Geometrical and Statistical Properties of Systems of Linear Inequalities with Applications in Pattern Recognition

This paper develops the separating capacities of families of nonlinear decision surfaces by a direct application of a theorem in classical combinatorial geometry. It is shown that a family of surfaces having d degrees of freedom has a natural separating capacity of 2d pattern vectors, thus extending and unifying results of Winder and others on the pattern-separating capacity of hyperplanes. Applying these ideas to the vertices of a binary n-cube yields bounds on the number of spherically, quadratically, and, in general, nonlinearly separable Boolean functions of n variables. It is shown that the set of all surfaces which separate a dichotomy of an infinite, random, separable set of pattern vectors can be characterized , on the average, by a subset of only 2d extreme pattern vectors. In addition, the problem of generalizing the classifications on a labeled set of pattern points to the classification of a new point is defined, and it is found that the probability of ambiguous generalization is large unless the number of training patterns exceeds the capacity of the set of separating surfaces.