A generalized Hopfield model to store and retrieve mismatched memory patterns

TL;DR

This paper investigates a generalized Hopfield model with mixed Gaussian and binary memory patterns, analyzing how mismatched variables affect retrieval capacity, basins of attraction, and overall network performance through theoretical and numerical methods.

Contribution

It introduces a hybrid Hopfield model with mixed variable types and characterizes how mismatched patterns influence retrieval properties and capacity.

Findings

Retrieval capacity decreases as the fraction of mismatched variables increases.

Pure Gaussian patterns lead to loss of retrieval due to spherical symmetry.

Even with near-zero capacity, the network maintains large basins of attraction.

Abstract

We study a class of Hopfield models where the memories are represented by a mixture of Gaussian and binary variables and the neurons are Ising spins. We study the properties of this family of models as the relative weight of the two kinds of variables in the patterns varies. We quantitatively determine how the retrieval phase squeezes towards zero as the memory patterns contain a larger fraction of mismatched variables. As the memory is purely Gaussian retrieval is lost for any positive storage capacity. It is shown that this comes about because of the spherical symmetry of the free energy in the Gaussian case. Introducing two different memory pattern overlaps between spin configurations and each contribution to the pattern from the two kinds of variables one can observe that the Gaussian parts of the patterns act as a noise, making retrieval more difficult. The basins of attraction of the states, the accuracy of the retrieval and the storage capacity are studied by means of Monte Carlo numerical simulations. We uncover that even in the limit where the network capacity shrinks to zero, the (few) retrieval states maintain a large basin of attraction and large overlaps with the mismatched patterns. So the network can be used for retrieval, but with a very small capacity.