Robust Pinball Twin Bounded Support Vector Machine for Data Classification

In this paper, a novel robust L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{1}$$\end{document}-norm based twin bounded support vector machine with pinball loss- having regularization term, scatter loss and misclassification loss- is proposed to enhance robustness in the presence of feature noise and outliers. Unlike in twin bounded support vector machine (TBSVM), pinball is used as the misclassification loss in place of hinge loss to reduce noise sensitivity. To further boost robustness, the scatter loss of the class of vectors is minimized using L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{1}$$\end{document}-norm. As an equivalent problem in simple form, a pair of quadratic programming problems (QPPs) is constructed (L1-Pin-TBSVM) with m variables where m is the number of training vectors. Unlike TBSVM, the proposed L1-Pin-TBSVM is free from inverse kernel matrix and the non-linear problem can be obtained directly from its linear formulation by applying the kernel trick. The efficacy and robustness of L1-Pin-TBSVM has been demonstrated by experiments performed on synthetic and UCI datasets in the presence of noise.