When Diffusion Models Memorize: Inductive Biases in Probability Flow of Minimum-Norm Shallow Neural Nets

TL;DR

This paper investigates how diffusion models' probability flow can memorize training data or generate novel points, analyzing shallow neural network denoisers and showing how early stopping influences convergence to data or manifold points.

Contribution

The study introduces a simplified score flow model, analyzes probability flow trajectories in shallow neural nets, and reveals how early stopping affects memorization and generalization in diffusion models.

Findings

Probability flow converges to training points or their combinations.

Early stopping enables flow to reach more general data manifold points.

Memorization decreases as training sample size increases.

Abstract

While diffusion models generate high-quality images via probability flow, the theoretical understanding of this process remains incomplete. A key question is when probability flow converges to training samples or more general points on the data manifold. We analyze this by studying the probability flow of shallow ReLU neural network denoisers trained with minimal $\ell^2$ norm. For intuition, we introduce a simpler score flow and show that for orthogonal datasets, both flows follow similar trajectories, converging to a training point or a sum of training points. However, early stopping by the diffusion time scheduler allows probability flow to reach more general manifold points. This reflects the tendency of diffusion models to both memorize training samples and generate novel points that combine aspects of multiple samples, motivating our study of such behavior in simplified settings. We extend these results to obtuse simplex data and, through simulations in the orthogonal case, confirm that probability flow converges to a training point, a sum of training points, or a manifold point. Moreover, memorization decreases when the number of training samples grows, as fewer samples accumulate near training points.