On Optimal Transport Maps Between 1 /d-Concave Densities
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910402500100096 |
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| author | Carlier, Guillaume Figalli, Alessio Santambrogio, Filippo |
| author_facet | Carlier, Guillaume Figalli, Alessio Santambrogio, Filippo |
| contents | In this paper, we extend the scope of Caffarelli's contraction theorem, which provides a measure of the Lipschitz constant for optimal transport maps between log-concave probability densities in $\R^d$. Our focus is on a broader category of densities, specifically those that are $\nicefrac{1}{d}$-concave and can be represented as $V^{-d}$, where $V$ is convex. By setting appropriate conditions, we derive linear or sublinear limitations for the optimal transport map. This leads us to a comprehensive Lipschitz estimate that aligns with the principles established in Caffarelli's theorem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_05456 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Optimal Transport Maps Between 1 /d-Concave Densities Carlier, Guillaume Figalli, Alessio Santambrogio, Filippo Analysis of PDEs In this paper, we extend the scope of Caffarelli's contraction theorem, which provides a measure of the Lipschitz constant for optimal transport maps between log-concave probability densities in $\R^d$. Our focus is on a broader category of densities, specifically those that are $\nicefrac{1}{d}$-concave and can be represented as $V^{-d}$, where $V$ is convex. By setting appropriate conditions, we derive linear or sublinear limitations for the optimal transport map. This leads us to a comprehensive Lipschitz estimate that aligns with the principles established in Caffarelli's theorem. |
| title | On Optimal Transport Maps Between 1 /d-Concave Densities |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2404.05456 |