On the number of solutions of a class of nonlinear equations related to neural networks with tapered connections

Tetsuo Nishi, Norikazu Takahashi

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

The number of solutions of a nonlinear equation x = sgn(Wx) is discussed. The equation is derived for the determination of equilibrium points of a kind of Hopfield neural networks. We impose some conditions on W. The conditions correspond to the case where a Hopfield neural network has n neurons arranged on a ring, each neuron has connections only from k preceding neurons and the magnitude of k connections decrease as the distance between two neurons increases. We show that the maximum number of solutions for the above case is extremely few and is independent of the number of neurons, n, if k is less than or equal to 4. We also show that the number of solutions generally increases exponentially with n by considering the case where k = n - 1.

Original languageEnglish
Pages (from-to)1299-1305
Number of pages7
JournalIEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
VolumeE78-A
Issue number10
Publication statusPublished - Oct 1995
Externally publishedYes

Fingerprint

Number of Solutions
Nonlinear equations
Neurons
Neuron
Nonlinear Equations
Neural Networks
Neural networks
Hopfield neural networks
Hopfield Neural Network
sgn
Less than or equal to
Equilibrium Point
Class
Ring
Decrease

ASJC Scopus subject areas

  • Hardware and Architecture
  • Information Systems
  • Electrical and Electronic Engineering

Cite this

@article{3f905a2e50504ed9a75531b97b3eeffb,
title = "On the number of solutions of a class of nonlinear equations related to neural networks with tapered connections",
abstract = "The number of solutions of a nonlinear equation x = sgn(Wx) is discussed. The equation is derived for the determination of equilibrium points of a kind of Hopfield neural networks. We impose some conditions on W. The conditions correspond to the case where a Hopfield neural network has n neurons arranged on a ring, each neuron has connections only from k preceding neurons and the magnitude of k connections decrease as the distance between two neurons increases. We show that the maximum number of solutions for the above case is extremely few and is independent of the number of neurons, n, if k is less than or equal to 4. We also show that the number of solutions generally increases exponentially with n by considering the case where k = n - 1.",
author = "Tetsuo Nishi and Norikazu Takahashi",
year = "1995",
month = "10",
language = "English",
volume = "E78-A",
pages = "1299--1305",
journal = "IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences",
issn = "0916-8508",
publisher = "Maruzen Co., Ltd/Maruzen Kabushikikaisha",
number = "10",

}

TY - JOUR

T1 - On the number of solutions of a class of nonlinear equations related to neural networks with tapered connections

AU - Nishi, Tetsuo

AU - Takahashi, Norikazu

PY - 1995/10

Y1 - 1995/10

N2 - The number of solutions of a nonlinear equation x = sgn(Wx) is discussed. The equation is derived for the determination of equilibrium points of a kind of Hopfield neural networks. We impose some conditions on W. The conditions correspond to the case where a Hopfield neural network has n neurons arranged on a ring, each neuron has connections only from k preceding neurons and the magnitude of k connections decrease as the distance between two neurons increases. We show that the maximum number of solutions for the above case is extremely few and is independent of the number of neurons, n, if k is less than or equal to 4. We also show that the number of solutions generally increases exponentially with n by considering the case where k = n - 1.

AB - The number of solutions of a nonlinear equation x = sgn(Wx) is discussed. The equation is derived for the determination of equilibrium points of a kind of Hopfield neural networks. We impose some conditions on W. The conditions correspond to the case where a Hopfield neural network has n neurons arranged on a ring, each neuron has connections only from k preceding neurons and the magnitude of k connections decrease as the distance between two neurons increases. We show that the maximum number of solutions for the above case is extremely few and is independent of the number of neurons, n, if k is less than or equal to 4. We also show that the number of solutions generally increases exponentially with n by considering the case where k = n - 1.

UR - http://www.scopus.com/inward/record.url?scp=0029387804&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0029387804&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0029387804

VL - E78-A

SP - 1299

EP - 1305

JO - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

JF - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

SN - 0916-8508

IS - 10

ER -