用SignReLU网络分析扩散模型的生成原理，给出理论保障。
Understanding Diffusion Models via Ratio-Based Function Approximation with SignReLU Networks
- 用SignReLU网络逼近比值形式的条件密度函数
- 证明了扩散模型反向过程的估计误差有界
- 适合研究生成模型理论的学者阅读
针对条件生成建模中目标条件密度呈现为f1/f2比值形式的挑战，本文构建了一个理论框架，用于近似此类比值型函数。其中，f1和f2为基于核的边缘密度，刻画结构化交互关系，这正是基于扩散的生成模型的核心设定。我们通过利用SignReLU激活函数的分段特性，给出了深度神经网络逼近该类函数的简洁证明，并在标准正则性假设下，建立了L^p(Omega)空间中的逼近界与收敛速率。进一步针对去噪扩散概率模型（DDPM），构造了基于SignReLU的反向过程神经估计器，并推导出生成分布与真实数据分布之间超额KL散度风险的上界。分析表明该风险可分解为逼近误差与估计误差两部分。这些结果为扩散生成模型的有限样本训练提供了泛化保证。
原文摘要 · Abstract (English)
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.
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