A coupling approach to Lipschitz transport maps
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866910810846003200 |
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| author | Conforti, Giovanni Eichinger, Katharina |
| author_facet | Conforti, Giovanni Eichinger, Katharina |
| contents | In this note, we propose a probabilistic approach to bound the (dimension-free) Lipschitz constant of the Langevin flow map on $\mathbb{R}^d$ introduced by Kim and Milman (2012). As example of application, we construct Lipschitz maps from a uniformly $\log$-concave probability measure to $\log$-Lipschitz perturbations as in Fathi, Mikulincer, Shenfeld (2024). Our proof is based on coupling techniques applied to the stochastic representation of the family of vector fields inducing the transport map. This method is robust enough to relax the uniform convexity to a weak asymptotic convexity condition and to remove the bound on the third derivative of the potential of the source measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_01353 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A coupling approach to Lipschitz transport maps Conforti, Giovanni Eichinger, Katharina Probability Analysis of PDEs Functional Analysis In this note, we propose a probabilistic approach to bound the (dimension-free) Lipschitz constant of the Langevin flow map on $\mathbb{R}^d$ introduced by Kim and Milman (2012). As example of application, we construct Lipschitz maps from a uniformly $\log$-concave probability measure to $\log$-Lipschitz perturbations as in Fathi, Mikulincer, Shenfeld (2024). Our proof is based on coupling techniques applied to the stochastic representation of the family of vector fields inducing the transport map. This method is robust enough to relax the uniform convexity to a weak asymptotic convexity condition and to remove the bound on the third derivative of the potential of the source measure. |
| title | A coupling approach to Lipschitz transport maps |
| topic | Probability Analysis of PDEs Functional Analysis |
| url | https://arxiv.org/abs/2502.01353 |