Minimax Distribution Estimation in Wasserstein Distance

TL;DR

This paper establishes bounds on the minimax rates for estimating probability distributions under the Wasserstein metric, leveraging metric properties and moment assumptions, advancing theoretical understanding in distribution estimation.

Contribution

It provides the first general bounds on minimax rates for Wasserstein distribution estimation based solely on metric and moment conditions.

Findings

Derived upper bounds on estimation rates

Established lower bounds matching the upper bounds

Applicable to a wide class of metric spaces and distributions

Abstract

The Wasserstein metric is an important measure of distance between probability distributions, with applications in machine learning, statistics, probability theory, and data analysis. This paper provides upper and lower bounds on statistical minimax rates for the problem of estimating a probability distribution under Wasserstein loss, using only metric properties, such as covering and packing numbers, of the sample space, and weak moment assumptions on the probability distributions.