An improved robust and sparse twin support vector regression via linear programming

Twin support vector regression (TSVR) was proposed recently as a novel regressor that tries to find a pair of nonparallel planes, i.e. ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon $$\end{document}-insensitive up- and down-bounds, by solving two related SVM-type problems. Though TSVR exhibits good performance compared with conventional methods like SVR, it suffers from the following issues: (1) it lacks model complexity control and thus may incur overfitting and suboptimal solution; (2) it needs to solve a pair of quadratic programming problems which are relatively complex to implement; (3) it is sensitive to outliers; and (4) its solution is not sparse. To address these problems, we propose in this paper a novel regression algorithm termed as robust and sparse twin support vector regression. The central idea is to reformulate TSVR as a convex problem by introducing regularization technique first and then derive a linear programming (LP) formulation which is not only simple but also allows robustness and sparseness. Instead of solving the resulting LP problem in the primal, we present a Newton algorithm with Armijo step-size to resolve the corresponding exact exterior penalty problem. The experimental results on several publicly available benchmark data sets show the feasibility and effectiveness of the proposed method.