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

TL;DR

This paper develops a sharp Lipschitz regularity theory for flow-matching and diffusion models, leading to optimal sampling rates and functional inequalities under broad conditions.

Contribution

It introduces a novel Lipschitz regularity framework with optimal dependence on time and dimension, improving understanding of diffusion-based sampling methods.

Findings

Optimal Wasserstein discretization error rate of  rac{ d}{N} for Euler-type samplers.

Constants remain stable and do not grow exponentially with the spatial extent of the target distribution.

Establishes globally Lipschitz transport maps implying Poincare9 and log-Sobolev inequalities.

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 Poincar\'e and log-Sobolev inequalities for a broad class of probability measures.