[2601.19285] Smoothing the Score Function for Generalization in Diffusion Models: An Optimization-based Explanation Framework

Computer Science > Machine Learning arXiv:2601.19285 (cs) [Submitted on 27 Jan 2026 (v1), last revised 30 Mar 2026 (this version, v2)] Title:Smoothing the Score Function for Generalization in Diffusion Models: An Optimization-based Explanation Framework Authors:Xinyu Zhou, Jiawei Zhang, Stephen J. Wright View a PDF of the paper titled Smoothing the Score Function for Generalization in Diffusion Models: An Optimization-based Explanation Framework, by Xinyu Zhou and 2 other authors View PDF HTML (experimental) Abstract:Diffusion models achieve remarkable generation quality, yet face a fundamental challenge known as memorization, where generated samples can replicate training samples exactly. We develop a theoretical framework to explain this phenomenon by showing that the empirical score function (the score function corresponding to the empirical distribution) is a weighted sum of the score functions of Gaussian distributions, in which the weights are sharp softmax functions. This structure causes individual training samples to dominate the score function, resulting in sampling collapse. In practice, approximating the empirical score function with a neural network can partially alleviate this issue and improve generalization. Our theoretical framework explains why: In training, the neural network learns a smoother approximation of the weighted sum, allowing the sampling process to be influenced by local manifolds rather than single points. Leveraging this insight, we propose...