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

Jonathan Weed, Francis Bach

Research output: Contribution to journalArticlepeer-review

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 μ approaches μ 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.

Original languageEnglish (US)
Pages (from-to)2620-2648
Number of pages29
JournalBernoulli
Volume25
Issue number4 A
DOIs
StatePublished - 2019

Keywords

  • Optimal transport
  • Quantization
  • Wasserstein metrics

ASJC Scopus subject areas

  • Statistics and Probability

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