This graduate-level text teaches students how to use a small number of powerful mathematical tools for analyzing and designing a wide variety of artificial neural network (ANN) systems, including their own customized neural networks. Mathematical Methods for Neural Network Analysis and Design offers an original, broad, and integrated approach that explains each tool in a manner that is independent of specific ANN systems. Although most of the methods presented are familiar, their systematic application to neural networks is new. Included are helpful chapter summaries and detailed solutions to over 100 ANN system analysis and design problems. For convenience, many of the proofs of the key theorems have been rewritten so that the entire book uses a relatively uniform notion. This text is unique in several ways. It is organized according to categories of mathematical tools -- for investigating the behavior of an ANN system, for comparing (and improving) the efficiency of system computations, and for evaluating its computational goals -- that correspond respectively to David Marr's implementational, algorithmic, and computational levels of description. And instead of devoting separate chapters to different types of ANN systems, it analyzes the same group of ANN systems from the perspective of different mathematical methodologies. A Bradford Book

1 Introduction 
1.1 The General Theoretical Framework 
1.2 Introduction to ANN Systems 
1.3 ANN System Applications 
1.4 Formal Definition of ANN Systems 
1.5 Relevant Mathematical Concepts 
1.6 Chapter Summary 
1.7 Additional Reading 
1.8 Elementary Problems 
1.9 Math Review Problems 
I IMPLEMENTATIONAL LEVEL
2 ANN Dynamical Systems 
2.1 ANN Classification Dynamical Systems 
2.2 ANN Learning Dynamical Systems 
2.3 Chapter Summary 
2.4 Problems 
3 Deterministic Nonlinear Dynamical Systems Analysis 
3.1 Autonomous Dynamical Systems 
3.2 Invariant Sets 
3.3 Invariant Set Theorem 
3.4 ANN Applications 
3.5 Proof of the Invariant Set Theorem 
3.6 Chapter Summary 
3.7 Elementary Problems 
3.8 Problems 
4 Stochastic Nonlinear Dynamical Systems Analysis 
4.1 Stochastic Convergence Concepts 
4.2 Stochastic Approximation Theorem 
4.3 ANN Applications 
4.4 Stochastic Approximation Theorem Proof 
4.5 Chapter Summary 
4.6 Elementary Problems 
4.7 Problems 
II ALGORITHMIC LEVEL
5 Nonlinear Optimization Theory 
5.1 Optimization Goals 
5.2 Deterministic Nonlinear Optimization 
5.3 Stochastic Nonlinear Optimization 
5.4 Convergence Rate Analysis 
5.5 Classical and ANN Applications 
5.6 Chapter Summary 
5.7 Elementary Problems 
5.8 Problems 
III COMPUTATIONAL LEVEL
6 Rational Inference Measures 
6.1 Measures for Relational Systems 
6.2 Special Measures for Relational Systems 
6.3 Gibbs Probability Measures (Markov Random Fields) 
6.4 Decision Rules 
6.5 The Generalization Problem 
6.6 Historical Perspective on Decision Making 
6.7 Chapter Summary 
6.8 Elementary Problems 
7 Expected Risk Classification and Learning Theory 
7.1 The Optimal Classification Assumption 
7.2 Rational ANN Learning Goals 
7.3 ANN Applications 
7.4 Chapter Summary 
7.5 Elementary Problems 
7.6 Problems 
8 Statistical Model Evaluation 
8.1 Asymptotic Distribution of Sampling Error 
8.2 Model Selection 
8.3 ANN Applications 
8.4 Chapter Summary 
8.5 Elementary Problems 
8.6 Problems
Epilogue 
Problem Solutions 
References 
Author Index 
Subject Index

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