Vector valued optimal transport: from dynamic to static formulations
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866913823219253248 |
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| author | Craig, Katy Trillos, Nicolás García Nikolić, Đorđe |
| author_facet | Craig, Katy Trillos, Nicolás García Nikolić, Đorđe |
| contents | Motivated by applications in classification of vector valued measures and multispecies PDE, we develop a theory that unifies existing notions of vector valued optimal transport, from dynamic formulations (à la Benamou-Brenier) to static formulations (à la Kantorovich). In our framework, vector valued measures are modeled as probability measures on a product space $\mathbb{R}^d \times G$, where $G$ is a weighted graph over a finite set of nodes and the graph geometry strongly influences the associated dynamic and static distances. We obtain sharp inequalities relating four notions of vector valued optimal transport and prove that the distances are mutually bi-Hölder equivalent. We discuss the theoretical and practical advantages of each metric and indicate potential applications in multispecies PDE and data analysis. In particular, one of the static formulations discussed in the paper is amenable to linearization, a technique that has been explored in recent years to accelerate the computation of pairwise optimal transport distances. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_03670 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Vector valued optimal transport: from dynamic to static formulations Craig, Katy Trillos, Nicolás García Nikolić, Đorđe Analysis of PDEs Machine Learning Metric Geometry Motivated by applications in classification of vector valued measures and multispecies PDE, we develop a theory that unifies existing notions of vector valued optimal transport, from dynamic formulations (à la Benamou-Brenier) to static formulations (à la Kantorovich). In our framework, vector valued measures are modeled as probability measures on a product space $\mathbb{R}^d \times G$, where $G$ is a weighted graph over a finite set of nodes and the graph geometry strongly influences the associated dynamic and static distances. We obtain sharp inequalities relating four notions of vector valued optimal transport and prove that the distances are mutually bi-Hölder equivalent. We discuss the theoretical and practical advantages of each metric and indicate potential applications in multispecies PDE and data analysis. In particular, one of the static formulations discussed in the paper is amenable to linearization, a technique that has been explored in recent years to accelerate the computation of pairwise optimal transport distances. |
| title | Vector valued optimal transport: from dynamic to static formulations |
| topic | Analysis of PDEs Machine Learning Metric Geometry |
| url | https://arxiv.org/abs/2505.03670 |