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

Jonathan Weed (1) , Francis Bach (2)

1 MIT - Massachusetts Institute of Technology
2 SIERRA - Statistical Machine Learning and Parsimony

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.

Domaines: Apprentissage [cs.LG]

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