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A combinatorial approach to understanding perceptron capabilities

This work investigates the classification capabilities of perceptrons which incorporate a single hidden layer of nodes from a theoretical viewpoint. In particular, the question of determining whether a given set can be realized as the decision region of such a network is considered. The main theoretic result demonstrates that the realizability of a set can be determined by restricting attention to any neighborhood of its boundary. This result is then used to identify general classes of realizable sets, and an example is given which shows that even though the realizability of a set might be readily discerned, the construction of an appropriate perceptron architecture may be complicated.