When Optimal Transport Meets Information Geometry

TL;DR

This paper surveys the emerging intersection of information geometry and optimal transport, highlighting recent advances, geometric structures, and open questions in the combined framework for probability measures.

Contribution

It provides a comprehensive overview of recent research linking information geometry and optimal transport, including key geometric concepts and open problems.

Findings

Discussion of entropy-regularized transport methods

Analysis of divergence functions from c-duality

Exploration of geometric structures like para-Kähler and Kähler geometries

Abstract

Information geometry and optimal transport are two distinct geometric frameworks for modeling families of probability measures. During the recent years, there has been a surge of research endeavors that cut across these two areas and explore their links and interactions. This paper is intended to provide an (incomplete) survey of these works, including entropy-regularized transport, divergence functions arising from $c$-duality, density manifolds and transport information geometry, the para-K\"ahler and K\"ahler geometries underlying optimal transport and the regularity theory for its solutions. Some outstanding questions that would be of interest to audience of both these two disciplines are posed. Our piece also serves as an introduction to the Special Issue on Optimal Transport of the journal Information Geometry.