Strong equivalence between metrics of Wasserstein type

Bayraktar, Erhan; Guo, Gaoyue

doi:10.1214/21-ecp383

Cited by 11 publications

(12 citation statements)

References 12 publications

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“…h ≥ 0 (RBF kernels are positive definite). We conclude that k − k is positive definite, hence (23) holds for RBF kernels.…”

Section: A3 Proof Of Theoremmentioning

confidence: 67%

“…We conclude that (23) holds with F defined as the unit ball of the RKHS associated with the linear kernel k(t i , t j ) = t i t j for t i , t j ∈ R, and F being the unit ball of the RKHS associated with the rescaled linear kernel k(x i , x j ) = x i x j /d for x i , x j ∈ R d .…”

Section: A3 Proof Of Theoremmentioning

confidence: 80%

“…[21] improved this result and showed that convergence in SW implies weak convergence in general. Finally, very recently [23] proved that SW and Wasserstein distances are weakly equivalent in general, meaning that convergence in one of them implies convergence in the other one. Combined with the statistical properties of SW, these results show that while inducing a similar topology to the one of Wasserstein distance, SW can further enjoy nice statistical properties, whereas the Wasserstein distance cannot.…”

Section: Introductionmentioning

confidence: 99%

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Statistical and Topological Properties of Sliced Probability Divergences

Nadjahi¹,

Durmus²,

Chizat³

et al. 2020

Preprint

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The idea of slicing divergences has been proven to be successful when comparing two probability measures in various machine learning applications including generative modeling, and consists in computing the expected value of a 'base divergence' between one-dimensional random projections of the two measures. However, the computational and statistical consequences of such a technique have not yet been well-established. In this paper, we aim at bridging this gap and derive some properties of sliced divergence functions. First, we show that slicing preserves the metric axioms and the weak continuity of the divergence, implying that the sliced divergence will share similar topological properties. We then precise the results in the case where the base divergence belongs to the class of integral probability metrics. On the other hand, we establish that, under mild conditions, the sample complexity of the sliced divergence does not depend on the dimension, even when the base divergence suffers from the curse of dimensionality. We finally apply our general results to the Wasserstein distance and Sinkhorn divergences, and illustrate our theory on both synthetic and real data experiments.

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“…h ≥ 0 (RBF kernels are positive definite). We conclude that k − k is positive definite, hence (23) holds for RBF kernels.…”

Section: A3 Proof Of Theoremmentioning

confidence: 67%

Section: A3 Proof Of Theoremmentioning

confidence: 80%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Statistical and Topological Properties of Sliced Probability Divergences

Nadjahi¹,

Durmus²,

Chizat³

et al. 2020

Preprint

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“…The sliced distances W p and W p are metrics on P p (R d ) and, in fact, induce the same topology as W p [BG21].…”

Section: Sliced Wasserstein Distancesmentioning

confidence: 99%

Statistical inference with regularized optimal transport

Goldfeld¹,

Kato²,

Rioux³

et al. 2022

Preprint

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Optimal transport (OT) is a versatile framework for comparing probability measures, with many applications to statistics, machine learning, and applied mathematics. However, OT distances suffer from computational and statistical scalability issues to high dimensions, which motivated the study of regularized OT methods like slicing, smoothing, and entropic penalty. This work establishes a unified framework for deriving limit distributions of empirical regularized OT distances, semiparametric efficiency of the plug-in empirical estimator, and bootstrap consistency. We apply the unified framework to provide a comprehensive statistical treatment of: (i) average-and max-sliced p-Wasserstein distances, for which several gaps in existing literature are closed; (ii) smooth distances with compactly supported kernels, the analysis of which is motivated by computational considerations; and (iii) entropic OT, for which our method generalizes existing limit distribution results and establishes, for the first time, efficiency and bootstrap consistency. While our focus is on these three regularized OT distances as applications, the flexibility of the proposed framework renders it applicable to broad classes of functionals beyond these examples.

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“…Besides, SW has been shown to offer nice theoretical properties as well. Indeed, it satisfies the metric axioms [13], the estimators obtained by minimizing SW are asymptotically consistent [14], the convergence in SW is equivalent to the convergence in Wasserstein [14,15], and even though the sample complexity of Wasserstein grows exponentially with the data dimension [16,17,18], the sample complexity of SW does not depend on the dimension [19]. However, the latter study also demonstrated with a theoretical error bound, that the quality of the Monte Carlo estimate of SW depends on the number of projections and the variance of the one-dimensional Wasserstein distances [19,Theorem 6].…”

Section: Introductionmentioning

confidence: 99%

Fast Approximation of the Sliced-Wasserstein Distance Using Concentration of Random Projections

Nadjahi¹,

Durmus²,

Jacob³

et al. 2021

Preprint

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The Sliced-Wasserstein distance (SW) is being increasingly used in machine learning applications as an alternative to the Wasserstein distance and offers significant computational and statistical benefits. Since it is defined as an expectation over random projections, SW is commonly approximated by Monte Carlo. We adopt a new perspective to approximate SW by making use of the concentration of measure phenomenon: under mild assumptions, one-dimensional projections of a high-dimensional random vector are approximately Gaussian. Based on this observation, we develop a simple deterministic approximation for SW. Our method does not require sampling a number of random projections, and is therefore both accurate and easy to use compared to the usual Monte Carlo approximation. We derive nonasymptotical guarantees for our approach, and show that the approximation error goes to zero as the dimension increases, under a weak dependence condition on the data distribution. We validate our theoretical findings on synthetic datasets, and illustrate the proposed approximation on a generative modeling problem.

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Strong equivalence between metrics of Wasserstein type

Cited by 11 publications

References 12 publications

Statistical and Topological Properties of Sliced Probability Divergences

Statistical and Topological Properties of Sliced Probability Divergences

Statistical inference with regularized optimal transport

Fast Approximation of the Sliced-Wasserstein Distance Using Concentration of Random Projections

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