Mathematical Perspectives on Neural Networks

Mathematical Perspectives on Neural Networks

Mathematical Perspectives on Neural Networks

Mathematical Perspectives on Neural Networks


Recent years have seen an explosion of new mathematical results on learning and processing in neural networks. This body of results rests on a breadth of mathematical background which even few specialists possess. In a format intermediate between a textbook and a collection of research articles, this book has been assembled to present a sample of these results, and to fill in the necessary background, in such areas as computability theory, computational complexity theory, the theory of analog computation, stochastic processes, dynamical systems, control theory, time-series analysis, Bayesian analysis, regularization theory, information theory, computational learning theory, and mathematical statistics.

Mathematical models of neural networks display an amazing richness and diversity. Neural networks can be formally modeled as computational systems, as physical or dynamical systems, and as statistical analyzers. Within each of these three broad perspectives, there are a number of particular approaches. For each of 16 particular mathematical perspectives on neural networks, the contributing authors provide introductions to the background mathematics, and address questions such as:

• Exactly what mathematical systems are used to model neural networks from the given perspective?

• What formal questions about neural networks can then be addressed?

• What are typical results that can be obtained? and

• What are the outstanding open problems?

A distinctive feature of this volume is that for each perspective presented in one of the contributed chapters, the first editor has provided a moderately detailed summary of the formal results and the requisite mathematical concepts. These summaries are presented in four chapters that tie together the 16 contributed chapters: three develop a coherent view of the three general perspectives -- computational, dynamical, and statistical; the other assembles these three perspectives into a unified overview of the neural networks field.


The last several years have witnessed a remarkable growth in interest in the study of brain-style computation. This effort has variously been characterized as the study of neural networks, connectionist architectures, parallel distributed processing systems, neuromorphic computation, artificial neural systems, and other names as well. For purposes of the present series we have chosen the phrase connectionist theory. the common theme to all of these efforts has been an interest in looking at the brain as a model of a parallel computational device very different from that of a traditional serial computer. the strategy has been to develop simplified mathematical models of brain-like systems, and then to study these models to understand how various computation problems can be solved by such devices. the work has attracted scientists from a number of disciplines. It has attracted neuroscientists who are interested in making models of the neural circuitry found in specific areas of the brains of various animals, physicists who see analogies between the dynamical behavior of brain-like systems and the kinds of nonlinear dynamical systems familiar in physics, computer engineers who are interested in fabricating brain-like computers, workers in artificial intelligence who are interested in building machines with the intelligence of biological organisms, psychologists who are interested in the mechanisms of human information processing, mathematicians who are interested in the mathematics of such brain-style systems, philosophers who are interested in how such systems change our view of the nature of mind and its relationship to brain, and many others. the wealth of talent and the breadth of interest have made the area a magnet for bright young students.

In the face of this exponential growth and multidisciplinary set of contributions, it is difficult for workers in the area, let alone those outside of the area interested in current developments, to keep up with the work or to find the important new papers . . .

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