Statistical Inference for Optimal Transport Maps: Recent Advances and Perspectives

TL;DR

This paper reviews recent theoretical advances in statistical inference for optimal transport maps, focusing on estimation, limit theorems, and future research directions to aid practical applications.

Contribution

It synthesizes recent progress in estimating OT maps and establishing limit theorems, highlighting new methods and perspectives for future research.

Findings

Recent limit theorems for OT map estimation

Development of inferential tools for OT maps

Extensions to special cases and variants

Abstract

In many applications of optimal transport (OT), the object of primary interest is the optimal transport map. This map rearranges mass from one probability distribution to another in the most efficient way possible by minimizing a specified cost. In this paper we review recent advances in estimating and developing limit theorems for the OT map, using samples from the underlying distributions. We also review parallel lines of work that establish similar results for special cases and variants of the basic OT setup. We conclude with a discussion of key directions for future research with the goal of providing practitioners with reliable inferential tools.