Complex-Valued.Neural.Networks.Advances.and.Applications

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Abstract This chapter contains sections titled: Introduction Derivatives in Wirtinger Calculus Complex Gradient Learning Algorithms for Feedforward CVNNs Learning Algorithms for Recurrent CVNNs Conclusion
IEEE Press 445 Hoes L Piscataway, NJ08854 IEEE Press Editorial Board 2013 John Anderson, Editor in Chief Linda shafer Saeid nahavandi George Zobrist George w. Amold David Jacobson Tariq Samad Ekram Hossain Mary lanzerotti Dmitry goldgof Om P. malik Kenneth Moore, Director ofieee Book and information Services(BIS Technical reviewers George M. georgion Gouhei Tanaka Complex-Valued Neural networks Advances and Applications Edited by Akira Hirose The University of Tokyo IEEE Press Series on Computational Intelligence David B Fogel, Series Editor ◆EE IEEE PRESS WILEY Copyright@ 2013 by The Institue of Electrical and Electronics Engineers. All rights reserved Published by John Wiley Sons, Inc, Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc, 222 Rosewood Drive, Danvers, MA 01923, (978)750-8400, fax (978)750-4470,oronthewebatwww.copyright.comRequeststothePublisherforpermissionshould be addressed to the Permissions Department, John Wiley &e sons, Inc., III River Street, Hoboken, NJ 07030,(201)748-6011,fax(201)748-6008,oronlineathttp://www.wiley.com/go/permission Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representation or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages For general information on our other products and services please contact our Customer Care Department within the United States at( 800)762-2974, outside the United States at (317)572-3993 or fax(3l7)572-4002 Wiley also publishes its books in a variety of electronic formats. Some content that appears in print, however, may not be available in electronic formats. For more information about wiley products, visit ourwebsiteatwww.wiley.col Library of Congress Cataloging-in-Publication Data is available ISBN978ll18344606 Printed in the United States of America 10987654321 CONTRIBUTORS ChAPTeR AKIRA HIROSE, The University of Tokyo, Tokyo, Japan CHAPTER 2 SIMONE FIORI, Universita Politecnica delle marche, Ancona, Italy ChaPter 3 TOHRU NITTA, National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Japan CHAPTER 4 MD. FAIUL AMIN, University of Fukui, Fukui, Japan; Khulna University of Engi- neering and Technology, Khulna, Bangladesh KAZUYUKI MURASE, University of Fukui, Fukui, Japan CONTRIBUTORS CHAPTER 5 TEIjIRO ISOKAWA, University of Hyogo, hyogo, japan HARUHIKO NISHIMURA, University of Hyogo, Hyogo, Japat NOBUYUKI MATSUI, University of Hyogo, Hyogo, Japan ChaPTeR 6 YASUAKI KUROE, Kyoto Institute of Technology, Kyoto, Japan CHAPTER 7 RAMASAMY SAVITHA, Nanyang Technological University, Singapore SUNDARAM SURESH, Nanyang Technological University, Singapore NARASIMHAN SUNDARARAJAN, Sri Jaya Chamarajendra College of Engineering (SJCE), Mysore, India chapter 8 NIKOLAY MANYAKOV, KU Leuven, Leuven, Belgium IGOR AIZENBERG, Texas A&M University-Texarkana, Texarkana, Texas, U.S.A NILOLAY CHUMERIN, KU Leuven, Leuven, Belgium MARK M. VAN HULLE, KU Leuven, Leuven, Belgium chapter 9 XIA HONG, University of Reading, Reading, U. K SHENG CHEN, University of Southampton, Southampton, U. K, King Abdulaziz University, Jeddah, Saudi arabia CHRIS J. HARRIS, University of