Denoising Diffusion Probabilistic Models (DDPM) are powerful state-of-the-art methods used to generate synthetic data from high-dimensional data distributions and are widely used for image, audio and video generation as well as many more applications in science and beyond. The manifold hypothesis states that high-dimensional data often lie on lower-dimensional manifolds within the ambient space, and is widely believed to hold in provided examples. While recent results has provided invaluable insight into how diffusion models adapt to the manifold hypothesis, they do not capture the great empirical success of these models, making this a very fruitful research direction. In this work, we study DDPMs under the manifold hypothesis and prove that they achieve rates independent of the ambient dimension in terms of learning the score. In terms of sampling, we obtain rates independent of the ambient dimension w.r.t. the Kullback-Leibler divergence, and $O(\sqrt{D})$ w.r.t. the Wasserstein distance. We do this by developing a new framework connecting diffusion models to the well-studied theory of extrema of Gaussian Processes.

翻译：去噪扩散概率模型（DDPM）是当前最先进的强大方法，用于从高维数据分布中生成合成数据，广泛应用于图像、音频和视频生成以及科学等领域的众多应用。流形假设指出，高维数据通常位于环境空间内的低维流形上，这一假设在现有案例中被广泛认为成立。尽管近期研究成果为扩散模型如何适应流形假设提供了宝贵见解，但这些研究尚未充分解释这些模型取得的巨大实证成功，使得该方向成为极具价值的研究领域。本工作研究流形假设下的DDPM，证明其在学习得分函数时能达到与环境维度无关的收敛速率。在采样方面，我们获得了相对于Kullback-Leibler散度与环境维度无关的收敛速率，以及相对于Wasserstein距离$O(\sqrt{D})$的收敛速率。我们通过建立连接扩散模型与高斯过程极值理论的新框架来实现这一目标。