Local Minima of a Quadratic Binary Functional with a Quasi-Hebbian Connection Matrix

TL;DR

This paper analyzes the local minima of a quadratic binary functional with a quasi-Hebbian connection matrix, deriving equations for minima and identifying critical weights through analytical and simulation methods.

Contribution

It introduces a quasi-Hebbian expansion for connection matrices and derives analytical equations for local minima, including critical weight values, confirmed by simulations.

Findings

Derived equations describing local minima.

Identified critical weight values for energy landscape reconstruction.

Confirmed analytical results with computer simulations.

Abstract

The local minima of a quadratic functional depending on binary variables are discussed. An arbitrary connection matrix can be presented in the form of quasi-Hebbian expansion where each pattern is supplied with its own individual weight. For such matrices statistical physics methods allow one to derive an equation describing local minima of the functional. A model where only one weight differs from other ones is discussed in detail. In this case the equation can be solved analytically. The critical values of the weight, for which the energy landscape is reconstructed, are obtained. Obtained results are confirmed by computer simulations.