证明了扩散采样器能利用数据内在低维结构，提速采样效率。
Adaptivity and Convergence of Probability Flow ODEs in Diffusion Generative Models
- 基于精确得分估计，构建概率流ODE采样器的理论收敛分析。
- 收敛速度达O(k/T)，k为数据内在维度，优于传统高维依赖方法。
- 适合关注生成模型理论保障与低维结构利用的研究者。
基于得分的生成模型通过学习逆转扩散过程将噪声转化为数据，已成为现代生成式AI的核心。本文为广泛使用的概率流ODE采样器提供了理论保障，该方法在实际中效率高。尽管已有研究探讨其收敛性，但其能否适应自然图像数据中常见的低维结构尚不明确。我们证明：在得分函数估计准确的前提下，概率流ODE采样器在总变差距离下的收敛率为$O(k/T)$（忽略对数因子），其中$k$为目标分布的内在维度，$T$为迭代次数。该维度无关的收敛率优于现有依赖于通常更大环境维度的结果，表明概率流ODE采样器能有效利用目标分布的内在低维结构，实现更快速的采样。
原文摘要 · Abstract (English)
Score-based generative models, which transform noise into data by learning to reverse a diffusion process, have become a cornerstone of modern generative AI. This paper contributes to establishing theoretical guarantees for the probability flow ODE, a widely used diffusion-based sampler known for its practical efficiency. While a number of prior works address its general convergence theory, it remains unclear whether the probability flow ODE sampler can adapt to the low-dimensional structures commonly present in natural image data. We demonstrate that, with accurate score function estimation, the probability flow ODE sampler achieves a convergence rate of $O(k/T)$ in total variation distance (ignoring logarithmic factors), where $k$ is the intrinsic dimension of the target distribution and $T$ is the number of iterations. This dimension-free convergence rate improves upon existing results that scale with the typically much larger ambient dimension, highlighting the ability of the probability flow ODE sampler to exploit intrinsic low-dimensional structures in the target distribution for faster sampling.
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