Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance

Jonathan Weed 1 Francis Bach 2
2 SIERRA - Statistical Machine Learning and Parsimony
DI-ENS - Département d'informatique de l'École normale supérieure, CNRS - Centre National de la Recherche Scientifique, Inria de Paris
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.
Type de document :
Pré-publication, Document de travail
2017
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https://hal.archives-ouvertes.fr/hal-01555307
Contributeur : Francis Bach <>
Soumis le : mardi 4 juillet 2017 - 07:41:25
Dernière modification le : jeudi 26 avril 2018 - 10:29:04

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  • HAL Id : hal-01555307, version 1
  • ARXIV : 1707.00087

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Jonathan Weed, Francis Bach. Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance. 2017. 〈hal-01555307〉

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