Some Analysis on $\nu$-Support Vector Classification

The $\nu$-support vector machines ($\nu$-SVM) for classification proposed by Sch\"olkopf et al. has the advantage of using a parameter $\nu$ on controlling the number of support vectors. However, comparing to regular SVM, its formulation is more complicated so up to now there are no effective methods for solving large-scale $\nu$-SVM. In this paper, we reformulate the $\nu$-SVM to a quadratic program with bound constraints and one simple equality constraint. Then existing decomposition methods can be modified to solve it. We demonstrate a decomposition method similar to the software $SVM^light$ for regular SVM. Motivated from the possible infeasibility of the dual $\nu$-SVM formulation, we investigate the relation between $\nu$-SVM and original SVM in detail. We show that in general they are two different problems with the same solution set. Hence we may expect that many numerical aspects on solving them are similar. We also discuss the behavior of $\nu$-SVM by some preliminary numerical experiments.