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

2025-06-23

AI Summary

This work investigates the convergence mechanisms of probability flows in diffusion models—specifically, when such flows converge to training samples, convex combinations thereof, or novel points interior to the data manifold—to unify the coexistence of memorization and generalization. We propose a simplified score-flow analysis framework leveraging orthogonal datasets and obtuse simplicial structures, combined with shallow ReLU denoisers, minimum ℓ²-norm constrained training, and controllable diffusion time scheduling, enabling systematic characterization of probability flow trajectories. Theoretically and empirically, we identify three distinct convergence regimes; prove that early stopping suppresses overfitting and encourages manifold-interior generation; and demonstrate a transition from memorization-dominated to manifold-generalizing behavior as sample size increases. Crucially, we provide the first characterization of the essential role of shallow networks in diffusion modeling through the lens of inductive bias.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Large pretrained models with web data
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

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.

Problem

• Understanding when probability flow converges to training samples
• Analyzing behavior of shallow ReLU neural network denoisers
• Exploring memorization and generalization in diffusion models

Innovation

• Shallow ReLU neural network denoisers
• Probability flow with minimal l2 norm
• Early stopping by diffusion scheduler