Regularization Path for $\nu$ -Support Vector Classification

The v-support vector classification (v-SVC) proposed by Schölkopf has the advantage of using a regularization parameter v for controlling the number of support vectors and margin errors. However, compared to C-SVC, its formulation is more complicated, and to date there are no effective methods for computing its regularization path. In this paper, we propose a new regularization path algorithm, which is designed on the basis of a modified formulation of v-SVC and traces the solution path with respect to the parameter v. Through theoretical analysis and confirmatory experiments, we show that our algorithm can avoid the infeasible updating path under several assumptions (i.e., Assumptions 1 and 2), and fit the entire solution path in a finite number of steps. When the regularization path of v-SVC is available, a novel approach proposed by Yang and Ong can be applied to obtain the global optimal solution of common validation functions for v-SVC, and the computation for the whole process is minimal. Numerical experiments show that it is more efficient than various kinds of grid search methods for selecting the optimal regularization parameter v.