Complex-Valued Neural Networks

AKIRA HIROSE, PhD, is a Professor in the Department of Electrical Engineering and Information Systems, the University of Tokyo, Japan. His main fields of interest are wireless electronics and neural networks on which he has published several books. Dr. Hirose is a Fellow of the IEEE, a senior member of the IEICE, and Vice President of the Japanese Neural Network Society.All contributors are members of the Task Force on Complex-Valued Neural Networks, IEEE Computational Intelligence Society Neural Network Technical Committee.

• Preface xv1 Application Fields and Fundamental Merits 1Akira Hirose1.1 Introduction 11.2 Applications of Complex-Valued Neural Networks 21.3 What is a complex number? 51.4 Complex numbers in feedforward neural networks 81.5 Metric in complex domain 121.6 Experiments to elucidate the generalization characteristics 161.7 Conclusions 262 Neural System Learning on Complex-Valued Manifolds 33Simone Fiori2.1 Introduction 342.2 Learning Averages over the Lie Group of Unitary Matrices 352.3 Riemannian-Gradient-Based Learning on the Complex Matrix-Hypersphere 412.4 Complex ICA Applied to Telecommunications 492.5 Conclusion 533 N-Dimensional Vector Neuron and Its Application to the N-Bit Parity Problem 59Tohru Nitta3.1 Introduction 593.2 Neuron Models with High-Dimensional Parameters 603.3 N-Dimensional Vector Neuron 653.4 Discussion 693.5 Conclusion 704 Learning Algorithms in Complex-Valued Neural Networks using Wirtinger Calculus 75Md. Faijul Amin and Kazuyuki Murase4.1 Introduction 764.2 Derivatives in Wirtinger Calculus 784.3 Complex Gradient 804.4 Learning Algorithms for Feedforward CVNNs 824.5 Learning Algorithms for Recurrent CVNNs 914.6 Conclusion 995 Quaternionic Neural Networks for Associative Memories 103Teijiro Isokawa, Haruhiko Nishimura, and Nobuyuki Matsui5.1 Introduction 1045.2 Quaternionic Algebra 1055.3 Stability of Quaternionic Neural Networks 1085.4 Learning Schemes for Embedding Patterns 1245.5 Conclusion 1286 Models of Recurrent Clifford Neural Networks and Their Dynamics 133Yasuaki Kuroe6.1 Introduction 1346.2 Clifford Algebra 1346.3 Hopfield-Type Neural Networks and Their Energy Functions 1376.4 Models of Hopfield-Type Clifford Neural Networks 1396.5 Definition of Energy Functions 1406.6 Existence Conditions of Energy Functions 1426.7 Conclusion 1497 Meta-cognitive Complex-valued Relaxation Network and its Sequential Learning Algorithm 153Ramasamy Savitha, Sundaram Suresh, and Narasimhan Sundararajan7.1 Meta-cognition in Machine Learning 1547.2 Meta-cognition in Complex-valued Neural Networks 1567.3 Meta-cognitive Fully Complex-valued Relaxation Network 1647.4 Performance Evaluation of McFCRN: Synthetic Complexvalued Function Approximation Problem 1717.5 Performance Evaluation of McFCRN: Real-valued Classification Problems 1727.6 Conclusion 1788 Multilayer Feedforward Neural Network with Multi-Valued Neurons for Brain-Computer Interfacing 185Nikolay V. Manyakov, Igor Aizenberg, Nikolay Chumerin, and Marc M. Van Hulle8.1 Brain-Computer Interface (BCI) 1858.2 BCI Based on Steady-State Visual Evoked Potentials 1888.3 EEG Signal Preprocessing 1928.4 Decoding Based on MLMVN for Phase-Coded SSVEP BCI 1968.5 System Validation 2018.6 Discussion 2039 Complex-Valued B-Spline Neural Networks for Modeling and Inverse of Wiener Systems 209Xia Hong, Sheng Chen and Chris J. Harris9.1 Introduction 2109.2 Identification and Inverse of Complex-Valued Wiener Systems 2119.3 Application to Digital Predistorter Design 2229.4 Conclusions 22910 Quaternionic Fuzzy Neural Network for View-invariant Color Face Image Recognition 235Wai Kit Wong, Gin Chong Lee, Chu Kiong Loo, Way Soong Lim, and Raymond Lock10.1 Introduction 23610.2 Face Recognition System 23810.3 Quaternion-Based View-invariant Color Face Image Recognition 24410.4 Enrollment Stage and Recognition Stage for Quaternion- Based Color Face Image Correlator 25510.5 Max-Product Fuzzy Neural Network Classifier 26010.6 Experimental Results 26610.7 Conclusion and Future Research Directions 274References 274Index 279

“In summary, this book contains a wide variety of hot topics on advanced computational intelligence methods which incorporate the concept of complex and hypercomplex number systems into the framework of artificial neural networks . . . Nevertheless, it seems that the applications of CVNNs and hypercomplex-valued neural networks are very promising.”  (IEEE Computational intelligence magazine, 1 May 2013)