Discovering and decoding latent mean-field structure with variational autoencoders

2026-06-07

AI Summary

This work proposes an unsupervised method to extract physically interpretable latent variables from generative models, revealing the underlying mean-field structure in many-body systems. By reconstructing the joint probability distribution using a variational autoencoder (VAE) and introducing a mutual information–based capacity criterion, the study establishes—for the first time—a rigorous correspondence between the VAE latent space and finite-size mean-field theory. The approach accurately recovers full microscopic mean-field parameters in exactly solvable models, including the Hopfield pattern matrix in the Hopfield model, as well as parameters in the Curie-Weiss and Maier-Saupe models. When applied to neural population recordings from salamander retina, the method reproduces key collective statistical features using only two latent variables and successfully constructs a generalized Hopfield model that quantitatively captures the experimental data.

Abstract

Generative models are increasingly used to capture correlations in many-body systems, but the representations they learn remain largely opaque to physical interpretation. Here, we establish an intuitive criterion that quantifies the capacity of a variational autoencoder (VAE) to faithfully reconstruct the joint probability distribution of a many body system. In a nutshell, a bound on the VAE capacity is obtained by comparing the rate of the latent channel to the bipartite mutual information of the data. Using this bound, we show that the conditionally independent decoder of any successful VAE is structurally identical to a finite-size mean-field factorization. Hence, a successful reconstruction is direct evidence for a latent mean-field theory and the microscopic parameters of that theory can be read off the trained decoder. We validate these conclusions on a hierarchy of solvable models with scalar (Curie-Weiss), vector (Hopfield) and tensor (Maier-Saupe) order parameters, recovering the full Hopfield pattern matrix from equilibrium samples alone. We find that, when applied to Salamander retinal recordings, a two-latent VAE reproduces the population statistics with only two effective collective variables allowing us to recover the `stored patterns' of the neural population and write a generalized Hopfield model which correctly models the experimental data.

Problem

• variational autoencoders
• many-body systems
• mean-field theory
• latent structure
• generative models

Innovation

• variational autoencoder
• mean-field theory
• many-body systems
• mutual information
• Hopfield model