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.

通过对高斯混合分布的详细研究，我们证明了扩散过程的光滑性质，并得出了与混合分布成分数无关的Lipschitz常数和二阶矩的严格上界。最后，我们将分析应用于各种扩散求解器，从而在目标分布和学习分布之间的总变异距离和KL散度方面建立具体的误差保证。

揭示扩散模型的平滑性质: 一个高斯混合视角