Complex-Valued Neural Network and Inverse Problems

Complex-Valued Neural Network and Inverse Problems

Takehiko Ogawa (Takushoku University, Japan)
DOI: 10.4018/978-1-60566-214-5.ch002
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Abstract

Network inversion solves inverse problems to estimate cause from result using a multilayer neural network. The original network inversion has been applied to usual multilayer neural networks with real-valued inputs and outputs. The solution by a neural network with complex-valued inputs and outputs is necessary for general inverse problems with complex numbers. In this chapter, we introduce the complex-valued network inversion method to solve inverse problems with complex numbers. In general, difficulties attributable to the ill-posedness of inverse problems appear. Regularization is used to solve this ill-posedness by adding some conditions to the solution. In this chapter, we also explain regularization for complex-valued network inversion.
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Introduction

It is necessary to solve inverse problems for estimating causes from observed results in various engineering fields. In particular, inverse problems have been studied in the field of mathematical science (Groetsch, 1993). The inverse problem determines the inner mechanisms or causes of an observed phenomenon. The cause is estimated from the fixed model and the given result in the inverse problem, while the result is determined from the given cause by using a certain fixed mathematical model in the forward problem. As a solution of inverse problems, the neural network based method has been proposed while other method such as the statistical method (Kaipio & Somersalo, 2005) and parametric method (Aster, Borchers, & Thurber, 2005) have also been studied.

The idea of inverting network mapping was proposed by Williams (1986). Then, Linden and Kindermann proposed a method of network inversion (Linden & Kindermann, 1989). Also, the algorithms and applications of network inversion are summarized by Jansen et al. (1999). In this method, inverse problems are solved by the inverse use of the input-output relation of trained multilayer neural networks. In other words, the corresponding input is estimated from the provided output via fixed weights, after finding the forward relation by network training. The direction of the input-output relation between the training and the inverse estimation is important in this method. The estimation process in multilayer neural networks is considered from the viewpoint of forward and inverse problems. The usual estimation process of multilayer neural networks provides a solution for forward problems because the network estimates the output from the input provided by the forward relation obtained in the training. On the other hand, we can solve inverse problems using multilayer neural networks that learn the forward relation by estimating the input from the given output inversely. Network inversion has been applied to actual problems; e.g., medical image processing (Valova, Kameyama, & Kosugi, 1995), robot control (Lu & Ito, 1995; Ogawa, Matsuura, & Kanada, 2005), optimization problems, and so on (Murray, Heg, & Pohlhammer, 1993; Ogawa, Jitsukawa, Kanada, Mori, & Sakata, 2002; Takeuchi & Kosugi, 1994). Moreover, the answer-in-weights scheme has been proposed to solve the difficulty of ill-posed inverse problems, as a related model of network inversion (Kosugi & Kameyama, 1993).

The original network inversion method proposed by Linden and Kindermann solves an inverse problem by using a usual multilayer neural network that handles the relation between real-valued input and output. However, a network method for complex-valued input and output is required to solve the general inverse problem whose cause and result extend to the complex domain. On the other hand, there exists an extension of the multilayer neural network to the complex domain (Benvenuto & Piazza, 1992; Hirose, 2005; Nitta, 1997). The complex-valued neural network learns the relations between complex-valued input and output in the form of complex-valued weights. This complex-valued network inversion was considered to solve inverse problems that extended to complex-valued input and output. In this method, the complex-valued input is inversely estimated from the provided complex-valued output by extending the input correction of the original network inversion method to the complex domain. Actually, the complex-valued input is estimated from the complex-valued output by giving a random input to the trained network, back-propagating the output error to the input, and correcting the input (Ogawa & Kanada, 2005a).

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Editorial Advisory Board
Table of Contents
Foreword
Sven Buchholz
Acknowledgment
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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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