Minimax estimation of smooth densities in Wasserstein distance

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