Universal vector and matrix optimal transport
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911189109309440 |
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| author | Khesin, Boris Modin, Klas |
| author_facet | Khesin, Boris Modin, Klas |
| contents | In this paper we propose a gauge-theoretic approach to the problems of optimal mass transport for vector and matrix densities. This resolves both the issues of positivity and action transitivity constraints. Bures-type metrics on the corresponding semi-direct product groups of diffeomorphisms and gauge transformations are related to Wasserstein-type metrics on vector half-densities and matrix densities via Riemannian submersions. We also describe their relation to Poisson geometry and demonstrate how the momentum map allows one to prove the Riemannian submersion properties. The obtained geodesic equations turn out to be vector versions of the Burgers equations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_02039 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Universal vector and matrix optimal transport Khesin, Boris Modin, Klas Differential Geometry Optimization and Control In this paper we propose a gauge-theoretic approach to the problems of optimal mass transport for vector and matrix densities. This resolves both the issues of positivity and action transitivity constraints. Bures-type metrics on the corresponding semi-direct product groups of diffeomorphisms and gauge transformations are related to Wasserstein-type metrics on vector half-densities and matrix densities via Riemannian submersions. We also describe their relation to Poisson geometry and demonstrate how the momentum map allows one to prove the Riemannian submersion properties. The obtained geodesic equations turn out to be vector versions of the Burgers equations. |
| title | Universal vector and matrix optimal transport |
| topic | Differential Geometry Optimization and Control |
| url | https://arxiv.org/abs/2510.02039 |