Diffusion Model's Generalization Can Be Characterized by Inductive Biases toward a Data-Dependent Ridge Manifold

TL;DR

This paper characterizes the generalization of diffusion models by analyzing their geometric behavior relative to data-dependent ridge manifolds, revealing a reach-align-slide mechanism during sample generation.

Contribution

It introduces a novel geometric framework using time-dependent ridge manifolds to understand diffusion model generalization and connects it to training dynamics, supported by experiments.

Findings

Generated samples first approach the ridge manifold

Distance to the ridge is influenced by training error

Motion along the ridge is governed by learned error components

Abstract

We study a data-dependent notion of diffusion-model generalization: when a model does not memorize the training set, where do its generated samples go relative to the geometry induced by the data? To answer this, we introduce a time-dependent family of log-density ridge manifolds constructed from the smoothed empirical distribution, and use it to characterize reverse-time inference. Our main result shows that generated samples evolve by a reach-align-slide mechanism: they first enter a neighborhood of the ridge, then their distance to the ridge is controlled by the normal component of training error, and finally their motion along the ridge is controlled by the tangential component. We further connect this geometric picture to training dynamics through directional decompositions of the learned error, and make this link explicit for random feature models, where architectural bias and optimization error can be separated quantitatively. Experiments on synthetic multimodal data and MNIST latent diffusion support the predicted geometric behavior in both low and high dimensions.