On optimal transport of matrix-valued measures

TL;DR

This paper introduces a novel framework for optimal transport of matrix-valued measures, extending classical metrics to a matricial setting inspired by fluid dynamics and establishing foundational properties of this new space.

Contribution

It defines the Kantorovich-Bures metric space for matrix-valued measures and proves key topological, metric, and geometric properties, including the existence of optimal transport paths.

Findings

Defined the Kantorovich-Bures metric space for matrix-valued measures

Proved existence of optimal transportation paths in this space

Established topological and geometric properties of the space

Abstract

We suggest a new way of defining optimal transport of positive-semidefinite matrix-valued measures. It is inspired by a recent rendering of the incompressible Euler equations and related conservative systems as concave maximization problems. The main object of our attention is the Kantorovich-Bures metric space, which is a matricial analogue of the Wasserstein and Hellinger-Kantorovich metric spaces. We establish some topological, metric and geometric properties of this space, which includes the existence of the optimal transportation path.