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.

翻译：在目标分布 \(p^\star\) 的一般性假设下，我们建立了流匹配向量场与扩散模型评分函数的尖锐Lipschitz正则性理论，该理论对时间和维度的依赖达到最优。作为应用，我们获得了维度 \(d\) 中欧拉型采样器的Wasserstein离散化误差界：在 \(N\) 个离散化步骤下，误差达到最优速率 \(\sqrt{d}/N\)（忽略对数因子）。此外，常数不会随 \(p^\star\) 的空间范围呈指数恶化。我们还证明，单侧Lipschitz控制可生成从标准高斯分布到 \(p^\star\) 的全局Lipschitz传输映射，这蕴含了广泛概率测度类的Poincaré不等式与对数Sobolev不等式。