Convergence Analysis of Recurrent Neural Networks
Since the outstanding and pioneering research work of Hopfield on recurrent neural networks (RNNs) in the early 80s of the last century, neural networks have rekindled strong interests in scientists and researchers. Recent years have recorded a remarkable
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Network Theory and Applications Volume 13 Managing Editors: Ding-Zhu Du University ofMinnesota, U.S.A. Cauligi Raghavendra University ofSouthern eali/orina, U.SA.
CONVERGENCE ANALYSIS OF RECURRENT NEURAL NETWORKS
ZHANGYI
School of Computer Science and Engineering University of Electronic Science and Technology of China Chengdu 610054 People's Republic of China
K. K. TAN
Department of Electrical and Computer Engineering The National University of Singapore 4 Engineering Drive 3 117576 Singapore
Springer-Science+Business Media, B.V.
Library or Congress Cataloging-in-Publication Vi, Zhangl Tan, KK Convergence Analysis ofRecurrent Neural Networks
ISBN 978-1-4757-3821-6 DOI 10.1007/978-1-4757-3819-3
ISBN 978-1-4757-3819-3 (eBook)
Copyright © 2004 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2004. Softcover reprint of the hardcover 1st edition 2004 All rights reserved. No part ofthis publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photo-copying, microfilming, recording, or otherwise, without the prior written permission ofthe publisher, with the exception of any material supplied specifically for the purpose ofbeing entered and executed on a computer system, for exclusive use by the purchaser ofthe work. Permissions forbooks published in the USA: perm; ss; ons@wkap com Permissions for books published in Europe: [email protected] Printed on acid-free paper.
Contents
ix xi xv
List of Figures Prefaee Aeknowledgments
1. INTRODUCTION Introduetion 1 1.1 Reeurrent Neural Networks 1.2 Convergenee of RNNs 1.3 Outline of the Book 2 Some Notations and Terminologies Energy Funetions Method for Convergenee Analysis 3 Energy Funetions 3.1 Relationships Between Minimums ofEnergy Funetions 3.2 4
and Equilibria of Networks Conclusions
2. HOPFIELD RECURRENT NEURAL NETWORKS 1 Introduetion 2 Equilibria Analysis 3 Complete Stability 4 Global Convergenee 4.1 Global Asymptotie Stability 4.2 Global Exponential Stability 5 Loeal Exponential Convergenee 5.1 Example 6
Diseussions and Conclusions
v
1 1 1 2 4 7 8 8 11
14 15 15 17 20 21 22 23 27 29 31
vi
CONVERGENCE ANALYSIS OF RECURRENT NEURAL NETWORKS
3. CELLULAR NEURAL NETWORKS Introduciton 1 Properties of Output function 2 Standard CNNs 3 The Standard CNN Model 3.1 Equilibria Analysis 3.2 Complete Stability of CNNs 3.3 Global Exponential Stability and Convergence Rate 3.4 Estimation
4
5
CNNs with Constant Delays 4.1 Model of CNNs with Constant Delays 4.2 Conditions for GES CNNs with Infinite Delay Model of CNNs with Infinite Delay Prelirninaries Relations Between State Stability and Output Stability Global Convergence Analysis Examples
5.1 5.2 5.3 5.4 5.5 6
Conclusions
4. RECURRENT NEURAL NETWORKS WITH UNSATURATING PIECEWISE LINEAR ACTIVATION FUNCTIONS 1 Introduction 2 Prelirninaries Multistability Analysis 3 Boundedness and Global Attractivity 3.1 Complete Stability 3.2
4 5
Simulation Examples Conclusions
5. LOTKA-VOLTERRA RECUR
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