Riemannian block SPD coupling manifold and its application to optimal transport

• URL: http://arxiv.org/abs/2201.12933v1
• Date: Sun, 30 Jan 2022 22:44:47 GMT
• Title: Riemannian block SPD coupling manifold and its application to optimal transport
• Abstract summary: We show that the OT problem can be viewed as an instance of a general symmetric positive definite (SPD) matrix-valued OT problem. The summation of row blocks and column blocks in the coupling matrix are constrained by the given block-SPD marginals.
• Score: 40.23168342389821
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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