Section: New Results
Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance
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 [39], we consider the fundamental question of how quickly the empirical measure obtained from independent samples from approaches 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 grows. Collaboration with Jonathan Weed, Francis Bach)