# Minimax estimation of smooth densities in Wasserstein distance

## TL;DR

This paper establishes the first minimax-optimal rates for nonparametric density estimation under Wasserstein distance, revealing dependence on density bounds and providing computationally feasible measures.

## Contribution

It introduces the first minimax rates for Wasserstein-based density estimation and constructs discretely supported measures achieving these rates.

## Key findings

- Derived minimax-optimal rates for general Wasserstein distances.
- Showed the dependence of rates on whether densities are bounded below.
- Provided discretely supported measures suitable for computation.

## 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, and show that, unlike classical nonparametric density estimation, these rates depend on whether the densities in question are bounded below. 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.