Mathematical Perspectives on Neural Networks

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

2 Overview: Computational Perspectives on Neural Networks

Paul Smolensky Johns Hopkins University

The processing problem in neural networks, as discussed in Chapter 1, is to understand the functions computed by neural networks as they convert input activation patterns into output patterns. Computation theory concerns the relation between the structure of (abstract) machines and the functions they compute, and so it obviously provides a natural perspective from which to investigate the processing problem in neural networks.

Probably the most basic property of neural networks viewed as computing machines is their parallelism. Thus, many of the questions explored in the computational perspective concern the consequences of parallelism on the functions computed, often by comparison with the functions computed by serial machines or machines less parallel than a particular kind of neural network.

For example, the first chapter in this part, Chapter 3 by Franklin and Garzon, considers an extreme kind of parallelism possible within neural networks: nets that have an infinite number of input units, over which can be represented, for example, a real number. Such nets can compute functions over the reals, whereas merely getting a real-valued input into a serial machine such as a Turing machine would take infinite time. What kind of functions over the reals can such nets compute? And how do the functions computed compare with those computed by models of parallel computation that are not neural nets (or not typical ones, at least)? (These other models include networks of general finite-state machines and regular cellular arrays of identical processors.)

In Chapter 4, Parberry asks what functions neural networks can compute with limited resources. Following the perspective of computational complexity theory, the basic questions concern how fast the necessary resources grow as the size of the input grows. How much do different kinds of parallelism help? For example,

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