A Statistical Theory of Frequency Recovery in Overparameterized Diffusion Models

Diffusion models have achieved broad success in image, video, and sequence generation, yet they often learn global layouts and temporal trends more faithfully than fine local details. Existing theory establishes diffusion models as powerful distribution estimators, but largely concerns full distribution estimation rates, leaving unclear how data properties and network capacity govern accuracy at different resolutions. We address this gap by developing a statistical theory of frequency recovery in overparameterized diffusion models, using the discrete Fourier transform to describe data at different resolutions. For a prescribed frequency cutoff, we establish approximation and estimation guarantees for the score function associated with the low-frequency component of the data distribution. We further prove a distribution recovery bound separating a statistical error, whose rate depends on an effective dimension determined by the number of retained frequencies, from a remainder reflecting dependence between low and high frequencies and the spectral energy of the latter. Crucially, our guarantees use weight matrix norm constraints tailored to the frequency cutoff and, at fixed depth and width, remain unchanged as the number of available parallel networks grows. Synthetic and CIFAR-10 experiments illustrate how sample size and weight decay affect frequency recovery.