Vector valued optimal transport: from dynamic to static formulations

TL;DR

This paper develops a unified theory of vector valued optimal transport, connecting dynamic and static formulations, with applications in PDEs and data analysis, and introduces a linearizable static formulation for efficient computation.

Contribution

It unifies dynamic and static vector valued optimal transport notions and introduces a linearizable static formulation for faster computations.

Findings

Established sharp inequalities relating four notions of vector valued optimal transport.

Proved mutual bi-Hölder equivalence of the distances.

Identified practical advantages and potential applications in PDEs and data analysis.

Abstract

Motivated by applications in classification of vector valued measures and multispecies PDE, we develop a theory that unifies existing notions of vector valued optimal transport, from dynamic formulations (\`a la Benamou-Brenier) to static formulations (\`a la Kantorovich). In our framework, vector valued measures are modeled as probability measures on a product space $\mathbb{R}^d \times G$, where $G$ is a weighted graph over a finite set of nodes and the graph geometry strongly influences the associated dynamic and static distances. We obtain sharp inequalities relating four notions of vector valued optimal transport and prove that the distances are mutually bi-H\"older equivalent. We discuss the theoretical and practical advantages of each metric and indicate potential applications in multispecies PDE and data analysis. In particular, one of the static formulations discussed in the paper is amenable to linearization, a technique that has been explored in recent years to accelerate the computation of pairwise optimal transport distances.