Evaluating the Vapnik-Chervonenkis dimension of artificial neural networks using 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 Poincare polynomial of the implied hyperplane arrangement. This ANN geometrically is a hyperplane arrangement configured to dichotomize a signed set (i.e., a two-class set). Since it is known that the cut- intersections of the hyperplane arrangement forms a semi- lattice, then 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 comes a stable formula to compute the V-C dimension values.