A Note on the Convergence of Denoising Diffusion Probabilistic Models

TL;DR

This paper provides a new theoretical upper bound on the Wasserstein distance between true data distributions and those learned by diffusion models, without restrictive assumptions and with elementary proofs.

Contribution

It introduces a novel, assumption-free upper bound on the Wasserstein distance for diffusion models applicable to arbitrary data distributions.

Findings

Bound does not depend exponentially on data complexity

Applicable to distributions without density w.r.t. Lebesgue measure

Builds on recent theoretical advances with elementary proofs

Abstract

Diffusion models are one of the most important families of deep generative models. In this note, we derive a quantitative upper bound on the Wasserstein distance between the data-generating distribution and the distribution learned by a diffusion model. Unlike previous works in this field, our result does not make assumptions on the learned score function. Moreover, our bound holds for arbitrary data-generating distributions on bounded instance spaces, even those without a density w.r.t. the Lebesgue measure, and the upper bound does not suffer from exponential dependencies. Our main result builds upon the recent work of Mbacke et al. (2023) and our proofs are elementary.