Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions
Iskander Azangulov, George Deligiannidis, Judith Rousseau
TL;DR
This work tackles diffusion-models under the manifold hypothesis, showing that score learning can achieve ambient-dimension–free rates by tying diffusion processes to extrema of Gaussian processes. It introduces a dimension-reduction strategy that replaces the high-dimensional manifold with a union of low-dimensional polynomial pieces, enabling neural-network-based score estimation with polylog-sized networks and near-optimal statistical rates. The results yield ambient-dimension–independent score-learning rates and dimension-affected but controlled Wasserstein bounds, along with KL guarantees for sampling, thereby explaining DDPMs’ strong performance on data with intrinsic low-dimensional structure. The framework paves the way for scalable diffusion modeling in very high dimensions and suggests practical NN architectures and manifold-approximation schemes for efficient training and sampling.
Abstract
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 the ambient space, 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, 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 score learning. In terms of sampling complexity, 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.