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Discrete Neural Computation: A Theoretical Foundation [Book Review]

Discrete neural computation is about the complexity of discrete threshold circuits. Although many fundamental issues in the complexity of threshold circuits are yet to be resolved, researchers recently obtained a number of impressive results applying a wide variety of techniques : Fourier methods, rational approximations, communication complexity, probabilistic constructions, and algebraic techniques. In this book, Siu, Roychowdhury, and Kailath bring together a number of such results on the resources (gate count, depth, size of weights) required of threshold circuits that compute specific as well as arbitrary Boolean functions. The book does a remarkable job of presenting most of the recent research concerning the complexity of discrete threshold circuits. These results include many interesting constructions for computing a number of specific Boolean functions as well as lower bounds on the resources. In particular, the authors present their constructions for depth-efficient threshold circuits for such arithmetic functions as addition, multiplication, and division. This book should be of interest to researchers in the areas of neural networks and circuit complexity, although it does not offer a broader context for the study of threshold circuits from the perspective of either of these areas. The coverage of the main topic of this book, complexity of threshold circuits, is comprehensive. In addition to presenting a number of classical results, the book includes a vast array of recent results from a wide variety of sources to provide the reader with the current state of the research in the area. The presentation of these results is fairly self-contained. I wish that the authors paid more attention to pedagogical matters. In certain places, the book is lacking in guiding the reader regarding what is significant. It is sometimes hard to wade through the unimportant or obvious theorems to find fundamental or elegant results. The book starts with the basic results regarding the capabilities of a linear threshold element. The coverage is not limited to classical