Riemannian block SPD coupling manifold and its application to optimal transport

TL;DR

This paper introduces a Riemannian manifold framework for solving optimal transport problems involving block-structured SPD matrix measures, enabling efficient Riemannian optimization for complex matrix-valued OT tasks.

Contribution

It formulates a novel Riemannian manifold structure for block SPD coupling matrices in optimal transport, facilitating advanced optimization techniques for matrix-valued measures.

Findings

Effective Riemannian optimization for SPD matrix OT problems

Versatile framework applicable to various applications

Improved computational efficiency in matrix-valued OT

Abstract

In this work, we study the optimal transport (OT) problem between symmetric positive definite (SPD) matrix-valued measures. We formulate the above as a generalized optimal transport problem where the cost, the marginals, and the coupling are represented as block matrices and each component block is a SPD matrix. The summation of row blocks and column blocks in the coupling matrix are constrained by the given block-SPD marginals. We endow the set of such block-coupling matrices with a novel Riemannian manifold structure. This allows to exploit the versatile Riemannian optimization framework to solve generic SPD matrix-valued OT problems. We illustrate the usefulness of the proposed approach in several applications.