Generalization Dynamics of Linear Diffusion Models

Diffusion models are powerful generative models that produce high-quality samples from complex data. While their infinite-data behavior is well understood, their generalization with finite data remains less clear. Classical learning theory predicts that generalization occurs at a sample complexity that is exponential in the dimension, far exceeding practical needs. We address this gap by analyzing diffusion models through the lens of data covariance spectra, which often follow power-law decays, reflecting the structure of real data. To understand whether such a power-law structure can benefit learning in diffusion models, we develop a theoretical framework based on linear neural networks, congruent with a Gaussian hypothesis on the data. We quantify how the covariance spectra of data and regularization impact generalization. We find two regimes: When $N