Neural Networks and Learning Machines

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1 Neural Networks and Learning Machines Third Edition Simon Haykin McMaster University Hamilton, Ontario, Canada Upper Saddle River Boston Columbus San Francisco New York Indianapolis London Toronto Sydney Singapore Tokyo Montreal Dubai Madrid Hong Kong Mexico City Munich Paris Amsterdam Cape Town

2 Contents Preface 10 Introduction 1 1. What is a Neural Network? The Human Brain Models of a Neuron 40 A. Neural Networks Viewed As Directed Graphs Feedback Network Architectures Knowledge Representation Learning Processes Learning Tasks Concluding Remarks 75 Notes and References 76 Chapter 1 Rosenblatt's Perceptron Introduction Perceptron The Perceptron Convergence Theorem Relation Between the Perceptron and Bayes Classifier for a Gaussian Environment Computer Experiment: Pattern Classification The Batch Perceptron Algorithm Summary and Discussion 95 Notes and References 96 Problems 96 Chapter 2 Model Building through Regression Introduction Linear Regression Model: Preliminary Considerations Maximum a Posteriori Estimation of the Parameter Vector Relationship Between Regularized Least-Squares Estimation and MAP Estimation Computer Experiment: Pattern Classification The Minimum-Description-Length Principle Finite Sample-Size Considerations The Instrumental-Variables Method Summary and Discussion 118 Notes and References 119 Problems 119 5

3 6 Contents Chapter 3 The Least-Mean-Square Algorithm Introduction Filtering Structure of the LMS Algorithm Unconstrained Optimization: a Review The Wiener Filter The Least-Mean-Square Algorithm Markov Model Portraying the Deviation of the LMS Algorithm from the Wiener Filter The Langevin Equation: Characterization of Brownian Motion Kushner's Direct-Averaging Method Statistical LMS Learning Theory for Small Learning-Rate Parameter Computer Experiment I: Linear Prediction Computer Experiment II: Pattern Classification Virtues and Limitations of the LMS Algorithm Learning-Rate Annealing Schedules Summary and Discussion 147 Notes and References 148 Problems 149 Chapter 4 Multilayer Perceptrons Introduction Some Preliminaries Batch Learning and On-Line Learning The Back-Propagation Algorithm XOR Problem Heuristics for Making the Back-Propagation Algorithm Perform Better Computer Experiment: Pattern Classification Back Propagation and Differentiation The Hessian and Its Role in On-Line Learning Optimal Annealing and Adaptive Control of the Learning Rate Generalization Approximations of Functions Cross-Validation Complexity Regularization and Network Pruning Virtues and Limitations of Back-Propagation Learning Supervised Learning Viewed as an Optimization Problem Convolutional Networks Nonlinear Filtering Small-Scale Versus Large-Scale Learning Problems Summary and Discussion 247 Notes and References 249 Problems 251 Chapter 5 Kernel Methods and Radial-Basis Function Networks Introduction Cover's Theorem on the Separability of Patterns The Interpolation Problem Radial-Basis-Function Networks K-Means Clustering Recursive Least-Squares Estimation of the Weight Vector Hybrid Learning Procedure for RBF Networks Computer Experiment: Pattern Classification Interpretations of the Gaussian Hidden Units 280

4 Contents Kerne] Regression and Its Relation to RBF Networks Summary and Discussion 287 Notes and References 289 Problems 291 Chapter 6 Support Vector Machines Introduction Optimal Hyperplane for Linearly Separable Patterns Optimal Hyperplane for Nonseparable Patterns The Support Vector Machine Viewed as a Kernel Machine Design of Support Vector Machines XOR Problem Computer Experiment: Pattern Classification Regression: Robustness Considerations Optimal Solution of the Linear Regression Problem The Representer Theorem and Related Issues Summary and Discussion 330 Notes and References 332 Problems 335 Chapter 7 Regularization Theory Introduction Hadamard's Conditions for Well-Posedness Tikhonov's Regularization Theory Regularization Networks Generalized Radial-Basis-Function Networks The Regularized Least-Squares Estimator: Revisited Additional Notes of Interest on Regularization Estimation of the Regularization Parameter Semisupervised Learning Manifold Regularization: Preliminary Considerations Differentiable Manifolds Generalized Regularization Theory Spectral Graph Theory Generalized Representer Theorem Laplacian Regularized Least-Squares Algorithm Experiments on Pattern Classification Using Semisupervised Learning Summary and Discussion 387 Notes and References 389 Problems 391 Chapter 8 Principal-Components Analysis Introduction Principles of Self-Organization Self-Organized Feature Analysis Principal-Components Analysis: Perturbation Theory Hebbian-Based Maximum Eigenfilter Hebbian-Based Principal-Components Analysis Case Study: Image Coding Kernel Principal-Components Analysis Basic Issues Involved in the Coding of Natural Images Kernel Hebbian Algorithm Summary and Discussion 440 Notes and References 443 Problems 446

