Low-Dimensional Adaptation of Diffusion Models: Convergence in Total Variation (extended Abstract)

Abstract

This paper presents new theoretical insights into how diffusion generative models adapt to low-dimensional structure in data distributions. We study two widely used samplers — the denoising diffusion probabilistic model (DDPM) and the denoising diffusion implicit model (DDIM) — and analyze their convergence behavior under the assumption of accurate score estimates. Our main result shows that both DDPM and DDIM require at most $O(k/\varepsilon)$ iterations (up to logarithmic factors) to generate samples that are $\varepsilon$-close to the target distribution in total variation distance, where $k$ captures an intrinsic low-dimensional structure of the distribution. Importantly, our theory holds without assuming smoothness or log-concavity. These results provide the first rigorous guarantees for the low-dimensional adaptation capability of DDIM-type samplers, and significantly improve upon prior TV-based convergence bounds for DDPM. Our analysis also highlights the role of discretization coefficients in exploiting low-dimensional structure, and establishes lower bounds that justify the optimality of commonly used parameter choices originally proposed by Ho et al. (2020); Song et al. (2020).