Dynamical Regimes of Diffusion Models
Giulio Biroli Tony Bonnaire Valentin de Bortoli Marc Mézard
Using statistical physics methods, we study 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. Our analysis reveals 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 any 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. The dependence of the collapse time on the dimension and
number of data provides a thorough characterization of the curse of
dimensionality for diffusion models. 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.