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

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.

Machine Learning (cs.LG); Artificial Intelligence (cs.AI)

arXiv:2601.21242 [cs.LG]

(or arXiv:2601.21242v2 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2601.21242 Focus to learn more arXiv-issued DOI via DataCite

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