Estimation of smooth densities in Wasserstein distance.

The Wasserstein distances are a set of metrics on probability distributions supported on $mathbb{R}^d$ with applications throughout statistics and machine learning. Often, such distances are used in the context of variational problems, in which the statistician employs in place of an unknown measure a proxy constructed on the basis of independent samples. This raises the basic question of how well measures can be approximated in Wasserstein distance. While it is known that an empirical measure comprising i.i.d. samples is rate-optimal for general measures, no improved results were known for measures possessing smooth densities. We prove the first minimax rates for estimation of smooth densities for general Wasserstein distances, thereby showing how the curse of dimensionality can be alleviated for sufficiently regular measures. We also show how to construct discretely supported measures, suitable for computational purposes, which enjoy improved rates. Our approach is based on novel bounds between the Wasserstein distances and suitable Besov norms, which may be of independent interest.