Replica symmetry breaking in neural networks: a few steps toward rigorous results

TL;DR

This paper adapts the broken replica interpolation technique to analyze the Hopfield neural network model, providing explicit free energy expressions at finite steps of replica-symmetry-breaking and recovering known results.

Contribution

It extends Guerra's interpolation method to the Hopfield model, enabling rigorous analysis of its free energy with finite-step replica symmetry breaking.

Findings

Explicit free energy expressions at finite RSB steps for Hopfield model.

Recovery of known 1RSB and 2RSB results for the Hopfield model.

Application of the technique to the Sherrington-Kirkpatrick model with a signal parameter.

Abstract

In this paper we adapt the broken replica interpolation technique (developed by Francesco Guerra to deal with the Sherrington-Kirkpatrick model, namely a pairwise mean-field spin-glass whose couplings are i.i.d. standard Gaussian variables) in order to work also with the Hopfield model (i.e., a pairwise mean-field neural-network whose couplings are drawn according to Hebb's learning rule): this is accomplished by grafting Guerra's telescopic averages on the transport equation technique, recently developed by some of the Authors. As an overture, we apply the technique to solve the Sherrington-Kirkpatrick model with i.i.d. Gaussian couplings centered at $J_0$ and with finite variance $J$; the mean $J_0$ plays the role of a signal to be detected in a noisy environment tuned by $J$, hence making this model a natural test-case to be investigated before addressing the Hopfield model. For both the models, an explicit expression of their quenched free energy in terms of their natural order parameters is obtained at the K-th step (K arbitrary, but finite) of replica-symmetry-breaking. In particular, for the Hopfield model, by assuming that the overlaps respect Parisi's decomposition (following the ziqqurat ansatz) and that the Mattis magnetization is self-averaging, we recover previous results obtained via replica-trick by Amit, Crisanti and Gutfreund (1RSB) and by Steffan and K\"{u}hn (2RSB).