发表机构
Nanyang Technological University（南洋理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究为基于分数的扩散模型中的逆向热流建立了任意阶先验正则性估计，并据此证明了高阶Taylor和Runge--Kutta采样器在总变差和Wasserstein距离下的维度相关复杂度上界。
AI 中文摘要
我们在Ornstein--Uhlenbeck背景下，即基于分数的扩散模型的概率流常微分方程，建立了逆向热流的任意阶先验正则性估计。对于满足支撑质量测度上的维度一致加倍条件的紧支撑、可能奇异的靶分布，我们逐点界定了流的迭代物质导数及其一阶空间导数，常数与维度$d$无关，且关于时间和支撑半径是显式的。证明依赖于两个要素：法向方向后验波动的空间一致界（这是靶分布几何性质唯一介入之处），以及一个在物质微分下封闭的中心化后验矩正规形。这一代数结构，连同加权正则性演算，在每一阶都产生定量界，包括Runge--Kutta阶段所需的混合时空方向导数。作为应用，对于在正向时间$\delta>0$停止的精确分数采样，$p$阶Taylor和显式Runge--Kutta格式在总变差距离下达到精度$\varepsilon$所需的步数为$\widetilde O(d^{1/p}\varepsilon^{-1/p})$，而Taylor格式在$W_2$距离下所需的步数为$\widetilde O(d^{1/(2p)}\varepsilon^{-1/p})$。
英文摘要
We establish arbitrary-order a priori regularity estimates for the reverse heat flow in the Ornstein--Uhlenbeck setting, that is, for the probability-flow ODE of score-based diffusion models. For compactly supported, possibly singular targets satisfying a dimension-uniform doubling condition on supporting-cap masses, we bound the iterated material derivatives of the flow and their first spatial derivatives pointwise, with constants independent of the dimension $d$ and explicit in time and support radius. The proof rests on two ingredients: a spatially uniform bound on normal-direction posterior fluctuations, which is the only point where the geometry of the target enters, and a centered posterior-moment normal form closed under material differentiation. This algebraic structure, together with a weighted regularity calculus, yields quantitative bounds at every order, including the mixed space--time directional derivatives required by Runge--Kutta stages. As an application, for exact-score sampling stopped at forward time $δ>0$, order-$p$ Taylor and explicit Runge--Kutta schemes reach accuracy $\varepsilon$ in $\widetilde O(d^{1/p}\varepsilon^{-1/p})$ steps in total variation, and the Taylor scheme in $\widetilde O(d^{1/(2p)}\varepsilon^{-1/p})$ steps in $W_2$.
