Analysis of the Hopfield Model Incorporating the Effects of Unlearning

TL;DR

This paper provides a detailed analytical study of a Hopfield model with unlearning, revealing how unlearning improves memory capacity and reduces spurious memories through replica analysis and numerical validation.

Contribution

It introduces a systematic replica-based framework to analyze the unlearning-augmented Hopfield model, offering new insights into its stability, capacity, and optimal parameters.

Findings

Unlearning enhances memory capacity.

Theoretical predictions match numerical simulations.

Unlearning suppresses spurious memories.

Abstract

We analyze a variant of the Hopfield model that incorporates an unlearning mechanism based on spin correlations in the high-temperature regime. In the large system limit where extensively many patterns are stored, we employ the replica method under the replica symmetric ansatz to characterize the model analytically. Our analysis provides a systematic and self-consistent framework that yields order-parameter equations and stability conditions at finite temperatures over a wide range of parameter settings. The resulting theory accurately captures the behavior of the signal-to-noise ratio, the memory capacity, and the criteria for selecting optimal hyperparameters, in agreement with the qualitative findings of Nokura (1996 \textit{J. Phys. A: Math. Gen.} \textbf{29} 3871). Moreover, the theoretical predictions show good agreement with numerical simulations, supporting the conclusion that unlearning enhances memory capacity by suppressing spurious memories.