Riemannian Block SPD Coupling Manifold and Its Application to Optimal Transport
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.
Cite
Text
Han et al. "Riemannian Block SPD Coupling Manifold and Its Application to Optimal Transport." Machine Learning, 2024. doi:10.1007/S10994-022-06258-WMarkdown
[Han et al. "Riemannian Block SPD Coupling Manifold and Its Application to Optimal Transport." Machine Learning, 2024.](https://mlanthology.org/mlj/2024/han2024mlj-riemannian/) doi:10.1007/S10994-022-06258-WBibTeX
@article{han2024mlj-riemannian,
title = {{Riemannian Block SPD Coupling Manifold and Its Application to Optimal Transport}},
author = {Han, Andi and Mishra, Bamdev and Jawanpuria, Pratik and Gao, Junbin},
journal = {Machine Learning},
year = {2024},
pages = {1595-1622},
doi = {10.1007/S10994-022-06258-W},
volume = {113},
url = {https://mlanthology.org/mlj/2024/han2024mlj-riemannian/}
}