Lipschitz regularity in Flow Matching and Diffusion Models: sharp sampling rates and functional inequalities

• URL: http://arxiv.org/abs/2604.06065v1
• Date: Tue, 07 Apr 2026 16:46:40 GMT
• Title: Lipschitz regularity in Flow Matching and Diffusion Models: sharp sampling rates and functional inequalities
• Abstract summary: We establish a sharp Lipschitz regularity theory for flow-matching vector fields and diffusion-model scores.<n>We show that the one-sided Lipschitz control yields a globally Lipschitz transport map from the standard Gaussian to $pstar$.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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