Neural Based Orthogonal Data Fitting: The EXIN Neural Networks / Edition 1

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Overview

The presentation of a novel theory in orthogonal regression

The literature about neural-based algorithms is often dedicated to principal component analysis (PCA) and considers minor component analysis (MCA) a mere consequence. Breaking the mold, Neural-Based Orthogonal Data Fitting is the first book to start with the MCA problem and arrive at important conclusions about the PCA problem.

The book proposes several neural networks, all endowed with a complete theory that not only explains their behavior, but also compares them with the existing neural and traditional algorithms. EXIN neurons, which are of the authors' invention, are introduced, explained, and analyzed. Further, it studies the algorithms as a differential geometry problem, a dynamic problem, a stochastic problem, and a numerical problem. It demonstrates the novel aspects of its main theory, including its applications in computer vision and linear system identification. The book shows both the derivation of the TLS EXIN from the MCA EXIN and the original derivation, as well as:

  • Shows TLS problems and gives a sketch of their history and applications
  • Presents MCA EXIN and compares it with the other existing approaches
  • Introduces the TLS EXIN neuron and the SCG and BFGS acceleration techniques and compares them with TLS GAO
  • Outlines the GeTLS EXIN theory for generalizing and unifying the regression problems
  • Establishes the GeMCA theory, starting with the identification of GeTLS EXIN as a generalization eigenvalue problem

In dealing with mathematical and numerical aspects of EXIN neurons, the book is mainly theoretical. All the algorithms, however, have been used in analyzing real-time problems and show accurate solutions. Neural-Based Orthogonal Data Fitting is useful for statisticians, applied mathematics experts, and engineers.

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Product Details

Meet the Author

GIANSALVO CIRRINCIONE, PHD, is an assistant professor at the University of Picardie-Jules Verne, Amiens, France. His current research interests are neural networks, data analysis, computer vision, intelligent control, applied mathematics, brain models, and system identification. E-mail address: exin@u-picardie.fr

MAURIZIO CIRRINCIONE, PHD, is a full professor of control and signal processing at the University of Technology of Belfort-Montbéliard, France. His current research interests are neural networks, modeling and control, system identification, data analysis, intelligent control, and electrical machines and drives. E-mail address: maurizio.cirrincione@utbm.fr

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Table of Contents

Foreword ix

Preface xi

1 Total Least Squares Problems 1

1.1 Introduction 1

1.2 Some TLS Applications 2

1.3 Preliminaries 3

1.4 Ordinary Least Squares Problems 4

1.5 Basic TLS Problem 5

1.6 Multidimensional TLS Problem 9

1.7 Nongeneric Unidimensional TLS Problem 11

1.8 Mixed OLS-TLS Problem 14

1.9 Algebraic Comparisons Between TLS and OLS 14

1.10 Statistical Properties and Validity 15

1.11 Basic Data Least Squares Problem 18

1.12 Partial TLS Algorithm 19

1.13 Iterative Computation Methods 19

1.14 Rayleigh Quotient Minimization Nonneural and Neural Methods 22

2 MCA EXIN Neuron 25

2.1 Rayleigh Quotient 25

2.2 Minor Components Analysis 28

2.3 MCA EXIN Linear Neuron 32

2.4 Rayleigh Quotient Gradient Flows 34

2.5 MCA EXIN ODE Stability Analysis 36

2.6 Dynamics of the MCA Neurons 50

2.7 Fluctuations (Dynamic Stability) and Learning Rate 66

2.8 Numerical Considerations 73

2.9 TLS Hyperplane Fitting 77

2.10 Simulations for the MCA EXIN Neuron 78

2.11 Conclusions 86

3 Variants of the MCA EXIN Neuron 89

3.1 High-Order MCA Neurons 89

3.2 Robust MCA EXIN Nonlinear Neuron (NMCA EXIN) 90

3.3 Extensions of the Neural MCA 96

4 Introduction to the TLS EXIN Neuron 117

4.1 From MCA EXIN to TLS EXIN 117

4.2 Deterministic Proof and Batch Mode 119

4.3 Acceleration Techniques 120

4.4 Comparison with TLS GAO 125

4.5 TLS Application: Adaptive IIR Filtering 126

4.6 Numerical Considerations 132

4.7 TLS Cost Landscape: Geometric Approach 135

4.8 First Considerations on the TLS Stability Analysis 139

5 Generalization of Linear Regression Problems 141

5.1 Introduction 141

5.2 Generalized Total Least Squares (GETLS EXIN) Approach 142

5.3 GeTLS Stability Analysis 149

5.4 Neural Nongeneric Unidimensional TLS 178

5.5 Scheduling 184

5.6 Accelerated MCA EXIN Neuron (MCA EXIN+) 188

5.7 Further Considerations 194

5.8 Simulations for the GeTLS EXIN Neuron 198

6 GeMCA EXIN Theory 205

6.1 GeMCA Approach 205

6.2 Analysis of Matrix K 210

6.3 Analysis of the Derivative of the Eigensystem of GeTLS EXIN 213

6.4 Rank One Analysis Around the TLS Solution 218

6.5 GeMCA spectra 219

6.6 Qualitative Analysis of the Critical Points of the GeMCA EXIN Error Function 224

6.7 Conclusions 225

References 227

Index 239

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