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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2411.10235 |
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| _version_ | 1866912385714880512 |
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| author | Stéphanovitch, Arthur |
| author_facet | Stéphanovitch, Arthur |
| contents | We extend the classical regularity theory of optimal transport to non-optimal transport maps generated by heat flow for perturbations of Gaussian measures. Considering probability measures of the form $dμ(x) = \exp\left(-\frac{|x|^2}{2} + a(x)\right)dx$ on $\mathbb{R}^d$ where $a$ has Hölder regularity $C^β$ with $β\geq 0$; we show that the Langevin map transporting the $d$-dimensional Gaussian distribution onto $μ$ achieves Hölder regularity $C^{β+ 1}$, up to a logarithmic factor. We additionally present applications of this result to functional inequalities and generative modelling. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_10235 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Smooth transport map via diffusion process Stéphanovitch, Arthur Probability We extend the classical regularity theory of optimal transport to non-optimal transport maps generated by heat flow for perturbations of Gaussian measures. Considering probability measures of the form $dμ(x) = \exp\left(-\frac{|x|^2}{2} + a(x)\right)dx$ on $\mathbb{R}^d$ where $a$ has Hölder regularity $C^β$ with $β\geq 0$; we show that the Langevin map transporting the $d$-dimensional Gaussian distribution onto $μ$ achieves Hölder regularity $C^{β+ 1}$, up to a logarithmic factor. We additionally present applications of this result to functional inequalities and generative modelling. |
| title | Smooth transport map via diffusion process |
| topic | Probability |
| url | https://arxiv.org/abs/2411.10235 |