Low-dimensional adaptation of diffusion models: Convergence in total variation

Abstract

This paper investigates how diffusion generative models leverage (unknown) low-dimensional structure to accelerate sampling. Focusing on two mainstream samplers -- the denoising diffusion implicit model (DDIM) and the denoising diffusion probabilistic model (DDPM), we prove that their iteration complexities under exact score functions are at most the order of $k/\varepsilon$ (up to log factor), where $\varepsilon$ is the precision in total variation distance and $k$ is some intrinsic dimension of the target distribution. We further extend these convergence guarantees to the setting in which the score functions are learned from data rather than known exactly, showing that the convergence performance degrades gracefully under suitable score estimation assumptions. We then show that these assumptions are attainable via kernel-based score estimators with finite-sample guarantees that also adapt to the low-dimensional structure. Our results apply to a broad family of target distributions without requiring smoothness or log-concavity. Our findings provide the first rigorous evidence for the adaptivity of the DDIM-type samplers to unknown low-dimensional structure, and improve over the state-of-the-art DDPM theory regarding total variation convergence.