🤖 AI Summary
This work investigates the regularity control of vector fields and score functions in flow matching and diffusion models, aiming to derive sampling error bounds that do not deteriorate exponentially with dimension or spatial scale. Under general assumptions on the target distribution, the authors establish time- and dimension-optimal Lipschitz regularity estimates and introduce a one-sided Lipschitz condition to construct a globally Lipschitz transport map, thereby obtaining the first Wasserstein error bounds free from exponential degradation. By combining Euler discretization with functional inequalities—specifically Poincaré and log-Sobolev inequalities—they achieve an optimal sampling error rate of \(O(\sqrt{d}/N)\) (up to logarithmic factors) and establish corresponding functional inequalities for a broad class of probability measures.
📝 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.