Geometric Intuition and Algorithms for Ev--SVM

Abstract

In this work we address the E$\nu$--SVM model proposed by Pérez --Cruz et al. as an extension of the traditional $\nu$ support vector classification model ($\nu$--SVM). Through an enhancement of the range of admissible values for the regularization parameter $\nu$, the E$\nu$--SVM has been shown to be able to produce a wider variety of decision functions, giving rise to a better adaptability to the data. However, while a clear and intuitive geometric interpretation can be given for the $\nu$--SVM model as a nearest--point problem in reduced convex hulls (RCH--NPP), no previous work has been made in developing such intuition for the E$\nu$--SVM model. In this paper we show how E$\nu$--SVM can be reformulated as a geometrical problem that generalizes RCH--NPP, providing new insights into this model. Under this novel point of view, we propose the rapminos algorithm, able to solve E$\nu$--SVM more efficiently than the current methods. Furthermore, we show how rapminos is able to address the E$\nu$--SVM model for any choice of regularization norm $\ell_{p \geq 1}$ seamlessly, which further extends the SVM model flexibility beyond the usual E$\nu$--SVM models.