Understanding Diffusion Models via Ratio-Based Function Approximation with SignReLU Networks
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908796056502272 |
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| author | Sun, Luwei Shen, Dongrui Li, Jianfe Zhao, Yulong Feng, Han |
| author_facet | Sun, Luwei Shen, Dongrui Li, Jianfe Zhao, Yulong Feng, Han |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_21242 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Understanding Diffusion Models via Ratio-Based Function Approximation with SignReLU Networks Sun, Luwei Shen, Dongrui Li, Jianfe Zhao, Yulong Feng, Han Machine Learning Artificial Intelligence 41A25, 60B10 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. |
| title | Understanding Diffusion Models via Ratio-Based Function Approximation with SignReLU Networks |
| topic | Machine Learning Artificial Intelligence 41A25, 60B10 |
| url | https://arxiv.org/abs/2601.21242 |