mlsanityData to 14 Oct 2026

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

arXiv 2601.21242 ↗First posted 29 Jan 2026cs.LG cs.AIcited by 0

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

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.

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 16 of its 22 references that are in the map. First posted on arXiv 29 Jan 2026.

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16 of 22 references.

Closest ancestors · 6

Same direction, newest first

  1. 2023Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data14 Feb
  2. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions22 Sep
  3. 2022Convergence for score-based generative modeling with polynomial complexity13 Jun
  4. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  5. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  6. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 10

  1. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,101
  2. 2022Video Diffusion Modelscited by 2,305
  3. 2014Conditional Generative Adversarial Netscited by 2,162
  4. 2016Density estimation using Real NVPcited by 2,042
  5. 2020DiffWave: A Versatile Diffusion Model for Audio Synthesiscited by 1,354
  6. 2020WaveGrad: Estimating Gradients for Waveform Generationcited by 622
  7. 2018Adaptivity of deep ReLU network for learning in Besov and mixed smooth Besov spaces: optimal rate and curse of dimensionalitycited by 162
  8. 2021Sharp Bounds on the Approximation Rates, Metric Entropy, and n-widths of Shallow Neural Networkscited by 86
  9. 2021An error analysis of generative adversarial networks for learning distributionscited by 36
  10. 2022SignReLU neural network and its approximation abilitycited by 4

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  1. 2025Diffusion Models: A Mathematical Introduction
  2. 2023Deep Networks as Denoising Algorithms: Sample-Efficient Learning of Diffusion Models in High-Dimensional Graphical Models
  3. 2025Low-dimensional adaptation of diffusion models: Convergence in total variation
  4. 2022Diffusion models as plug-and-play priors
  5. 2026From Score Approximation to Distribution Approximation in Score-Based Diffusion Models
  6. 2026A Tutorial on Diffusion Theory: From Differential Equations to Diffusion Models
  7. 2024O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptions
  8. 2026From Score Learning to Discretized Sampling: An End-to-End Generalization Analysis of Diffusion Models