Unraveling the Smoothness Properties of Diffusion Models: A Gaussian Mixture Perspective

TL;DR

This paper provides a theoretical analysis of diffusion models, focusing on their smoothness properties when the target distribution is a Gaussian mixture, and offers bounds on Lipschitz constants and error guarantees.

Contribution

It establishes that diffusion processes for Gaussian mixture targets are also Gaussian mixtures and derives bounds on their smoothness properties independent of mixture complexity.

Findings

Diffusion process densities are Gaussian mixtures if the target is a Gaussian mixture.

Derived tight bounds on Lipschitz constants independent of the number of mixture components.

Provided error guarantees for diffusion solvers in terms of total variation and KL divergence.

Abstract

Diffusion models have made rapid progress in generating high-quality samples across various domains. However, a theoretical understanding of the Lipschitz continuity and second momentum properties of the diffusion process is still lacking. In this paper, we bridge this gap by providing a detailed examination of these smoothness properties for the case where the target data distribution is a mixture of Gaussians, which serves as a universal approximator for smooth densities such as image data. We prove that if the target distribution is a $k$-mixture of Gaussians, the density of the entire diffusion process will also be a $k$-mixture of Gaussians. We then derive tight upper bounds on the Lipschitz constant and second momentum that are independent of the number of mixture components $k$. Finally, we apply our analysis to various diffusion solvers, both SDE and ODE based, to establish concrete error guarantees in terms of the total variation distance and KL divergence between the target and learned distributions. Our results provide deeper theoretical insights into the dynamics of the diffusion process under common data distributions.