Geometric algorithms for parametric-margin ν-supportν-support vector machine

The parametric-margin ν-supportν-support vector machine (par-ν-SVM)(par-ν-SVM) is a useful classifier in many cases, especially when the noise is heteroscedastic. In this paper, the geometric interpretation for the par-ν-SVMpar-ν-SVM is described, which is equivalent to finding a couple of points in two disjoint μ-reducedμ-reduced convex hulls (μ-RCHs)(μ-RCHs) by simultaneously minimizing the square distance and maximizing the square norm of their sum with a weight factor 1/(cν)1/(cν) given by users. Motivated by the Gilbert–Schlesinger–Kozinec (GSK) and Mitchell–Dem'yanov–Malozemov (MDM) algorithms, two geometric algorithms, called the parametric μ-GSK(par-μ-GSK) and parametric μ-MDM(par-μ-MDM) algorithms, are introduced to solve the par-ν-SVMpar-ν-SVM. Computational results on several synthetic as well as benchmark datasets demonstrate the significant performance of the proposed algorithms in terms of both kernel operations and classification accuracy.