Understanding Generalizability of Diffusion Models Requires Rethinking the Hidden Gaussian Structure
This work addresses the problem of understanding why diffusion models generalize well, which is crucial for researchers and practitioners in machine learning, though it is incremental in providing new insights into existing phenomena.
The authors investigated the generalizability of diffusion models by analyzing their learned score functions, discovering that as models transition from memorization to generalization, their denoisers become increasingly linear, revealing an inductive bias towards capturing Gaussian structure from training data. They empirically showed this bias is unique to diffusion models in generalization regimes, particularly with limited model capacity or during early training phases.
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.