A phase transition in diffusion models reveals the hierarchical nature of data

Significance The success of deep learning is often attributed to its ability to harness the hierarchical and compositional structure of data. However, formalizing and testing this notion remained a challenge. This work shows how diffusion models—generative AI techniques producing high-resolution images—operate at different hierarchical levels of features over different time scales of the diffusion process. This phenomenon allows for the generation of images of various classes by recombining low-level features. We study a hierarchical model of data that reproduces this phenomenology and provides a theoretical explanation for this compositional behavior. Overall, the present framework provides a description of how generative models operate, and put forward diffusion models as powerful lenses to probe data structure.