@article{da05c302435b486389e089aba65d80ba,
title = "Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance",
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.",
keywords = "Optimal transport, Quantization, Wasserstein metrics",
author = "Jonathan Weed and Francis Bach",
note = "Funding Information: JW and FB acknowledge support from the Chaire {\'E}conomie des nouvelles donn{\'e}es, with the data science Joint Research Initiative with the Fonds AXA pour la recherche, and the Initiative de Recherche “Machine Learning for Large-Scale Insurance” from the Institut Louis Bachelier. JW acknowledges support from NSF Graduate Research Fellowship 1122374 and FB{\textquoteright}s hospitality at INRIA, where this research was conducted. Funding Information: JW and FB acknowledge support from the Chaire ?conomie des nouvelles donn?es, with the data science Joint Research Initiative with the Fonds AXA pour la recherche, and the Initiative de Recherche ?Machine Learning for Large-Scale Insurance? from the Institut Louis Bachelier. JW acknowledges support from NSF Graduate Research Fellowship 1122374 and FB's hospitality at INRIA, where this research was conducted. We thank Guillaume Carlier, Marco Cuturi, and Gabriel Peyr? for discussions related to this work, and we thank the anonymous reviewers for suggesting several references and improvements. Publisher Copyright: {\textcopyright} 2019 ISI/BS.",
year = "2019",
doi = "10.3150/18-BEJ1065",
language = "English (US)",
volume = "25",
pages = "2620--2648",
journal = "Bernoulli",
issn = "1350-7265",
publisher = "International Statistical Institute",
number = "4 A",
}