Hopfield model for patterns with internal structure

TL;DR

This paper analyzes a spherical Hopfield model incorporating internal pattern structures modeled by Gaussian variables, deriving the free energy analytically and exploring phase transitions including spin glass and glass phases.

Contribution

It introduces a spherical Hopfield model with internal pattern structures and derives the free energy using the replica method, revealing new phase behaviors.

Findings

Identification of spin glass phase at high temperatures

Emergence of a glass phase with patterns at low temperatures

Analytical derivation of free energy and overlap distribution

Abstract

The spherical version of the Hopfield model for pattern recognition is considered in the static limit. Structures inside the patterns are modeled by Gaussian random variables that reward correlation between pairs of spins in a given pattern. The free energy is derived analytically with the replica method. The overlap distribution obeys a self-consistent equation. Coming from high temperatures, a spin glass phase is entered, in which patterns and correlations appear at lower temperatures. For small enough loading capacity, also a glass phase with patterns and correlations appears.