Abstract
In this paper, we propose a new linear programming formulation of exact 1-norm twin support vector machine (TWSVM) for classification whose solution is obtained by solving a pair of dual exterior penalty problems as unconstrained minimization problems using Newton–Armijo algorithm. The idea of our formulation is to reformulate TWSVM as a strongly convex problem by incorporated regularization techniques and then derive an exact 1-norm linear programming formulation for TWSVM to improve robustness and sparsity. The solution of two modified unconstrained minimization problems reduces to solving just two systems of linear equations as opposed to solving two quadratic programming problems in TWSVM and TBSVM, which leads to extremely simple and fast algorithm. One significant advantage of our proposed method is the implementation of structural risk minimization principle. However, only empirical risk is considered in the primal problems of TWSVM due to its complex structure and thus may incur overfitting and suboptimal in some cases. Our approach has the advantage that a pair of matrix equation of order equals to the number of input examples is solved at each iteration of the algorithm. The algorithm converges from any starting point that can be easily implemented in MATLAB without using any optimization packages. Computational comparisons of our proposed method against original TWSVM, GEPSVM and SVM have been made on both synthetic and benchmark datasets. Experimental results show that our method is better or comparable in both computation time and classification accuracy.
Similar content being viewed by others
References
Balasundaram S, Tanveer M. On Lagrangian twin support vector regression. Neural Comput Appl. 2013;22(1):257–67.
Bradley PS, Mangasarian OL. Feature selection via concave minimization and support vector machines. In: Machine learning proceedings of the fifteenth international conference (ICML ’98). 1998; p. 82–90.
Brown MPS, Grundy WN, Lin D. Knowledge-based analysis of microarray gene expression data using support vector machine. In: Proceedings of the national academy of sciences of USA. 2000; 97(1):262–7.
Burges C. A tutorial on support vector machines for pattern recognition. Data Min Knowl Discov. 1998;2:1–43.
Cambria E, Hussain A. Sentic album: content-, concept-, and context-based online personal photo management system. Cogn Comput. 2012;4(4):477–96.
Cambria E, Hussain A. Sentic computing: techniques, tools, and applications. In: Springer briefs in Cogn Comput; 2012.
Cao FL, Xie TF, Xu ZB. The estimate for approximation error of neural networks: a constructive approach. Neurocomputing. 2009;71(4):626–30.
Chang CC, Lin CJ. A library for support vector machines. LIBSVM: ACM Trans Intell Syst Technol. 2011;2(3):27:1– 27.
Cortes C, Vapnik VN. Support vector networks. Mach Learn. 1995;20:273–97.
Cristianini N, Shawe-Taylor J. An introduction to support vector machines and other kernel based learning method. Cambridge: Cambridge University Press; 2000.
Duda RO, Hart PR, Stork DG. Pattern classification. 2nd ed. London: Wiley; 2001.
Fiacco AV, McCormick GP. Nonlinear programming: sequential unconstrained minimization techniques. London: Wiley; 1968.
Fung G, Mangasarian OL. Finite Newton method for Lagrangian support vector machine classification. Neurocomputing. 2003;55(1–2):39–55.
Gao S, Ye Q, Ye N. 1-norm least squares twin support vector machines. Neurocomputing. 2011;74:3590–7.
Grassi M, Cambria E, Hussain A, Piazza F. Sentic web: a new paradigm for managing social media affective information. Cogn Comput. 2011;3(3):480–9.
Hiriart-Urruty J-B, Strodiot JJ, Nguyen VH. Generalized hessian matrix and second order optimality conditions for problems with CL1 data. Appl Math Optim. 1984;11:4356.
Jayadeva, Khemchandani R, Chandra S. Twin support vector machines for pattern classification. IEEE Trans Pattern Anal Mach Intell. 2007;29(5):905–910.
Joachims T, Ndellec C, Rouveriol C. Text categorization with support vector machines: learning with many relevant features. In: European conference on machine learning. 1998; (10): 137–142.
Joachims T. Making large-scale support vector machine learning practical. In: Advances in Kernel methods: support vector learning. MIT Press, Cambridge; 1999.
Kumar MA, Gopal M. Application of smoothing technique on twin support vector machines. Pattern Recognit Lett. 2008;29:1842–8.
