Construction of convex hull classifiers in high dimensions

Abstract

We propose an algorithm to approximate each class region by a small number of approximated convex hulls and to use these for classification. The classifier is one of non-kernel maximum margin classifiers. It keeps the maximum margin in the original feature space, unlike support vector machines with a kernel. The construction of an exact convex hull requires an exponential time in dimension, so we find an approximate convex hull (a polyhedron) instead, which is constructed in linear time in dimension. We also propose a model selection procedure to control the number of faces of convex hulls for avoiding over-fitting, in which a fast procedure is adopted to calculate an upper-bound of the leave-one-out error. In comparison with support vector machines, the proposed approach is shown to be comparable in performance but more natural in the extension to multi-class problems.