基于得分的扩散模型中从得分近似到分布近似
From Score Approximation to Distribution Approximation in Score-Based Diffusion Models
浏览论文内容
中文总结 AI 辅助
研究基于得分的扩散模型中得分近似与分布近似的关系，利用经典神经网络近似理论等，通过推导显式上界，建立两者定量联系，给出从得分函数近似到反向扩散模型概率分布近似的简单明确保证。
中文摘要 AI 辅助
基于得分的扩散模型在生成建模中取得了显著的经验成功，但其近似理论基础仍不完整。经典通用近似定理虽保证神经网络可近似得分函数，但尚不清楚这种近似能否转化为对反向扩散过程生成的概率分布的近似。本文建立了这两个概念之间严格的定量联系。证明若神经网络足够准确地近似真实得分函数，则相应反向扩散模型生成的概率分布在Kullback-Leibler散度上接近目标数据分布，不过正向扩散过程的终端分布与用于初始化反向过程的先验之间存在不可约的不匹配。更确切地说，根据得分近似误差、扩散噪声调度和终端先验不匹配，推导了分布近似误差的显式上界。分析结合了Hornik通用近似定理、路径空间上的Girsanov定理和相对熵的数据处理不等式。与近期在有限样本统计设置和数据分布结构假设下研究得分近似的工作互补，本文基于经典神经网络近似理论进行了近似理论分析。所得定理提供了一个简单明确的保证，将得分函数的神经网络近似与反向扩散模型生成的概率分布近似联系起来。
英文摘要
Score-based diffusion models have achieved remarkable empirical success in generative modeling, yet their approximation-theoretic foundations remain incomplete. In particular, although classical universal approximation theorems guarantee that neural networks can approximate score functions, it remains unclear whether such approximation guarantees translate into approximation of the probability distributions generated by reverse diffusion processes. In this paper, we establish a rigorous quantitative connection between these two notions. Specifically, we prove that if a neural network approximates the true score function sufficiently accurately, then the probability distribution generated by the corresponding reverse diffusion model is close to the target data distribution in Kullback-Leibler (KL) divergence, up to an irreducible mismatch between the terminal distribution of the forward diffusion process and the prior used to initialize the reverse process. More precisely, we derive an explicit upper bound on the distribution approximation error in terms of the score approximation error, the diffusion noise schedule, and the terminal prior mismatch. Our analysis combines Hornik's universal approximation theorem, Girsanov's theorem on path space, and the data processing inequality for relative entropy. Complementary to recent work that studies score approximation under finite-sample statistical settings and structural assumptions on the data distribution, our work develops an approximation-theoretic analysis based on classical neural network approximation theory. The resulting theorem provides a simple and explicit guarantee linking neural network approximation of score functions to approximation of the probability distributions generated by reverse diffusion models.
