Bounding ground state energy of Hopfield models

TL;DR

This paper establishes theoretical bounds and proposes algorithms for estimating the ground state energy in Hopfield models, a class of random optimization problems relevant in neural networks and statistical physics.

Contribution

It introduces a simple mechanism to derive rigorous bounds and presents fast algorithms for approximating ground state energies in Hopfield models.

Findings

Theoretical bounds for positive and negative Hopfield models are derived.

Fast algorithms provide practical estimates for ground state energies.

Results demonstrate the effectiveness of bounds and algorithms in these models.

Abstract

In this paper we look at a class of random optimization problems that arise in the forms typically known as Hopfield models. We view two scenarios which we term as the positive Hopfield form and the negative Hopfield form. For both of these scenarios we define the binary optimization problems that essentially emulate what would typically be known as the ground state energy of these models. We then present a simple mechanism that can be used to create a set of theoretical rigorous bounds for these energies. In addition to purely theoretical bounds, we also present a couple of fast optimization algorithms that can also be used to provide solid (albeit a bit weaker) algorithmic bounds for the ground state energies.