Southampton, Southampton, U. K CHAPTER 10 WAI KIT WONG, Multimedia University, Melaka, Malaysia GIN CHONG LEE, Multimedia University, Melaka, Malaysia CHU KIONG LoO, University of Malaya, Kuala Lumpur, Malaysia WAY SOONG LIM, Multimedia University, Melaka, Malaysia RAYmOND LOCK, Multimedia University, Melaka, malaysia CONTENTS Preface 1 Application Fields and Fundamental Merits Akira hirose 1.1 Introduction 1.2 Applications of Complex-valued Neural Networks 1.2.1 Antenna design 2 1.2.2 Estimation of direction of Arrival and B eamforming 1.2.3 Radar Imaging 1. 2. 4 Acoustic Signal Processing and Ultrasonic Imaging 3 1.2.5 Communications Signal Processing 1.2.6 Image Processing 1.2.7 Social Systems Such as Traffic and Power Systems 4 1.2.8 Quantum Devices Such as Superconductive Devices 1.29O Optical /lightwave Information Processing Including Carrier-Frequency Multiplexing CONTENTS 2. 10 Hypercomplex-Valued Neural Networks 1.3 What is a complex number? 1.3.1 Geometric and intuitive definition 1.3.2 Definition as ordered Pair of real numbers 1.3.3 Real 2x2 Matrix Representation 1.4 Complex numbers in feedforward neural networks 4.1 Synapse and Network Function in Layered Feedforward Neural Networks 9 4.2 Circularity 1.5 Metric in complex domain 12 1.5.1 Metric in Complex-Valued Self-Organizing Map 12 1.5.2 Euclidean metric 12 1.5.3 Complex inner-Product Metric 14 1.5.4 Comparison Between Complex Inner Product and Euclidean distance 14 1.5.5 Metric in Correlation Learning 1.6 Experiments to elucidate the generalization characteristics 16 1.6.1 Forward Processing and Learning Dynamics 17 1.6.2 Experimental Setup 21 1.6.3 Results 24 1.7 Conclusions 26 Refe 27 2 Neural System Learning on Complex -Valued Manifolds 33 Simone fiori 2.1 Introduction 34 2.2 Learning Averages over the Lie Group of Unitary Matrices 35 2.2.1 Differential-Geometric Setting 36 2.2.2 An Averaging Procedure over the Lie group of Unitary matrices 37 2.3 Riemannian-Gradient-Based Learning on the Complex Matrix-Hypersphere 2.3.1 Geometric Characterization of the matrix H yp re 2.3.2 Geodesic-Stepping Optimization Method 45 2.3.3 Application to Optimal Precoding in MIMO Broadcast channels 46 2. 4 Complex ica applied to telecommunications 49 CONTENTS 2.4.1 Complex-Weighted Rigid-Body Learning Equations for Ica 51 2.4.2 Application to the Blind Separation of QAM/PSK ignalS 3 References 55 3 N-Dimensional Vector Neuron and Its application to the N-bit Parity Problem 59 ① thru nitta 3. 1 Introductio 59 3.2 Neuron Models with High-Dimensional Parameters 60 3.2.1 Complex-Valued Neuron 60 3.2.2 Hyperbolic Neuron 61 3.2.3 Three-Dimensional Vector Neuron 61 3.2.4 Three-Dimensional Vector Product Neuron 62 3.2.5 Quaternary Neuron 63 3.2.6 Clifford Neuron 3.3 N-Dimensional Vector Neuron 3.3.1 N-Dimensional Vector Neuron Model 65 3.3.2 Decision Boundary 65 3.3. 3 N-Bit Parity Problem 67 3.3.4 A Solution 67 3.4 Discussion 3.5 Conclusion 70 References 71 4 Learning Algorithms in Complex-Valued Neural Networks using Wirtinger Calculus 75 Md. Faijul amin and Kazuyuki murase 4.1 Introduction 76 4.2 Derivatives in Wirtinger Calculus 4.3 Complex gradient 4.4 Learning Algorithms for Feedforward CVNNs 82 4.4.1 Complex gradient descent algorithm 82 4.4.2 Complex Levenberg Marquardt algorithm 4.4.3 Computer Simulations 4.5 Learning Algorithms for Recurrent CVNns

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