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

Riemannian block SPD coupling manifold and its application to optimal transport