The replica symmetric behavior of the analogical neural network

TL;DR

This paper investigates the replica symmetric phase of an analogical neural network, deriving an exact free energy expression and analyzing its phase transition behavior through a novel interpolation method.

Contribution

It introduces a new interpolation scheme for analyzing the free energy of the analogical neural network, extending techniques from spin-glass models.

Findings

Derived an exact free energy expression at the replica symmetric level.

Identified divergence of order parameter fluctuations at the critical line.

Connected neural network behavior to bipartite spin-glass models.

Abstract

In this paper we continue our investigation of the analogical neural network, paying interest to its replica symmetric behavior in the absence of external fields of any type. Bridging the neural network to a bipartite spin-glass, we introduce and apply a new interpolation scheme to its free energy that naturally extends the interpolation via cavity fields or stochastic perturbations to these models. As a result we obtain the free energy of the system as a sum rule, which, at least at the replica symmetric level, can be solved exactly. As a next step we study its related self-consistent equations for the order parameters and their rescaled fluctuations, found to diverge on the same critical line of the standard Amit-Gutfreund-Sompolinsky theory.