Distributional Limit Theory for Optimal Transport

TL;DR

This paper reviews the statistical inference for optimal transport, focusing on the asymptotic behavior of empirical OT quantities and highlighting open problems in the field.

Contribution

It provides a comprehensive review of the limit theory for empirical optimal transport and discusses key open problems in the area.

Findings

Asymptotic distributions of empirical OT plans and costs are characterized.

Identification of key open problems in the statistical inference of OT.

Applications of OT in various fields are discussed with a focus on empirical measures.

Abstract

Optimal Transport (OT) is a resource allocation problem with applications in biology, data science, economics and statistics, among others. In some of the applications, practitioners have access to samples which approximate the continuous measure. Hence the quantities of interest derived from OT -- plans, maps and costs -- are only available in their empirical versions. Statistical inference on OT aims at finding confidence intervals of the population plans, maps and costs. In recent years this topic gained an increasing interest in the statistical community. In this paper we provide a comprehensive review of the most influential results on this research field, underlying the some of the applications. Finally, we provide a list of open problems.