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Abstract <p>We analyze a Hopfield neural network with a quasi-diagonal connection matrix. We use the term “quasi-diagonal matrix” to denote a matrix with all elements equal zero except the elements on the first super- and sub-diagonals of the principle diagonal. The nonzero elements are arbitrary real numbers. Such matrix generalizes the well-known connection matrix of the one dimensional Ising model with open boundary conditions where all nonzero elements equal <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12005_2025_5358_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\( + 1\)</EquationSource> <!--OptMem2570014Litinskii-m1--> </InlineEquation>. We present a simple description of the fixed points of the Hopfield neural network and their dependence on the matrix elements. The obtained results also allow us to analyze the cases of a) the nonzero elements constitute arbitrary super- and sub-diagonals and b) periodic boundary conditions.</p>

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Hopfield Model with Quasi-Diagonal Connection Matrix

  • 期刊论文

  • 发表日期：

    2025年9月17日

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  • Leonid Litinskii

摘要

Abstract

We analyze a Hopfield neural network with a quasi-diagonal connection matrix. We use the term “quasi-diagonal matrix” to denote a matrix with all elements equal zero except the elements on the first super- and sub-diagonals of the principle diagonal. The nonzero elements are arbitrary real numbers. Such matrix generalizes the well-known connection matrix of the one dimensional Ising model with open boundary conditions where all nonzero elements equal \( + 1\) . We present a simple description of the fixed points of the Hopfield neural network and their dependence on the matrix elements. The obtained results also allow us to analyze the cases of a) the nonzero elements constitute arbitrary super- and sub-diagonals and b) periodic boundary conditions.