A relativistic extension of Hopfield neural networks via the mechanical analogy

TL;DR

This paper introduces a relativistic extension of the Hopfield neural network model using a mechanical analogy, leading to improved stability and potential applications in deep learning and network pruning.

Contribution

It proposes a novel relativistic cost function for Hopfield networks, analyzed through a Hamilton-Jacobi mechanical analogy, with analytical solutions and improved stability demonstrated.

Findings

Reduced stability of spurious states in simulations

Analytical characterization of the free energy

Enhanced network performance with unlearning contributions

Abstract

We propose a modification of the cost function of the Hopfield model whose salient features shine in its Taylor expansion and result in more than pairwise interactions with alternate signs, suggesting a unified framework for handling both with deep learning and network pruning. In our analysis, we heavily rely on the Hamilton-Jacobi correspondence relating the statistical model with a mechanical system. In this picture, our model is nothing but the relativistic extension of the original Hopfield model (whose cost function is a quadratic form in the Mattis magnetization which mimics the non-relativistic Hamiltonian for a free particle). We focus on the low-storage regime and solve the model analytically by taking advantage of the mechanical analogy, thus obtaining a complete characterization of the free energy and the associated self-consistency equations in the thermodynamic limit. On the numerical side, we test the performances of our proposal with MC simulations, showing that the stability of spurious states (limiting the capabilities of the standard Hebbian construction) is sensibly reduced due to presence of unlearning contributions in this extended framework.