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

Abstract

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 Poincaré 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 Poincaré 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.