The thermodynamic limit in mean field neural networks

TL;DR

This paper proves the existence of the thermodynamic limit for mean-field neural networks, specifically the Hopfield model, by establishing a measure-concentration assumption and connecting it to known spin-glass free energies.

Contribution

It provides a rigorous proof of the thermodynamic limit for the Hopfield model's free energy using measure concentration and spin-glass equivalences, addressing a long-standing problem.

Findings

Established the existence of the free energy limit for the Hopfield model.

Linked the Hopfield free energy to spin-glass models with known limits.

Confirmed the replica-symmetry solution matches heuristic results from the 1980s.

Abstract

In the last five decades, mean-field neural-networks have played a crucial role in modelling associative memories and, in particular, the Hopfield model has been extensively studied using tools borrowed from the statistical mechanics of spin glasses. However, achieving mathematical control of the infinite-volume limit of the model's free-energy has remained elusive, as the standard treatments developed for spin-glasses have proven unfeasible. Here we address this long-standing problem by proving that a measure-concentration assumption for the order parameters of the theory is sufficient for the existence of the asymptotic limit of the model's free energy. The proof leverages the equivalence between the free energy of the Hopfield model and a linear combination of the free energies of a hard and a soft spin-glass, whose thermodynamic limits are rigorously known. Our work focuses on the replica-symmetry level of description (for which we recover the explicit expression of the free-energy found in the eighties via heuristic methods), yet, our scheme is expected to work also under (at least) the first step of replica symmetry breaking.