SVM-Maj: a majorization approach to linear support vector machines with different hinge errors Regular Article First Online: 27 March 2008 Received: 03 November 2007 Revised: 23 February 2008 Accepted: 25 February 2008 DOI :
10.1007/s11634-008-0020-9
Cite this article as: Groenen, P.J.F., Nalbantov, G. & Bioch, J.C. ADAC (2008) 2: 17. doi:10.1007/s11634-008-0020-9 Abstract Support vector machines (SVM) are becoming increasingly popular for the prediction of a binary dependent variable. SVMs perform very well with respect to competing techniques. Often, the solution of an SVM is obtained by switching to the dual. In this paper, we stick to the primal support vector machine problem, study its effective aspects, and propose varieties of convex loss functions such as the standard for SVM with the absolute hinge error as well as the quadratic hinge and the Huber hinge errors. We present an iterative majorization algorithm that minimizes each of the adaptations. In addition, we show that many of the features of an SVM are also obtained by an optimal scaling approach to regression. We illustrate this with an example from the literature and do a comparison of different methods on several empirical data sets.
Keywords Support vector machines Iterative majorization Absolute hinge error Quadratic hinge error Huber hinge error Optimal scaling Download to read the full article text
References Borg I, Groenen PJF (2005) Modern multidimensional scaling: theory and applications, 2nd edn. Springer, New York
MATH Google Scholar Burges CJC (1998) A tutorial on support vector machines for pattern recognition. Knowl Discov Data Min 2: 121–167
CrossRef Google Scholar Chang C-C, Lin C-J (2006) LIBSVM: a library for support vector machines (Software available at
http://www.csie.ntu.edu.tw/~cjlin/libsvm )
Chu W, Keerthi S, Ong C (2003) Bayesian trigonometric support vector classifier. Neural Comput 15(9): 2227–2254
MATH CrossRef Google Scholar Cristianini N, Shawe-Taylor J (2000) An introduction to support vector machines. Cambridge University Press, Cambridge
Google Scholar De Leeuw J (1994) Block relaxation algorithms in statistics. In: Bock H-H, Lenski W, Richter MM(eds) Information systems and data analysis. Springer, Berlin, pp 308–324
Google Scholar Gifi A (1990) Nonlinear multivariate analysis. Wiley, Chichester
MATH Google Scholar Groenen PJF, Nalbantov G, Bioch JC (2007) Nonlinear support vector machines through iterative majorization. In: Decker R, Lenz H-J(eds) Advances in data analysis. Springer, Berlin, pp 149–162
CrossRef Google Scholar Hastie T, Tibshirani R, Friedman J (2001) The elements of statistical learning. Springer, New York
MATH Google Scholar Heiser WJ (1995) Convergent computation by iterative majorization: theory and applications in multidimensional data analysis. In: Krzanowski WJ(eds) Recent advances in descriptive multivariate analysis. Oxford University Press, Oxford, pp 157–189
Google Scholar Hsu C-W, Lin C-J (2006) BSVM: bound-constrained support vector machines (Software available at
http://www.csie.ntu.edu.tw/~cjlin/bsvm/index.html )
Huber PJ (1981) Robust statistics. Wiley, New York
MATH Google Scholar Hunter DR, Lange K (2004) A tutorial on MM algorithms. Am Stat 39: 30–37
MathSciNet CrossRef Google Scholar Joachims T (1999) Making large-scale SVM learning practical. In: Schölkopf B, Burges C, Smola A (eds) Advances in kernel methods—support vector learning. MIT-Press, Cambridge (
http://www-ai.cs.uni-dortmund.de/DOKUMENTE/joachims_99a.pdf )
Joachims T (2006) Training linear SVMs in linear time. In: Proceedings of the ACM conference on knowledge discovery and data mining (KDD) (
http://www.cs.cornell.edu/People/tj/publications/joachims_06a.pdf )
Kiers HAL (2002) Setting up alternating least squares and iterative majorization algorithms for solving various matrix optimization problems. Comput Stat Data Anal 41: 157–170
MATH CrossRef MathSciNet Google Scholar Kruskal JB (1965) The analysis of factorial experiments by estimating monotone transformations of the data. J R Stat Soc Ser B 27: 251–263
MathSciNet Google Scholar Lange K, Hunter DR, Yang I (2000) Optimization transfer using surrogate objective functions. J Comput Graph Stat 9: 1–20
CrossRef MathSciNet Google Scholar Newman D, Hettich S, Blake C, Merz C (1998) UCI repository of machine learning databases (
http://www.ics.uci.edu/~mlearn/MLRepository.html University of California, Irvine, Department of Information and Computer Sciences)
Rosset S, Zhu J (2007) Piecewise linear regularized solution paths. Ann Stat 35: 1012–1030
MATH CrossRef MathSciNet Google Scholar Rousseeuw PJ, Leroy AM (2003) Robust regression and outlier detection. Wiley, New York
Google Scholar Suykens JAK, Van Gestel T, De Brabanter J, De Moor B, Vandewalle J (2002) Least squares support vector machines. World Scientific, Singapore
MATH Google Scholar Van der Kooij AJ (2007) Prediction accuracy and stability of regression with optimal scaling transformations. Unpublished doctoral dissertation, Leiden University
Van der Kooij AJ, Meulman JJ, Heiser WJ (2006) Local minima in categorical multiple regression. Comput Stat Data Anal 50: 446–462
CrossRef MATH Google Scholar Vapnik VN (2000) The nature of statistical learning theory. Springer, New York
MATH Google Scholar Young FW (1981) Quantitative analysis of qualitative data. Psychometrika 46: 357–388
MATH CrossRef MathSciNet Google Scholar Young FW, De Leeuw J, Takane Y (1976) Additive structure in qualitative data: an alternating least squares method with optimal scaling features. Psychometrika 41: 471–503
CrossRef Google Scholar Young FW, De Leeuw J, Takane Y (1976) Regression with qualitative and quantitative variables: an alternating least squares method with optimal scaling features. Psychometrika 41: 505–529
MATH CrossRef Google Scholar Authors and Affiliations 1. Econometric Institute Erasmus University Rotterdam Rotterdam The Netherlands 2. ERIM and Econometric Institute Erasmus University Rotterdam Rotterdam The Netherlands