The denoising diffusion probabilistic model (DDPM) has emerged as a mainstream generative model in generative AI. While sharp convergence guarantees have been established for the DDPM, the iteration complexity is, in general, proportional to the ambient data dimension, resulting in overly conservative theory that fails to explain its practical efficiency. This has motivated the recent work Li and Yan (2024a) to investigate how the DDPM can achieve sampling speed-ups through automatic exploitation of intrinsic low dimensionality of data. We strengthen this line of work by demonstrating, in some sense, optimal adaptivity to unknown low dimensionality. For a broad class of data distributions with intrinsic dimension $k$, we prove that the iteration complexity of the DDPM scales nearly linearly with $k$, which is optimal when using KL divergence to measure distributional discrepancy. Notably, our work is closely aligned with the independent concurrent work Potaptchik et al. (2024) -- posted two weeks prior to ours -- in establishing nearly linear-$k$ convergence guarantees for the DDPM.

翻译：去噪扩散概率模型（DDPM）已成为生成式人工智能中的主流生成模型。尽管已为DDPM建立了严格的收敛性保证，但其迭代复杂度通常与数据的环境维度成正比，导致理论过于保守，无法解释其实际效率。这促使近期研究Li and Yan (2024a) 探讨DDPM如何通过自动利用数据内在低维度来实现采样加速。我们通过证明其在某种意义上对未知低维度的最优适应性，强化了这一研究方向。对于一类具有内在维度$k$的广泛数据分布，我们证明了DDPM的迭代复杂度与$k$近似呈线性关系，这在用KL散度衡量分布差异时是最优的。值得注意的是，我们的工作与独立同期研究Potaptchik et al. (2024)（早于我们两周发布）高度一致，均为DDPM建立了近似线性于$k$的收敛性保证。