Lower bounds on the restitution error in the Hopfield model
type: Item
instance of: scholarly article
title: Lower bounds on the restitution error in the Hopfield model
zbMATH Open document ID: 0870.60021
author: Daria Loukianova
publication date: 1997-09-01T00:00:00Z
review text: This paper analyzes the question of the storage capacity of the Hopfield model. Non-rigorous analytic as well as numerical results suggest that the Hopfield model with \(N\) neurons can store up \(\alpha_cN\) patterns, where \(\alpha_c\approx 0.14\). While it has been proven by \textit{Ch. M. Newman} [Networks 1, 223-238 (1988)] that \(\alpha_c\geq 0.055\), there is no rigorous result that would show that \(\alpha_c<\infty\). The present paper takes a step into this direction by studying the local minima of the Hamiltonian in the neighborhood of one pattern. It is proven that for any \(\alpha>0\), there exists a \(\delta(\alpha)>0\), such that in a ball of radius \(\delta(\alpha)\) (distances are measured by the Hamming distance), there is no local minimum. Furthermore, it is shown than as \(\alpha\) tends to infinity, \(\delta(\alpha)\) becomes larger that 0.05. In spite of the relatively poor numerical result, the paper is interesting to read and contains a number of very nice and original ideas.
reviewed by: Anton Bovier
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60G17
Mathematics Subject Classification ID: 92B20
Mathematics Subject Classification ID: 68T10
zbMATH DE Number: 1002142
zbMATH Keywords: storage capacity
zbMATH Keywords: Hopfield model
zbMATH Keywords: local minima of the Hamiltonian
MaRDI profile type: Publication
full work available at URL: s004400050081
OpenAlex ID: W2082662184
DOI: 10.1007/S004400050081
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 1002142