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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.06065 |
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| _version_ | 1866915921402003456 |
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| author | Stéphanovitch, Arthur |
| author_facet | Stéphanovitch, Arthur |
| contents | Under general assumptions on the target distribution $p^\star$, we establish a sharp Lipschitz regularity theory for flow-matching vector fields and diffusion-model scores, with optimal dependence on time and dimension. As applications, we obtain Wasserstein discretization bounds for Euler-type samplers in dimension $d$: with $N$ discretization steps, the error achieves the optimal rate $\sqrt{d}/N$ up to logarithmic factors. Moreover, the constants do not deteriorate exponentially with the spatial extent of $p^\star$. We also show that the one-sided Lipschitz control yields a globally Lipschitz transport map from the standard Gaussian to $p^\star$, which implies Poincaré and log-Sobolev inequalities for a broad class of probability measures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_06065 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lipschitz regularity in Flow Matching and Diffusion Models: sharp sampling rates and functional inequalities Stéphanovitch, Arthur Statistics Theory Probability Machine Learning Under general assumptions on the target distribution $p^\star$, we establish a sharp Lipschitz regularity theory for flow-matching vector fields and diffusion-model scores, with optimal dependence on time and dimension. As applications, we obtain Wasserstein discretization bounds for Euler-type samplers in dimension $d$: with $N$ discretization steps, the error achieves the optimal rate $\sqrt{d}/N$ up to logarithmic factors. Moreover, the constants do not deteriorate exponentially with the spatial extent of $p^\star$. We also show that the one-sided Lipschitz control yields a globally Lipschitz transport map from the standard Gaussian to $p^\star$, which implies Poincaré and log-Sobolev inequalities for a broad class of probability measures. |
| title | Lipschitz regularity in Flow Matching and Diffusion Models: sharp sampling rates and functional inequalities |
| topic | Statistics Theory Probability Machine Learning |
| url | https://arxiv.org/abs/2604.06065 |