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November 2019 Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance
Jonathan Weed, Francis Bach
Bernoulli 25(4A): 2620-2648 (November 2019). DOI: 10.3150/18-BEJ1065

Abstract

The Wasserstein distance between two probability measures on a metric space is a measure of closeness with applications in statistics, probability, and machine learning. In this work, we consider the fundamental question of how quickly the empirical measure obtained from $n$ independent samples from $\mu$ approaches $\mu$ in the Wasserstein distance of any order. We prove sharp asymptotic and finite-sample results for this rate of convergence for general measures on general compact metric spaces. Our finite-sample results show the existence of multi-scale behavior, where measures can exhibit radically different rates of convergence as $n$ grows.

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Jonathan Weed. Francis Bach. "Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance." Bernoulli 25 (4A) 2620 - 2648, November 2019. https://doi.org/10.3150/18-BEJ1065

Information

Received: 1 August 2017; Revised: 1 May 2018; Published: November 2019
First available in Project Euclid: 13 September 2019

zbMATH: 07110107
MathSciNet: MR4003560
Digital Object Identifier: 10.3150/18-BEJ1065

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

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Vol.25 • No. 4A • November 2019
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