Understanding Generalizability of Diffusion Models Requires Rethinking the Hidden Gaussian Structure

TL;DR

This paper investigates the generalization of diffusion models, revealing that as they generalize, their denoisers become more linear and capture Gaussian structure, which explains their strong generalization capabilities.

Contribution

The study uncovers the linearity trend in diffusion denoisers during generalization and links it to Gaussian structure bias, providing new insights into diffusion models' inductive biases.

Findings

Diffusion denoisers become more linear as models generalize.

Linear denoisers approximate optimal Gaussian denoisers based on training data.

Diffusion models exhibit a Gaussian structure bias, especially with limited capacity.

Abstract

In this work, we study the generalizability of diffusion models by looking into the hidden properties of the learned score functions, which are essentially a series of deep denoisers trained on various noise levels. We observe that as diffusion models transition from memorization to generalization, their corresponding nonlinear diffusion denoisers exhibit increasing linearity. This discovery leads us to investigate the linear counterparts of the nonlinear diffusion models, which are a series of linear models trained to match the function mappings of the nonlinear diffusion denoisers. Surprisingly, these linear denoisers are approximately the optimal denoisers for a multivariate Gaussian distribution characterized by the empirical mean and covariance of the training dataset. This finding implies that diffusion models have the inductive bias towards capturing and utilizing the Gaussian structure (covariance information) of the training dataset for data generation. We empirically demonstrate that this inductive bias is a unique property of diffusion models in the generalization regime, which becomes increasingly evident when the model's capacity is relatively small compared to the training dataset size. In the case that the model is highly overparameterized, this inductive bias emerges during the initial training phases before the model fully memorizes its training data. Our study provides crucial insights into understanding the notable strong generalization phenomenon recently observed in real-world diffusion models.