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Hopfield model for patterns with internal structure

Theodorus Maria Nieuwenhuizen

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.

Hopfield model for patterns with internal structure

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.
Paper Structure (28 sections, 104 equations)

This paper contains 28 sections, 104 equations.

Table of Contents

  1. Introduction
  2. Correlation within patterns
  3. Setup
  4. The spherical Hopfield model
  5. The Hamiltonian
  6. The replica free energy
  7. On the sign of the quartic term
  8. High temperature phase
  9. Free energy in the glass phase
  10. Incorporating structure inside patterns
  11. The correlation Hamiltonian
  12. Contribution to the replicated free energy
  13. The full free replicated energy
  14. The free energy in the replica limit $n\to0$
  15. Replica symmetric free energy
  16. ...and 13 more sections