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

TL;DR

This paper analyzes the local minima of a quadratic binary functional with a quasi-Hebbian connection matrix, deriving analytical equations and confirming results through simulations.

Contribution

It introduces a quasi-Hebbian expansion for connection matrices and derives analytical equations for local minima, including a detailed solution for a specific case.

Findings

Derived an equation describing local minima for quasi-Hebbian matrices

Solved analytically a special case with one differing weight

Confirmed theoretical 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 details. In this case the above-mention equation can be solved analytically. Obtained results are confirmed by computer simulations.