Statistical Aspects of Wasserstein Distances

TL;DR

This paper reviews the mathematical foundations and statistical applications of Wasserstein distances, highlighting their versatility in analyzing probability distributions and their recent role in statistical inference.

Contribution

It provides a comprehensive overview of Wasserstein distances, emphasizing their mathematical properties and recent developments in statistical methodology and inference.

Findings

Wasserstein distances effectively measure distribution perturbations.

They facilitate weak convergence and moment convergence analysis.

Recent advances include their use as objects of statistical inference.

Abstract

Wasserstein distances are metrics on probability distributions inspired by the problem of optimal mass transportation. Roughly speaking, they measure the minimal effort required to reconfigure the probability mass of one distribution in order to recover the other distribution. They are ubiquitous in mathematics, with a long history that has seen them catalyse core developments in analysis, optimization, and probability. Beyond their intrinsic mathematical richness, they possess attractive features that make them a versatile tool for the statistician: they can be used to derive weak convergence and convergence of moments, and can be easily bounded; they are well-adapted to quantify a natural notion of perturbation of a probability distribution; and they seamlessly incorporate the geometry of the domain of the distributions in question, thus being useful for contrasting complex objects. Consequently, they frequently appear in the development of statistical theory and inferential methodology, and have recently become an object of inference in themselves. In this review, we provide a snapshot of the main concepts involved in Wasserstein distances and optimal transportation, and a succinct overview of some of their many statistical aspects.