5 8 Contents Chapter 9 Self-Organizing Maps Introduction Two Basic Feature-Mapping Models Self-Organizing Map Properties of the Feature Map Computer Experiments I: Disentangling Lattice Dynamics Using SOM Contextual Maps Hierarchical Vector Quantization Kernel Self-Organizing Map Computer Experiment II: Disentangling Lattice Dynamics Using Kernel SOM Relationship Between Kernel SOM and Kullback-Leibler Divergence Summary and Discussion 494 Notes and References 496 Problems 498 Chapter 10 Information-Theoretic Learning Models Introduction Entropy Maximum-Entropy Principle Mutual Information Kullback-Leibler Divergence Copulas Mutual Information as an Objective Function to be Optimized Maximum Mutual Information Principle Infomax and Redundancy Reduction Spatially Coherent Features Spatially Incoherent Features Independent-Components Analysis Sparse Coding of Natural Images and Comparison with ICA Coding Natural-Gradient Learning for Independent-Components Analysis Maximum-Likelihood Estimation for Independent-Components Analysis Maximum-Entropy Learning for Blind Source Separation Maximization of Negentropy for Independent-Components Analysis Coherent Independent-Components Analysis Rate Distortion Theory and Information Bottleneck Optimal Manifold Representation of Data Computer Experiment: Pattern Classification Summary and Discussion 589 Notes and References 592 Problems 600 Chapter 11 Stochastic Methods Rooted in Statistical Mechanics Introduction Statistical Mechanics Markov Chains Metropolis Algorithm Simulated Annealing Gibbs Sampling Boltzmann Machine Logistic Belief Nets Deep Belief Nets Deterministic Annealing 638

6 Contents Analogy of Deterministic Annealing with Expectation-Maximization Algorithm Summary and Discussion 645 Notes and References 647 Problems 649 Chapter 12 Dynamic Programming Introduction Markov Decision Process Bellman's Optimality Criterion Policy Iteration Value Iteration Approximate Dynamic Programming: Direct Methods Temporal-Difference Learning Q-Learning Approximate Dynamic Programming: Indirect Methods Least-Squares Policy Evaluation Approximate Policy Iteration Summary and Discussion 691 Notes and References 693 Problems 696 Chapter 13 Neurodynamics Introduction Dynamic Systems Stability of Equilibrium States Attractors Neurodynamic Models Manipulation of Attractors as a Recurrent Network Paradigm Hopfield Model The Cohen-Grossberg Theorem Brain-State-In-A-Box Model Strange Attractors and Chaos Dynamic Reconstruction of a Chaotic Process Summary and Discussion 750 Notes and References 752 Problems 755 Chapter 14 Bayseian Filtering for State Estimation of Dynamic Systems Introduction State-Space Models Kaiman Filters The Divergence-Phenomenon and Square-Root Filtering The Extended Kaiman Filter The Bayesian Filter Cubature Kaiman Filter: Building on the Kaiman Filter Particle Filters Computer Experiment: Comparative Evaluation of Extended Kaiman and Particle Filters Kaiman Filtering in Modeling of Brain Functions Summary and Discussion 808 Notes and References 810 Problems 812

7 10 Contents Chapter 15 Dynamically Driven Recurrent Networks Introduction Recurrent Network Architectures Universal Approximation Theorem Controllability and Observability Computational Power of Recurrent Networks Learning Algorithms Back Propagation Through Time Real-Time Recurrent Learning Vanishing Gradients in Recurrent Networks Supervised Training Framework for Recurrent Networks Using Nonlinear Sequential State Estimators Computer Experiment: Dynamic Reconstruction of Mackay-Glass Attractor Adaptivity Considerations Case Study: Model Reference Applied to Neurocontrol Summary and Discussion 863 Notes and References 867 Problems 870 Bibliography 875 Index 916

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