Riemannian block SPD coupling manifold and its application to optimal transport
Functional Analysis
2023-02-09 v2 Optimization and Control
Machine Learning
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.
Keywords
Cite
@article{arxiv.2201.12933,
title = {Riemannian block SPD coupling manifold and its application to optimal transport},
author = {Andi Han and Bamdev Mishra and Pratik Jawanpuria and Junbin Gao},
journal= {arXiv preprint arXiv:2201.12933},
year = {2023}
}