An effective method to determine whether a point is within a convex hull and its generalized convex polyhedron classifier

Abstract

A convex polyhedron classifier that encloses the minority class using a combination of hyperplanes is potentially effective in imbalanced classification. To construct an easy-to-use convex polyhedron classifier, this paper first presents a theoretical foundation for determining whether a point is within the convex hull of a finite point set. This foundation corresponds to a geometric method in which the result is expressed as a separating hyperplane. If the given point and the given convex hull are located on either side of the learned hyperplane, this indicates that the point is outside of the convex hull. Otherwise, the conclusion that the point is within the convex hull can be obtained. As a generalization of the geometric method, a convex polyhedron classifier is further proposed for binary classification. If two finite point sets are polyhedrally separable, a series of hyperplanes can be learned as a combined (piecewise linear) classifier, which surrounds a point set that is inside using a convex polyhedron and excludes the other point set that is outside. Experimental results on twelve real-world datasets show that the proposed classifier is generally better than the other two piecewise linear classifiers. Moreover, a comparison with several types of support vector machines confirms its competitiveness.