Riemannian block SPD coupling manifold and its application to optimal transport

Optimal transport (OT) has seen its popularity in various fields of applications. We start by observing that the OT problem can be viewed as an instance of a general symmetric positive definite (SPD) matrix-valued OT 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.