Parisi Phase in a Neuron

TL;DR

This paper extends Parisi's spin glass framework to analyze pattern storage in a single neuron, identifying multiple phases and symmetry breaking phenomena through a variational free energy approach.

Contribution

It introduces a novel variational free energy functional for a neuron based on spin glass theory, revealing complex phase behavior and symmetry breaking.

Findings

Identification of multiple thermodynamical phases

Analytical expressions for the order parameter function

Quantitative results for free energy, storage error, and entropy

Abstract

Pattern storage by a single neuron is revisited. Generalizing Parisi's framework for spin glasses we obtain a variational free energy functional for the neuron. The solution is demonstrated at high temperature and large relative number of examples, where several phases are identified by thermodynamical stability analysis, two of them exhibiting spontaneous full replica symmetry breaking. We give analytically the curved segments of the order parameter function and in representative cases compute the free energy, the storage error, and the entropy.