On a general matrix-valued unbalanced optimal transport problem

TL;DR

This paper introduces a comprehensive class of matrix-valued unbalanced optimal transport distances called weighted Wasserstein-Bures distance, generalizing existing models and analyzing their geometric properties in the space of positive semi-definite matrix measures.

Contribution

It defines a new class of transport distances with a generalized formulation, unifies existing models, and characterizes the geometric and topological structure of the measure space.

Findings

The space is a complete geodesic space.

The distance exhibits a conic structure.

Includes models like Kantorovich-Bures and Wasserstein-Fisher-Rao.

Abstract

We introduce a general class of transport distances ${\rm WB}_{\Lambda}$ over the space of positive semi-definite matrix-valued Radon measures $\mathcal{M}(\Omega,\mathbb{S}_+^n)$, called the weighted Wasserstein-Bures distance. Such a distance is defined via a generalized Benamou-Brenier formulation with a weighted action functional and an abstract matricial continuity equation, which leads to a convex optimization problem. Some recently proposed models, including the Kantorovich-Bures distance and the Wasserstein-Fisher-Rao distance, can naturally fit into ours. We give a complete characterization of the minimizer and explore the topological and geometrical properties of the space $(\mathcal{M}(\Omega,\mathbb{S}_+^n),{\rm WB}_{\Lambda})$. In particular, we show that $(\mathcal{M}(\Omega,\mathbb{S}_+^n),{\rm WB}_{\Lambda})$ is a complete geodesic space and exhibits a conic structure.