Neural-Based Orthogonal Data Fitting: The EXIN Neural Networks
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.
1 The Total Least Squares Problems.
1.2 Some TLS Applications.
1.4 Ordinary Least Squares Problems.
1.5 Basic TLS Problem.
1.6 Multidimensional TLS Problem.
1.7 Nongeneric Unidimensional TLS Problem.
1.8 Mixed OLS–TLS Problem.
1.9 Algebraic Comparisons Between TLS and OLS.
1.10 Statistical Properties and Validity.
1.11 Basic Data Least Squares Problem.
1.12 The Partial TLS Algorithm.
1.13 Iterative Computation Methods.
1.14 Rayleigh Quotient Minimization Non Neural and Neural Methods.
2 The MCA EXIN Neuron.
2.1 The Rayleigh Quotient.
2.2 The Minor Component Analysis.
2.3 The MCA EXIN Linear Neuron.
2.4 The Rayleigh Quotient Gradient Flows.
2.5 The MCA EXIN ODE Stability Analysis.
2.6 Dynamics of the MCA Neurons.
2.7 Fluctuations (Dynamic Stability) and Learning Rate.
2.8 Numerical Considerations.
2.9 TLS Hyperplane Fitting.
2.10 Simulations for the MCA EXIN Neuron.
3 Variants of the MCA EXIN Neuron.
3.1 High-Order MCA Neurons.
3.2 The Robust MCA EXIN Nonlinear Neuron (NMCA EXIN).
3.3 Extensions of the Neural MCA.
4 Introduction to the TLS EXIN Neuron.
4.1 From MCA EXIN to TLS EXIN.
4.2 Deterministic Proof and Batch Mode.
4.3 Acceleration Techniques.
4.4 Comparison with TLS GAO.
4.5 A TLS Application: Adaptive IIR Filtering.
4.6 Numerical Considerations.
4.7 The TLS Cost Landscape: Geometric Approach.
4.8 First Considerations on the TLS Stability Analysis.
5 Generalization of Linear Regression Problems.
5.2 The Generalized Total Least Squares (GeTLS EXIN) Approach.
5.3 The GeTLS Stability Analysis.
5.4 Neural Nongeneric Unidimensional TLS.
5.6 The Accelerated MCA EXIN Neuron (MCA EXIN+).
5.7 Further Considerations.
5.8 Simulations for the GeTLS EXIN Neuron.
6 The GeMCA EXIN Theory.
6.1 The GeMCA Approach.
6.2 Analysis of Matrix K.
6.3 Analysis of the Derivative of the Eigensystem of GeTLS EXIN.
6.4 Rank One Analysis Around the TLS Solution.
6.5 The GeMCA Spectra.
6.6 Qualitative Analysis of the Critical Points of the GeMCA EXIN Error Function.
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: firstname.lastname@example.org