证明扩散模型在多种系数下仍能高效利用低维结构采样。
Diffusion Models Adapt to Low-Dimensional Structure Under Flexible Coefficient Choices
- 提出新理论框架，适用于广泛更新系数。
- 仅需 $\widetilde{O}(k/\varepsilon)$ 次迭代即可生成 $\varepsilon$-准确样本。
- 为实际扩散采样器的鲁棒性提供理论支持，适合算法研究者。
扩散模型已知能利用未知的低维结构加速采样，但现有在低维数据结构下的收敛理论大多局限于特定系数选择。这引发一个根本问题：对低维结构的适应性是否对更新系数的精确选择敏感？本文表明，这种适应性是扩散模型的稳健性质。对于一大类更新系数，我们证明在总变差距离下，仅需 $\widetilde{O}(k/\varepsilon)$ 次迭代即可生成 $\varepsilon$-准确样本，且独立于环境维度。该框架显著拓展了已知具备低维适应性的扩散采样器类别，适用于多种实际常用方法。结果为扩散采样器在不同系数选择下应用于结构化高维数据时的实证有效性提供了理论依据。
原文摘要 · Abstract (English)
Diffusion models are known to exploit unknown low-dimensional structure to accelerate sampling. However, existing convergence theory under low-dimensional data structure has largely focused on update rules with narrowly prescribed coefficient choices. This raises a fundamental question: is adaptation to low-dimensional structure sensitive to the precise choice of update coefficients? In this paper, we show that such adaptation is a robust property of diffusion models. For a broad class of update coefficients, we prove that $\widetilde{O}(k/\varepsilon)$ iterations suffice to generate an $\varepsilon$-accurate sample in total variation (TV) distance, independently of the ambient dimension. Our framework substantially broadens the class of diffusion samplers known to enjoy low dimensional adaptation and applies to several commonly used methods in practice. These results provide a theoretical justification for the empirical effectiveness of diffusion samplers across different coefficient choices when applied to structured, high-dimensional data.
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