Minimax estimation of smooth densities in Wasserstein distance

Abstract

We study nonparametric density estimation problems where error is measured in the Wasserstein distance, a metric on probability distributions popular in many areas of statistics and machine learning. We give the first minimax-optimal rates for this problem for general Wasserstein distances for two classes of densities: smooth probability densities on ${[0,1]}^{\mathit{d}}$ bounded away from 0, and sub-Gaussian densities lying in the Hölder class ${\mathcal{C}}^{\mathit{s}}$, $\mathit{s}\in (0,1)$. Unlike classical nonparametric density estimation, these rates depend on whether the densities in question are bounded below, even in the compactly supported case. Motivated by variational problems involving the Wasserstein distance, we also show how to construct discretely supported measures, suitable for computational purposes, which achieve the minimax rates. Our main technical tool is an inequality giving a nearly tight dual characterization of the Wasserstein distances in terms of Besov norms.