric Bayesian approach for simulation optimization with input uncertainty

Wang,

Zhang,

2020

Preprint

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Stochastic simulation models are increasingly popular for analyzing complex stochastic systems. However, the input distributions required to drive the simulation are typically unknown in practice and are usually estimated from real world data. Since the amount of real world data tends to be limited, the resulting estimation of the input distribution will contain errors. This estimation error is commonly known as input uncertainty. In this paper, we consider the stochastic simulation optimization problem when input uncertainty is present and assume that both the input distribution family and the distribution parameters are unknown. We adopt a nonparametric Bayesian approach to model the input uncertainty and propose a nonparametric Bayesian risk optimization (NBRO) framework to study simulation optimization under input distribution uncertainty. We establish the consistency and asymptotic normality of the value of the objective function, the optimal solution, and the optimal value for the NBRO problem. We further propose a simulation optimization algorithm based on Efficient Global Optimization (EGO), which explicitly takes into account the input uncertainty so as to efficiently solve the NBRO problem. We study the consistency property of the modified EGO algorithm and empirically test its efficiency with an inventory example and a critical care facility problem.

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Section: Discussionmentioning

confidence: 99%

A nonparametric Bayesian approach for simulation optimization with input uncertainty

Wang,

Zhang,

2020

Preprint

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“…For more information on I 0 (µ, ν) see [14]. We remark that when α ≥ 2 then under A4,A5,A6, it has been shown I 0 (µ, ν) ≤ C for a positive constant C [14].…”

Section: Properties Of the Entropic Mapmentioning

confidence: 99%

“…For more information on I 0 (µ, ν) see [14]. We remark that when α ≥ 2 then under A4,A5,A6, it has been shown I 0 (µ, ν) ≤ C for a positive constant C [14]. The notation a b means that there exists positive constants c, C such that ca ≤ b ≤ Ca and a b means that there is a positive constant C such that a ≤ Cb.…”

Section: Properties Of the Entropic Mapmentioning

confidence: 99%

Measure Estimation in the Barycentric Coding Model

Werenski¹,

Jiang²,

Tasissa³

et al. 2022

Preprint

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This paper considers the problem of measure estimation under the barycentric coding model (BCM), in which an unknown measure is assumed to belong to the set of Wasserstein-2 barycenters of a finite set of known measures. Estimating a measure under this model is equivalent to estimating the unknown barycenteric coordinates. We provide novel geometrical, statistical, and computational insights for measure estimation under the BCM, consisting of three main results. Our first main result leverages the Riemannian geometry of Wasserstein-2 space to provide a procedure for recovering the barycentric coordinates as the solution to a quadratic optimization problem assuming access to the true reference measures. The essential geometric insight is that the parameters of this quadratic problem are determined by inner products between the optimal displacement maps from the given measure to the reference measures defining the BCM. Our second main result then establishes an algorithm for solving for the coordinates in the BCM when all the measures are observed empirically via i.i.d. samples. We prove precise rates of convergence for this algorithm-determined by the smoothness of the underlying measures and their dimensionality-thereby guaranteeing its statistical consistency. Finally, we demonstrate the utility of the BCM and associated estimation procedures in three application areas: (i) covariance estimation for Gaussian measures; (ii) image processing; and (iii) natural language processing.

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“…The approach taken in this paper and the techniques used for its analysis bear various connections to recent developments in the literature on optimal transport, e.g., on the estimation of (smooth) optimal transport maps [40,41,42,43,44]. Key steps in our proofs are based on adaptations of parts of the analysis in [42,43,44].…”

Section: Introductionmentioning

confidence: 99%

Permuted and Unlinked Monotone Regression in $\mathbb{R}^d$: an approach based on mixture modeling and optimal transport

Slawski¹,

Sen²

2022

Preprint

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Suppose that we have a regression problem with response variable Y ∈ R d and predictor X ∈ R d , for d ≥ 1. In permuted or unlinked regression we have access to separate unordered data on X and Y , as opposed to data on (X, Y )-pairs in usual regression. So far in the literature the case d = 1 has received attention, see e.g., the recent papers by Rigollet and Weed [Information & Inference,8, and Balabdaoui et al. [J. Mach. Learn. Res., 22(172), 1-60]. In this paper, we consider the general multivariate setting with d ≥ 1. We show that the notion of cyclical monotonicity of the regression function is sufficient for identification and estimation in the permuted/unlinked regression model. We study permutation recovery in the permuted regression setting and develop a computationally efficient and easy-to-use algorithm for denoising based on the Kiefer-Wolfowitz [Ann. Math. Statist.,27,[887][888][889][890][891][892][893][894][895][896][897][898][899][900][901][902][903][904][905][906] nonparametric maximum likelihood estimator and techniques from the theory of optimal transport. We provide explicit upper bounds on the associated mean squared denoising error for Gaussian noise. As in previous work on the case d = 1, the permuted/unlinked setting involves slow (logarithmic) rates of convergence rooting in the underlying deconvolution problem. Numerical studies corroborate our theoretical analysis and show that the proposed approach performs at least on par with the methods in the aforementioned prior work in the case d = 1 while achieving substantial reductions in terms of computational complexity.

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Faster Wasserstein Distance Estimation with the Sinkhorn Divergence

Cited by 7 publications

References 30 publications

A nonparametric Bayesian approach for simulation optimization with input uncertainty

A nonparametric Bayesian approach for simulation optimization with input uncertainty

Measure Estimation in the Barycentric Coding Model

Permuted and Unlinked Monotone Regression in $\mathbb{R}^d$: an approach based on mixture modeling and optimal transport

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