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

Yingyu Liang, Zhenmei Shi, Zhao Song, Yufa Zhou

TL;DR

This paper analyzes the theoretical smoothness of diffusion models when data are modeled as a $k$-mixture of Gaussians, proving that the forward diffusion preserves a $k$-Gaussian mixture form and deriving Lipschitz and second-moment bounds that are independent of $k$.$ The forward density at time $t$ remains $p_t(x)=\sum_{i=1}^k \frac{\alpha_i(t)}{(2\pi)^{d/2}\det(\Sigma_i(t))^{1/2}} \exp(-\tfrac{1}{2}(x-\mu_i(t))^\top\Sigma_i(t)^{-1}(x-\mu_i(t)))$, enabling tight, $k$-independent control of the score Lipschitz constant $L$ and second moment $m_2$.$ Using these bounds, the authors establish concrete error guarantees for both SDE-based (DDPM) and ODE-based (DPOM/DPUM) diffusion solvers in total variation and KL divergence under discretization, with the guarantees expressed in terms of discretization steps, horizon $T$, and score-estimation error $\epsilon_0$.$ These results provide deeper theoretical insight into diffusion dynamics under common data distributions and inform design choices for reliable diffusion-based generation.

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.

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

TL;DR

This paper analyzes the theoretical smoothness of diffusion models when data are modeled as a -mixture of Gaussians, proving that the forward diffusion preserves a -Gaussian mixture form and deriving Lipschitz and second-moment bounds that are independent of .tp_t(x)=\sum_{i=1}^k \frac{\alpha_i(t)}{(2\pi)^{d/2}\det(\Sigma_i(t))^{1/2}} \exp(-\tfrac{1}{2}(x-\mu_i(t))^\top\Sigma_i(t)^{-1}(x-\mu_i(t)))kLm_2 Using these bounds, the authors establish concrete error guarantees for both SDE-based (DDPM) and ODE-based (DPOM/DPUM) diffusion solvers in total variation and KL divergence under discretization, with the guarantees expressed in terms of discretization steps, horizon , and score-estimation error .$ These results provide deeper theoretical insight into diffusion dynamics under common data distributions and inform design choices for reliable diffusion-based generation.

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 -mixture of Gaussians, the density of the entire diffusion process will also be a -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 . 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.
Paper Structure (49 sections, 56 theorems, 169 equations, 1 figure, 1 table)

This paper contains 49 sections, 56 theorems, 169 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Our contribution:
  3. Other studies of the mixture of Gaussian under diffusion.
  4. Related Work
  5. Lipschitz and Second Momentum of Mixture of Gaussian
  6. Score Based Model and Diffusion Model
  7. Background on score based model and diffusion model
  8. Definitions of different solvers
  9. Main Result for Application
  10. Key definitions and assumptions
  11. Total variation
  12. KL divergence
  13. Probability flow ODE
  14. Tools From Previous Work
  15. Conclusion
  16. ...and 34 more sections

Key Result

Lemma 3.2

Let $a, b \in \mathbb{R}$. Let ${\cal D}$ be a $k$-mixture of Gaussian distribution, and let $p$ be its $\mathsf{pdf}$ defined in Definition def:p_t_k_gaussian_informal, i.e., Let $x \in \mathbb{R}^d$ sample from ${\cal D}$. Let $z\in \mathbb{R}^d$ and $z \sim {\cal N}(0,I)$, which is independent from $x$. Then, we have a new random variable $y = a x + b z$ which is also a $k$-mixture of Gaussian

Figures (1)

  • Figure 1: An illustration of discrete diffusion process for $8$ mixture of Gaussian as shown in Eq. \ref{['eq:foward_SDE_dicretized']}. The left figure represents the target 2-dimensional data distribution $p_0$. The right figure represents the standard normal distribution $p_T = \mathcal{N}(0, I_{2\times 2})$, where $T=3$.

Theorems & Definitions (125)

  • Definition 3.1: $k$ mixtures of Gaussian $\mathsf{pdf}$
  • Lemma 3.2: Informal version of Lemma \ref{['lem:pdf_of_k_gaussian_plus_single_gaussian:formal']}
  • Lemma 3.3: Lipschitz, informal version of Lemma \ref{['lem:lip_const_k_gaussian:formal']}
  • Remark 3.4
  • Lemma 3.5: Second momentum, informal version of Lemma \ref{['lem:second_moment:formal']}
  • Definition 4.1
  • Definition 4.2: Time discretization
  • Definition 4.3
  • Definition 4.4
  • Definition 4.5
  • ...and 115 more