Probing the Latent Hierarchical Structure of Data via Diffusion Models
Antonio Sclocchi Alessandro Favero Noam Itzhak Levi Matthieu Wyart
High-dimensional data must be highly structured to be learnable. Although the compositional and hierarchical nature of data is often put forward to explain learnability, quantitative measurements establishing these properties are scarce. Likewise, accessing the latent variables underlying such a data structure remains a challenge. In this work, we show that forward-backward experiments in diffusion-based models, where data is noised and then denoised to generate new samples, are a promising tool to probe the latent structure of data. We predict in simple hierarchical models that, in this process, changes in data occur by correlated chunks, with a length scale that diverges at a noise level where a phase transition is known to take place. Remarkably, we confirm this prediction in both text and image datasets using state-of-the-art diffusion models. Our results show how latent variable changes manifest in the data and establish how to measure these effects in real data using diffusion models.
data structure, hierarchical compositionality, diffusion models, statistical physics, phase transition
A hierarchical structure in the data induces a diverging correlation length at a phase transition in diffusion models, which is observed also in text and images.