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

Understand

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.