Evaluating the Vapnik-Chervonenkis dimension of artificial neural networksusing the Poincare polynomial

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 Poincar e polynomial of the implied hyperplane arrangement. This ANN is geometrical ly a hyperplane arrangement, which is configured to dichotomize a signed se t (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 b e used to evaluate certain geometric invariants of this semi-lattice, in pa rticular, 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 s table formula to compute the V-C dimension values. (C) 1999 Elsevier Scienc e Ltd. All rights reserved.