A Geometric Interpretation of v-SVM Classifiers

We show that the recently proposed variant of the Support Vector machine (SVM) algorithm, known as v-SVM, can be interpreted as a maximal separation between subsets of the convex hulls of the data, which we call soft convex hulls. The soft convex hulls are controlled by choice of the parameter v. The proposed geometric interpretation of v-SVM also leads to necessary and sufficient conditions for the existence of a choice of v for which the v-SVM solution is nontrivial. 1 Introduction Recently, SchOlkopf et al. [I) introduced a new class of SVM algorithms, called v-SVM, for both regression estimation and pattern recognition. The basic idea is to remove the user-chosen error penalty factor C that appears in SVM algorithms by introducing a new variable p which, in the pattern recognition case, adds another degree of freedom to the margin. For a given normal to the separating hyperplane, the size of the margin increases linearly with p.