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Evaluating the Vapnik–Chervonenkis dimension of artificial neural networks using the Poincaré polynomial

✍ Scribed by Martha A. Carter; Mark E. Oxley

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
90 KB
Volume
12
Category
Article
ISSN
0893-6080

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✦ Synopsis

The Vapnik-Chervonenkis (V-C) dimension of a set of functions representing a feed-forward, multi-layered, single output artificial neural network (ANN) with hard-limited activation functions can be evaluated using the Poincare ´polynomial of the implied hyperplane arrangement. This ANN is geometrically a hyperplane arrangement, which is configured to dichotomize a signed set (i.e., a two-class set). As it is known that the cut-intersections of the hyperplane arrangement forms a semi-lattice, the Poincare ´polynomial can be used to evaluate certain geometric invariants of this semi-lattice, in particular, the cardinality of the resultant chamber set of the arrangements, which is shown to be the V-C dimension. From this theory, we arrive at a stable formula to compute the V-C dimension values.

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