🤖 AI Summary
This work investigates the convergence of Denoising Diffusion Probabilistic Models (DDPMs) under the manifold hypothesis in high-dimensional settings, aiming to overcome the “curse of dimensionality” and achieve environment-dimension-independent efficient score estimation and sampling. Methodologically, it establishes, for the first time, a theoretical connection between diffusion models and extreme value theory of Gaussian processes, integrating manifold learning, score matching analysis, stochastic differential equations, and information geometry. Theoretical contributions include: (1) a dimension-independent convergence rate for score estimation; (2) a proof that the sampling complexity—measured by KL divergence—is entirely independent of the ambient dimension (D); and (3) a Wasserstein distance convergence rate of (O(sqrt{D})), which is optimal for this setting. These results uncover the intrinsic dimensionality-reduction capability of diffusion models on low-dimensional manifolds and provide rigorous theoretical foundations for high-dimensional generative modeling.
📝 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 extit{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.