Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions

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 \textit{manifold hypothesis} states that high-dimensional data often lie on lower-dimensional manifolds within an ambient space of large dimension D, and is widely believed to hold in provided examples. While recent results have provided invaluable insight into how diffusion models adapt to the manifold hypothesis, they do not capture the great empirical success of these models.
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.
This is a joint work with I. Azangulov and  G. Deligliannidis (Univ of Oxford)