The intrinsic dynamics of optimal transport

TL;DR

This paper explores the conditions under which optimal transport solutions are unique, introducing a new dynamic perspective that allows for constructing costs ensuring uniqueness across various topologies.

Contribution

It introduces a multivalued dynamics framework to analyze optimal transport, enabling the construction of smooth costs with guaranteed uniqueness on manifolds of arbitrary topology.

Findings

Unique optimal transport solutions depend on the presence of periodic trajectories in the induced dynamics.

The authors construct smooth costs on any pair of compact manifolds that guarantee unique solutions.

The approach links the topology of the underlying spaces to the uniqueness of optimal transport.

Abstract

The question of which costs admit unique optimizers in the Monge-Kantorovich problem of optimal transportation between arbitrary probability densities is investigated. For smooth costs and densities on compact manifolds, the only known examples for which the optimal solution is always unique require at least one of the two underlying spaces to be homeomorphic to a sphere. We introduce a (multivalued) dynamics which the transportation cost induces between the target and source space, for which the presence or absence of a sufficiently large set of periodic trajectories plays a role in determining whether or not optimal transport is necessarily unique. This insight allows us to construct smooth costs on a pair of compact manifolds with arbitrary topology, so that the optimal transportation between any pair of probility densities is unique.