Algorithms for the computation of reduced convex hulls

Abstract

Geometric interpretations of Support Vector Machines (SVMs) have introduced the concept of a reduced convex hull. A reduced convex hull is the set of all convex combinations of a set of points where the weight any single point can be assigned is bounded from above by a constant. This paper decouples reduced convex hulls from their origins in SVMs and allows them to be constructed independently. Two algorithms for the computation of reduced convex hulls are presented - a simple recursive algorithm for points in the plane and an algorithm for points in an arbitrary dimensional space. Upper bounds on the number of vertices and facets in a reduced convex hull are used to analyze the worst-case complexity of the algorithms. © Springer-Verlag Berlin Heidelberg 2009.