Models of Complex-Valued Hopfield-Type Neural Networks and Their Dynamics

Models of Complex-Valued Hopfield-Type Neural Networks and Their Dynamics

Yasuaki Kuroe (Kyoto Institute of Technology, Japan)
DOI: 10.4018/978-1-60566-214-5.ch006
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This chapter presents models of fully connected complex-valued neural networks which are complex-valued extension of Hopfield-type neural networks and discusses methods to study their dynamics. In particular the authors investigate existence conditions of energy functions for complex-valued Hopfield-type neural networks. Emphasized are the properties of the activation functions which assure the existence of an energy function for the networks. As an application of the energy function, qualitative analysis of the network by utilizing the energy function is shown and a synthesis method of complex-valued associative memories is discussed.
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In recent years, there have been increasing research interests of artificial neural networks and many efforts have been made on applications of neural networks to various fields. As applications of the neural networks spread more widely, developing neural network models which can directly deal with complex numbers is desired in various fields. Several models of complex-valued neural networks have been proposed and their abilities of information processing have been investigated.

The purpose of this chapter is to present models of fully connected complex-valued neural networks which are complex-valued extension of Hopfield-type neural networks and to discuss methods to investigate their dynamics. In particular, we investigate existence conditions of energy functions and propose a function as an energy function for the complex-valued neural networks. In the complex region there are several possibilities in choosing an activation function because of a variety of complex functions. Based on the existence condition we investigate the properties of the complex functions which assure the existence of an energy function and discuss about how to find them. Several classes of complex functions which are widely used as activation functions in the models of complex-valued neural networks proposed so far are considered.

Once energy functions are constructed for neural networks, they are expected to be applied to various problems of various fields such as qualitative analysis of the neural networks, synthesis of associative memories and several optimization problems, similar to the real-valued ones. As applications, this chapter presents qualitative analysis of dynamics of the complex-valued neural networks by using the energy function. Furthermore a synthesis method of complex-valued associative memories by utilizing the analysis results is discussed.

In the following, the imaginary unit is denoted by . The set of complex (real) numbers is denoted by (). The -dimensional complex (real) space is denoted by and the set of complex (real) matrices is denoted by . For (), its real and imaginary parts are denoted by () and (), respectively.

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Editorial Advisory Board
Table of Contents
Sven Buchholz
Tohru Nitta
Chapter 1
Masaki Kobayashi
Information geometry is one of the most effective tools to investigate stochastic learning models. In it, stochastic learning models are regarded as... Sample PDF
Complex-Valued Boltzmann Manifold
Chapter 2
Takehiko Ogawa
Network inversion solves inverse problems to estimate cause from result using a multilayer neural network. The original network inversion has been... Sample PDF
Complex-Valued Neural Network and Inverse Problems
Chapter 3
Boris Igelnik
This chapter describes the clustering ensemble method and the Kolmogorovs Spline Complex Network, in the context of adaptive dynamic modeling of... Sample PDF
Kolmogorovs Spline Complex Network and Adaptive Dynamic Modeling of Data
Chapter 4
V. Srinivasa Chakravarthy
This chapter describes Complex Hopfield Neural Network (CHNN), a complex-variable version of the Hopfield neural network, which can exist in both... Sample PDF
A Complex-Valued Hopfield Neural Network: Dynamics and Applications
Chapter 5
Mitsuo Yoshida, Takehiro Mori
Global stability analysis for complex-valued artificial recurrent neural networks seems to be one of yet-unchallenged topics in information science.... Sample PDF
Global Stability Analysis for Complex-Valued Recurrent Neural Networks and Its Application to Convex Optimization Problems
Chapter 6
Yasuaki Kuroe
This chapter presents models of fully connected complex-valued neural networks which are complex-valued extension of Hopfield-type neural networks... Sample PDF
Models of Complex-Valued Hopfield-Type Neural Networks and Their Dynamics
Chapter 7
Sheng Chen
The complex-valued radial basis function (RBF) network proposed by Chen et al. (1994) has found many applications for processing complex-valued... Sample PDF
Complex-Valued Symmetric Radial Basis Function Network for Beamforming
Chapter 8
Rajoo Pandey
The equalization of digital communication channel is an important task in high speed data transmission techniques. The multipath channels cause the... Sample PDF
Complex-Valued Neural Networks for Equalization of Communication Channels
Chapter 9
Cheolwoo You, Daesik Hong
In this chapter, the complex Backpropagation (BP) algorithm for the complex backpropagation neural networks (BPN) consisting of the suitable node... Sample PDF
Learning Algorithms for Complex-Valued Neural Networks in Communication Signal Processing and Adaptive Equalization as its Application
Chapter 10
Donq-Liang Lee
New design methods for the complex-valued multistate Hopfield associative memories (CVHAMs) are presented. The author of this chapter shows that the... Sample PDF
Image Reconstruction by the Complex-Valued Neural Networks: Design by Using Generalized Projection Rule
Chapter 11
Naoyuki Morita
The author proposes an automatic estimation method for nuclear magnetic resonance (NMR) spectra of the metabolites in the living body by magnetic... Sample PDF
A Method of Estimation for Magnetic Resonance Spectroscopy Using Complex-Valued Neural Networks
Chapter 12
Michele Scarpiniti, Daniele Vigliano, Raffaele Parisi, Aurelio Uncini
This chapter aims at introducing an Independent Component Analysis (ICA) approach to the separation of linear and nonlinear mixtures in complex... Sample PDF
Flexible Blind Signal Separation in the Complex Domain
Chapter 13
Nobuyuki Matsui, Haruhiko Nishimura, Teijiro Isokawa
Recently, quantum neural networks have been explored as one of the candidates for improving the computational efficiency of neural networks. In this... Sample PDF
Qubit Neural Network: Its Performance and Applications
Chapter 14
Shigeo Sato, Mitsunaga Kinjo
The advantage of quantum mechanical dynamics in information processing has attracted much interest, and dedicated studies on quantum computation... Sample PDF
Neuromorphic Adiabatic Quantum Computation
Chapter 15
G.G. Rigatos, S.G. Tzafestas
Neural computation based on principles of quantum mechanics can provide improved models of memory processes and brain functioning and is of primary... Sample PDF
Attractors and Energy Spectrum of Neural Structures Based on the Model of the Quantum Harmonic Oscillator
Chapter 16
Teijiro Isokawa, Nobuyuki Matsui, Haruhiko Nishimura
Quaternions are a class of hypercomplex number systems, a four-dimensional extension of imaginary numbers, which are extensively used in various... Sample PDF
Quaternionic Neural Networks: Fundamental Properties and Applications
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