Metric conditions that guarantee existence and uniqueness of Optimal Transport maps
Fuente:
arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912095421857792 |
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| author | Li, Shucheng Magnabosco, Mattia Schultz, Timo |
| author_facet | Li, Shucheng Magnabosco, Mattia Schultz, Timo |
| contents | We investigate metric conditions that allow to prove existence and uniqueness of a map solving the Monge problem between two marginals in a metric (measure) space, proving two main results. Firstly, we introduce a nonsmooth version of the Riemannian twist condition that we call local metric twist condition, showing, under this assumption on the cost function, existence and uniqueness of optimal transport maps. Secondly, we prove the same result for cost equal to $d^2$ in a metric space $(X, d)$ satisfying a quantitative non-branching assumption, that we call locally-uniformly non-branching. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22567 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Metric conditions that guarantee existence and uniqueness of Optimal Transport maps Li, Shucheng Magnabosco, Mattia Schultz, Timo Metric Geometry We investigate metric conditions that allow to prove existence and uniqueness of a map solving the Monge problem between two marginals in a metric (measure) space, proving two main results. Firstly, we introduce a nonsmooth version of the Riemannian twist condition that we call local metric twist condition, showing, under this assumption on the cost function, existence and uniqueness of optimal transport maps. Secondly, we prove the same result for cost equal to $d^2$ in a metric space $(X, d)$ satisfying a quantitative non-branching assumption, that we call locally-uniformly non-branching. |
| title | Metric conditions that guarantee existence and uniqueness of Optimal Transport maps |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2410.22567 |