$L^\infty$-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher's infinitesimal model
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908474113261568 |
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| author | Khudiakova, Ksenia A. Maas, Jan Pedrotti, Francesco |
| author_facet | Khudiakova, Ksenia A. Maas, Jan Pedrotti, Francesco |
| contents | We prove upper bounds on the $L^\infty$-Wasserstein distance from optimal transport between strongly log-concave probability densities and log-Lipschitz perturbations. In the simplest setting, such a bound amounts to a transport-information inequality involving the $L^\infty$-Wasserstein metric and the relative $L^\infty$-Fisher information. We show that this inequality can be sharpened significantly in situations where the involved densities are anisotropic. Our proof is based on probabilistic techniques using Langevin dynamics. As an application of these results, we obtain sharp exponential rates of convergence in Fisher's infinitesimal model from quantitative genetics, generalising recent results by Calvez, Poyato, and Santambrogio in dimension 1 to arbitrary dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_04151 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $L^\infty$-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher's infinitesimal model Khudiakova, Ksenia A. Maas, Jan Pedrotti, Francesco Probability 49Q22, 60J60, 92D25 We prove upper bounds on the $L^\infty$-Wasserstein distance from optimal transport between strongly log-concave probability densities and log-Lipschitz perturbations. In the simplest setting, such a bound amounts to a transport-information inequality involving the $L^\infty$-Wasserstein metric and the relative $L^\infty$-Fisher information. We show that this inequality can be sharpened significantly in situations where the involved densities are anisotropic. Our proof is based on probabilistic techniques using Langevin dynamics. As an application of these results, we obtain sharp exponential rates of convergence in Fisher's infinitesimal model from quantitative genetics, generalising recent results by Calvez, Poyato, and Santambrogio in dimension 1 to arbitrary dimensions. |
| title | $L^\infty$-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher's infinitesimal model |
| topic | Probability 49Q22, 60J60, 92D25 |
| url | https://arxiv.org/abs/2402.04151 |