A priori regularity of the reverse heat flow and dimension-dependent complexity of higher-order diffusion samplers
Abstract
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 ε in Õ(d1/pε−1/p) steps in total variation, and the Taylor scheme in Õ(d1/(2p)ε−1/p) steps in W2.
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BibTeX
@misc{wang2026priori,
title = {{A priori regularity of the reverse heat flow and dimension-dependent complexity of higher-order diffusion samplers}},
author = {Xixian Wang and Zhongjian Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24622},
eprint = {2609.24622},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24622},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - A priori regularity of the reverse heat flow and dimension-dependent complexity of higher-order diffusion samplers
AU - Xixian Wang
AU - Zhongjian Wang
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24622
UR - https://arxiv.org/abs/2609.24622
AB - 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 ε in Õ(d^1/pε^−1/p) steps in total variation, and the Taylor scheme in Õ(d^1/(2p)ε^−1/p) steps in W_2.
ER -