Dynamical Regimes of Diffusion Models

We present a study of generative diffusion models in the regime where the dimension of space and the number of data are large and the score function has been trained optimally. We reveals the existence of three distinct dynamical regimes during the backward generative diffusion process. The generative dynamics starting from pure noise encounters first a 'speciation' transition where the gross structure of data is unraveled through a mechanism similar to symmetry breaking in phase transitions. It is followed at later time by a 'collapse' transition where the trajectories of the dynamics become attracted to one of the memorized data points through a mechanism which is similar to the condensation in a glass phase. For realistic dataset the speciation time can be found from a spectral analysis of the correlation matrix and the collapse time can be found from the estimation of an 'excess entropy' in the data. Analytical solutions for simple models like high-dimensional Gaussian mixtures substantiate these findings and provide a theoretical framework while extensions to more complex scenarios and numerical validations with real datasets confirm the theoretical predictions.