Multi-marginal optimal transport: theory and applications

TL;DR

This paper surveys multi-marginal optimal transport, exploring its theoretical foundations, solution properties, and applications, highlighting differences from classical cases and the influence of cost functions.

Contribution

It provides a comprehensive overview of the theory, solution structure, and applications of multi-marginal optimal transport, emphasizing new insights and divergences from classical models.

Findings

Solutions' uniqueness and structure depend delicately on the cost function.

There are notable differences from the classical two-marginal case.

The paper discusses two specific applications of the multi-marginal problem.

Abstract

Over the past five years, multi-marginal optimal transport, a generalization of the well known optimal transport problem of Monge and Kantorovich, has begun to attract considerable attention, due in part to a wide variety of emerging applications. Here, we survey this problem, addressing fundamental theoretical questions including the uniqueness and structure of solutions. The (partial) answers to these questions uncover a surprising divergence from the classical two marginal setting, and reflect a delicate dependence on the cost function. We go one to describe two applications of the multi-marginal problem.