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Understanding Diffusion Models via Ratio-Based Function Approximation with SignReLU Networks

Luwei Sun, Dongrui Shen, Jianfe Li, Yulong Zhao, Han Feng

TL;DR

The paper addresses the challenge of approximating ratio-type functionals $\frac{f_1}{f_2}$ arising in conditional diffusion-model training. It introduces SignReLU neural networks to directly approximate such ratios, with explicit approximation bounds and stability control near small denominators. Specializing to diffusion models (DDPMs), the authors construct a neural estimator for the reverse process and decompose the excess KL risk into approximation and estimation components, providing finite-sample guarantees and a training-set augmentation strategy that reduces statistical error. Under Hölder-smooth assumptions, the work establishes near-minimax convergence rates and demonstrates how ratio-based analysis yields end-to-end guarantees for diffusion-based conditional generation with practical implications for sample efficiency and stability.

Abstract

Motivated by challenges in conditional generative modeling, where the target conditional density takes the form of a ratio f1 over f2, this paper develops a theoretical framework for approximating such ratio-type functionals. Here, f1 and f2 are kernel-based marginal densities that capture structured interactions, a setting central to diffusion-based generative models. We provide a concise proof for approximating these ratio-type functionals using deep neural networks with the SignReLU activation function, leveraging the activation's piecewise structure. Under standard regularity assumptions, we establish L^p(Omega) approximation bounds and convergence rates. Specializing to Denoising Diffusion Probabilistic Models (DDPMs), we construct a SignReLU-based neural estimator for the reverse process and derive bounds on the excess Kullback-Leibler (KL) risk between the generated and true data distributions. Our analysis decomposes this excess risk into approximation and estimation error components. These results provide generalization guarantees for finite-sample training of diffusion-based generative models.

Understanding Diffusion Models via Ratio-Based Function Approximation with SignReLU Networks

TL;DR

The paper addresses the challenge of approximating ratio-type functionals arising in conditional diffusion-model training. It introduces SignReLU neural networks to directly approximate such ratios, with explicit approximation bounds and stability control near small denominators. Specializing to diffusion models (DDPMs), the authors construct a neural estimator for the reverse process and decompose the excess KL risk into approximation and estimation components, providing finite-sample guarantees and a training-set augmentation strategy that reduces statistical error. Under Hölder-smooth assumptions, the work establishes near-minimax convergence rates and demonstrates how ratio-based analysis yields end-to-end guarantees for diffusion-based conditional generation with practical implications for sample efficiency and stability.

Abstract

Motivated by challenges in conditional generative modeling, where the target conditional density takes the form of a ratio f1 over f2, this paper develops a theoretical framework for approximating such ratio-type functionals. Here, f1 and f2 are kernel-based marginal densities that capture structured interactions, a setting central to diffusion-based generative models. We provide a concise proof for approximating these ratio-type functionals using deep neural networks with the SignReLU activation function, leveraging the activation's piecewise structure. Under standard regularity assumptions, we establish L^p(Omega) approximation bounds and convergence rates. Specializing to Denoising Diffusion Probabilistic Models (DDPMs), we construct a SignReLU-based neural estimator for the reverse process and derive bounds on the excess Kullback-Leibler (KL) risk between the generated and true data distributions. Our analysis decomposes this excess risk into approximation and estimation error components. These results provide generalization guarantees for finite-sample training of diffusion-based generative models.
Paper Structure (34 sections, 18 theorems, 132 equations)

This paper contains 34 sections, 18 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Related work and background
  3. Our approach and scope
  4. Our contributions
  5. Analysis on "$\frac{f_1}{f_2}$ - type" functions
  6. Error Analysis of diffusion model
  7. Problem Formulation
  8. The Forward Process.
  9. The Backward Process.
  10. Excess KL Risk
  11. Approximation Error
  12. Estimation Error
  13. Sketch proof of Theorem \ref{['thm:esti']}
  14. Discussion on the training set size
  15. Excess KL Risk
  16. ...and 19 more sections

Key Result

Theorem 1

Let $f_1$ and $f_2 \in \mathcal{S}$. Assume that $f_{1}(x)$ and $f_2(x)$ are uniformly bounded above and below by positive constants for all $x\in [-1,1]^d$. Then there exists $\phi$ realized by a SignReLU neural network $\mathcal{N}\mathcal{N}(\mathcal{W},\mathcal{L},\mathcal{M})$ with depth $\math

Theorems & Definitions (23)

  • Theorem 1
  • Proposition 1
  • Lemma 1
  • Theorem 2
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Theorem 3
  • Proposition 5
  • Definition 1: Covering number
  • ...and 13 more