Training v-Support Vector Classifiers: Theory and Algorithms

The nu-support vector machine (nu-SVM) for classification proposed by Schölkopf, Smola, Williamson, and Bartlett (2000) has the advantage of using a parameter nu on controlling the number of support vectors. In this article, we investigate the relation between nu-SVM and C-SVM in detail. We show that in general they are two different problems with the same optimal solution set. Hence, we may expect that many numerical aspects of solving them are similar. However, compared to regular C-SVM, the formulation of nu-SVM is more complicated, so up to now there have been no effective methods for solving large-scale nu-SVM. We propose a decomposition method for nu-SVM that is competitive with existing methods for C-SVM. We also discuss the behavior of nu-SVM by some numerical experiments.