Received: 31 July 2019 Revised: 5 October 2019 Accepted: 25 October 2019 DOI: 10.1002/ett.3820 SPECIAL ISSUE ARTICLE Sparse support vector machine with pinball loss M. Tanveer 1 S. Sharma 1 R. Rastogi 2 P. Anand 2 1 Discipline of Mathematics, Indian Institute of Technology Indore, Simrol, Indore, India 2 Department of Computer Science, Faculty of Mathematics and Computer Science, South Asian University, Delhi, India Correspondence M. Tanveer, Discipline of Mathematics, Indian Institute of Technology Indore, Simrol, Indore, 453552, India. Email: mtanveer@iiti.ac.in Funding information Early Career Research Award Scheme, Grant/Award Number: ECR/2017/000053; Research (EMR) Scheme, Grant/Award Number: 22(0751)/17/EMR-II; Indian Institute of Technology Indore Abstract The standard support vector machine (SVM) with a hinge loss function suffers from feature noise sensitivity and instability. Employing a pinball loss function instead of a hinge loss function in SVMs provides noise insensitivity to the model as it maximizes the quantile distance. However, the pinball loss function simul- taneously causes the model to lose sparsity by penalizing correctly classified samples. To overcome the aforementioned shortcomings, we propose a novel sparse SVM with pinball loss (Pin-SSVM) for solving classification problems. The proposed Pin-SSVM employs L1-norm in the SVM classifier with the pinball loss (Pin-SVM), which ensures the robustness, sparseness, and noise insensi- tivity of the model. The proposed Pin-SSVM eradicates the need to solve the dual as we simply obtain its solution by solving a linear programming problem (LPP). The proposed Pin-SSVM does not spend more computational time as that of Pin-SVM. Hence, solving an LPP with two linear inequality constraints does not affect the computational complexity. The numerical experiments on several real-world benchmark noise corrupted and imbalanced UCI datasets demon- strate that the proposed Pin-SSVM is suitable for noisy and imbalanced data sets, and in most cases, outperforms the results of the baseline models. 1 INTRODUCTION Support vector machines (SVMs) are introduced to machine learning community by Vapnik and other researchers for clas- sification and regression problems. 1-6 SVM has earned a great popularity as it has made substantial contribution to real-life applications such as face detection, 7 cancer diagnosis, 8 electroencephalogram signal classification, 9 text categorization, 10 feature extraction 11 and biomedicine. 12 SVM has excellently performed on a wide variety of problems 7,13 as it constructs a separating hyperplane such that the margin between the two supporting parallel hyperplanes is maximized. This idea leads to an introduction of regularization term. Finally, an optimal separating hyperplane is selected to be the plane lying in the middle of supporting hyperplanes. It incorporates the structural risk minimization principle of statistical learn- ing theory. 14 Thenceforth, various models have been proposed for multiclass classification, including binary decision tree SVM, 15 kernel method, 16 discriminative methods, 17 triclass SVM, 18 binary tree architecture, 19 decision directed acyclic graph technique, 20 and multiclass pattern recognition. 21 The most challenging part in SVM is its high computational complexity as solving a large quadratic programming problem (QPP) causes SVM to take longer training time. Later on, Fung and Mangasarian 22 introduced a Newton method for feature selection for linear programming SVM (NLPSVM) which employs 1-norm in SVM formulation that is known to generate a very sparse solution. NLPSVM suppresses the input feature space, which means that the separating hyper- plane depends on a very few input vectors. Afterwards, various solvers have been proposed for obtaining solution of SVM including SVMlight 13 and sequential minimal optimization. 23,24 Subsequently, Mangasarian 25 proposed an exact 1-norm SVM, which is formulated as an unconstrained minimization of a convex differentiable piecewise-quadratic objective Trans Emerging Tel Tech. 2020;e3820. wileyonlinelibrary.com/journal/ett © 2020 John Wiley & Sons, Ltd. 1 of 13 https://doi.org/10.1002/ett.3820