Dynamical regimes of diffusion models

We study generative diffusion models in the regime where both the data dimension and the sample size are large, and the score function is trained optimally. Using statistical physics methods, we identify three distinct dynamical regimes during the generative diffusion process. The generative dynamics, starting from pure noise, first encounters a speciation transition, where the broad structure of the data emerges, akin to symmetry breaking in phase transitions. This is followed by a collapse phase, where the dynamics is attracted to a specific training point through a mechanism similar to condensation in a glass phase. The speciation time can be obtained from a spectral analysis of the data's correlation matrix, while the collapse time relates to an excess entropy measure, and reveals the existence of a curse of dimensionality for diffusion models. These theoretical findings are supported by analytical solutions for Gaussian mixtures and confirmed by numerical experiments on real datasets.