Kumar MA, Gopal M. Least squares twin support vector machines for pattern classification. Expert Syst Appl. 2009;36:7535–43.
Kumar MA, Khemchandani R, Gopal M, Chandra S. Knowledge based least squares twin support vector machines. Inf Sci. 2010;180(23):4606–18.
Mangasarian OL. A finite Newton method for classification. Optim Methods Softw. 2002;17:913–29.
Mangasarian OL. Exact 1-norm support vector machines via unconstrained convex differentiable minimization. J Mach Learn Res. 2006;7:1517–30.
Mangasarian OL, Wild EW. Multisurface proximal support vector classification via generalized eigenvalues. IEEE Trans Pattern Anal Mach Intell. 2006;28(1):69–74.
Murphy PM, Aha DW. UCI repository of machine learning databases. University of California Irvine. http://www.ics.uci.edu/mlearn; 1992.
Musicant DR. NDC: Normally distributed clustered datasets. http://www.cs.wisc.edu/musicant/data/dc; 1998.
Osuna E, Freund R, Girosi F. Training support vector machines: an application to face detection. In: Proceedings of computer vision and pattern recognition. 1997; p. 130–136.
Peng X. TSVR: an efficient twin support vector machine for regression. Neural Netw. 2010;23(3):365–72.
Peng X. TPMSVM: a novel twin parametric-margin support vector machine for pattern recognition. Pattern Recognit. 2011;44:2678–92.
Peng X. Building sparse twin support vector machine classifiers in primal space. Inf Sci. 2011;181:3967–80.
Platt J. Fast training of support vector machines using sequential minimal optimization. In: Scholkopf B, Burges CJC, Smola AJ, editors. Advances in Kernel methods-support vector learning. Cambridge: MIT Press; 1999. p. 185–208.
Qi Z, Tian Y, Shi Y. A nonparallel support vector machine for a classification problem with universum learning. J Comput Appl Math. 2014;263:288–98.
Shao YH, Chen WJ, Deng NY. Nonparallel hyperplane support vector machine for binary classification problems. Inf Sci. 2014;263:22–35.
Shao YH, Zhang CH, Wang XB, Deng NY. Improvements on twin support vector machines. IEEE Trans Neural Netw. 2011;22(6):962–8.
Shao YH, Zhang CH, Yang ZM, Jing L, Deng NY. An \(\epsilon \)-twin support vector machine for regression. Neural Comput Appl. 2013;23(1):175–85.
Tanveer M. Smooth linear programming twin support vector machines. Int J Mach Learn Comput. 2013;3(2):240–4.
Tian Y, Ping Y. Large-scale linear nonparallel support vector machine solver. Neural Netw. 2014;50:166–74.
Tikhonov AN, Arsen VY. Solutions of ill-posed problems. New York: Wiley; 1977.
Vapnik VN. Statistical learning theory. New York: Wiley; 1998.
Vapnik VN. The nature of statistical learning theory. 2nd ed. New York: Springer; 2000.
Wang QF, Cambria E, Liu CL, Hussain A. Common sense knowledge for handwritten Chinese recognition. Cogn Comput. 2013;5(2):234–42.
Wu JD, Liu CT. Finger-vein pattern identification using svm and neural network technique. Expert Syst Appl. 2011;38(11):14284–9.
Xu Y, Guo R, Wang L. A twin multi-class classification support vector machine. Cogn Comput. 2013;5(4):580–8.
Xu Y, Wang L. A weighted twin support vector regression. Knowl-Based Syst. 2012;33:92–101.
Xu Y, Wang L, Zhong P. A rough margin-based \(\nu \)-twin support vector machine. Neural Comput Appl. 2012;21(6):1307–17.
Zhong P, Xu Y, Zhao Y. Training twin support vector regression via linear programming. Neural Comput Appl. 2012;21(2):399–407.
Acknowledgments
The author acknowledges the valuable comments of the anonymous reviewers and the Editor of Cognitive Computation whose enthusiasm is gladly appreciated. Also, the author would like to express his sincere gratitude to Professor S. Balasundaram for his help during the preparation of this manuscript.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tanveer, M. Robust and Sparse Linear Programming Twin Support Vector Machines. Cogn Comput 7, 137–149 (2015). https://doi.org/10.1007/s12559-014-9278-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12559-014-9278-8