Riemannian block SPD coupling manifold and its application to optimal transport
This work addresses optimal transport for matrix-valued data, which is incremental as it extends existing methods to a more complex setting.
The authors tackled the optimal transport problem between symmetric positive definite matrix-valued measures by formulating it as a generalized problem with block matrices and endowing the set of block-coupling matrices with a novel Riemannian manifold structure, enabling the use of Riemannian optimization to solve these problems.
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.