| Researchers rely on the mathematics of Vapnik and Chervonenkis to capture quantitative measures of the capabilities of specific artificial neural network (ANN) architectures. The measure is known as the V-C dimension of a set of sets, {dollar}{lcub}cal C{rcub},{dollar} and is defined as the largest cardinality l of a set of vectors in R{dollar}sp{lcub}d{rcub}{dollar} such that there is at least one set of vectors of cardinality l such that all partitions of that set into two sets can be represented by a set {dollar}fin{lcub}cal C{rcub}.{dollar} There is an abundance of research on determining the value of V-C dimensions of ANNs.; The fundamental thesis of this research is that the V-C dimension is not an appropriate measure of ANN capabilities. Consequently, the results of this research provide a basis of mathematics on which to build more intuitive quantifiers. Specifically, lattice structures for ANNs are established upon which a generalized method of invariant analysis of an arrangement of hyperplanes can be examined. In addition, a generalized function of invariants is presented as a mechanism for defining capability quantifiers. Moreover, a quantifier is defined based on an invariant, geometric complexity. The invariant is defined by concepts of combinatorial geometry.; Research on V-C dimension is refined and extended yielding formulas for evaluating V-C dimension for certain cases. As a consequence of the study of combinatorial geometry of hyperplane arrangements, it is shown that solutions to the chamber counting problem that are based on analysis of the Poincare polynomial also provide a closed form relation for determining the value of the V-C dimension of ANNs. |
