Lifting/lowering Hopfield models ground state energies

TL;DR

This paper develops advanced mechanisms to significantly improve bounds on ground state energies in Hopfield models, demonstrating that convexity-based bounds can be substantially refined in combinatorial optimization problems.

Contribution

It introduces more powerful mechanisms for bounding ground state energies in Hopfield models, surpassing previous simple bounds and showing convexity bounds can be greatly improved.

Findings

Convexity-based bounds can be substantially improved.

New mechanisms outperform previous simple bounds.

First results showing significant enhancement of bounds in these models.

Abstract

In our recent work \cite{StojnicHopBnds10} we looked at a class of random optimization problems that arise in the forms typically known as Hopfield models. We viewed two scenarios which we termed as the positive Hopfield form and the negative Hopfield form. For both of these scenarios we defined the binary optimization problems whose optimal values essentially emulate what would typically be known as the ground state energy of these models. We then presented a simple mechanisms that can be used to create a set of theoretical rigorous bounds for these energies. In this paper we create a way more powerful set of mechanisms that can substantially improve the simple bounds given in \cite{StojnicHopBnds10}. In fact, the mechanisms we create in this paper are the first set of results that show that convexity type of bounds can be substantially improved in this type of combinatorial problems.