证明扩散模型采样可利用未知低维结构，加速收敛。
Low-dimensional adaptation of diffusion models: Convergence in total variation
- 基于低维结构分析，推导出采样迭代复杂度为 k/ε 阶。
- 在学习得分函数条件下，收敛性能仍保持稳定。
- 适用于非光滑、非对数凹分布，适合研究生成模型理论者。
本文研究扩散生成模型如何利用未知的低维结构以加速采样。聚焦于两种主流采样器——去噪扩散隐式模型（DDIM）和去噪扩散概率模型（DDPM），我们证明在精确得分函数下，其迭代复杂度不超过 $k/\varepsilon$ 阶（含对数因子），其中 $\varepsilon$ 为总变差距离的精度要求，$k$ 为目标分布的内在维度。进一步将该收敛性扩展至得分函数从数据中学习的情形，表明在合适的得分估计假设下，收敛性能退化温和。我们还证明，通过核基得分估计器可在有限样本下满足这些假设，并自适应低维结构。结果适用于广泛的无光滑性或对数凹性要求的目标分布。本工作首次为 DDIM 类采样器对未知低维结构的适应性提供了严格证据，并改进了现有针对总变差收敛的 DDPM 理论。
原文摘要 · Abstract (English)
This paper investigates how diffusion generative models leverage (unknown) low-dimensional structure to accelerate sampling. Focusing on two mainstream samplers -- the denoising diffusion implicit model (DDIM) and the denoising diffusion probabilistic model (DDPM), we prove that their iteration complexities under exact score functions are at most the order of $k/\varepsilon$ (up to log factor), where $\varepsilon$ is the precision in total variation distance and $k$ is some intrinsic dimension of the target distribution. We further extend these convergence guarantees to the setting in which the score functions are learned from data rather than known exactly, showing that the convergence performance degrades gracefully under suitable score estimation assumptions. We then show that these assumptions are attainable via kernel-based score estimators with finite-sample guarantees that also adapt to the low-dimensional structure. Our results apply to a broad family of target distributions without requiring smoothness or log-concavity. Our findings provide the first rigorous evidence for the adaptivity of the DDIM-type samplers to unknown low-dimensional structure, and improve over the state-of-the-art DDPM theory regarding total variation convergence.
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