Mathematical Perspectives on Neural Networks

By Paul Smolensky; Michael C. Mozer et al. | Go to book overview

16 Information Theory and Neural Nets

J. Rissanen IBM Almaden Research Center

When asked to write a survey chapter on information theory and neural nets, I was both intrigued and concerned. Intrigued, because I was hoping that information theory might provide some guidance to the understanding of these tricky but apparently powerful modeling devices and possibly to suggest even new ideas for their design. Concerned, because I had, and still have, only a superficial knowledge of neural nets, and I had found the reading of their vast literature tough going, which the mixture of well-known principles with ad hoc techniques couched in an applications-oriented language, made no easier. Although information theoretic notions appear quite frequently in the neural net literature, there are not many major areas of contact between the two disciplines. The intersection is not empty, however, and I have selected three main themes where information theory has already made an impact or, as I hope, it will prove beneficial to the neural network community.

The first of these themes, the calculation of the storage capacity of the neural nets of Hopfield's type, utilizes the main idea in error correcting codes. The basic facts were found by McElice, Posner, Rodemich, and Venkatesh ( 1987); see also Amari and Maginu ( 1988), and Koml6s and Paturi ( 1988). The problem, which is still subject to research, is quite intricate, and in this chapter I give a simple but rigorous account of the main result only. Unlike in the original papers, where the central limit theorem is used, I derive in the brief Sec. 2 the main storage bound by a direct application of Chernoff's inequality.

The second main theme concerns the design of certain types of neural nets by the maximization of one of the most fundamental notions in information theory, the mutual information, and it is of a very wide scope. The procedure, introduced to the neural networks by Linsker ( 1986), (1988), (1989a), (1989b) and called the

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