SVM-Maj: a majorization approach to linear support vector machines with different hinge errors

Open Access
Regular Article

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 

Mathematics Subject Classification (2000)

90C30 62H30 68T05 

References

  1. Borg I, Groenen PJF (2005) Modern multidimensional scaling: theory and applications, 2nd edn. Springer, New YorkMATHGoogle Scholar
  2. Burges CJC (1998) A tutorial on support vector machines for pattern recognition. Knowl Discov Data Min 2: 121–167CrossRefGoogle Scholar
  3. 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)
  4. Chu W, Keerthi S, Ong C (2003) Bayesian trigonometric support vector classifier. Neural Comput 15(9): 2227–2254MATHCrossRefGoogle Scholar
  5. Cristianini N, Shawe-Taylor J (2000) An introduction to support vector machines. Cambridge University Press, CambridgeGoogle Scholar
  6. 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 308–324Google Scholar
  7. Gifi A (1990) Nonlinear multivariate analysis. Wiley, ChichesterMATHGoogle Scholar
  8. 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 149–162CrossRefGoogle Scholar
  9. Hastie T, Tibshirani R, Friedman J (2001) The elements of statistical learning. Springer, New YorkMATHGoogle Scholar
  10. 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 157–189Google Scholar
  11. 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)
  12. Huber PJ (1981) Robust statistics. Wiley, New YorkMATHGoogle Scholar
  13. Hunter DR, Lange K (2004) A tutorial on MM algorithms. Am Stat 39: 30–37MathSciNetCrossRefGoogle Scholar
  14. 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)
  15. 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)
  16. Kiers HAL (2002) Setting up alternating least squares and iterative majorization algorithms for solving various matrix optimization problems. Comput Stat Data Anal 41: 157–170MATHCrossRefMathSciNetGoogle Scholar
  17. Kruskal JB (1965) The analysis of factorial experiments by estimating monotone transformations of the data. J R Stat Soc Ser B 27: 251–263MathSciNetGoogle Scholar
  18. Lange K, Hunter DR, Yang I (2000) Optimization transfer using surrogate objective functions. J Comput Graph Stat 9: 1–20CrossRefMathSciNetGoogle Scholar
  19. 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)
  20. Rosset S, Zhu J (2007) Piecewise linear regularized solution paths. Ann Stat 35: 1012–1030MATHCrossRefMathSciNetGoogle Scholar
  21. Rousseeuw PJ, Leroy AM (2003) Robust regression and outlier detection. Wiley, New YorkGoogle Scholar
  22. Suykens JAK, Van Gestel T, De Brabanter J, De Moor B, Vandewalle J (2002) Least squares support vector machines. World Scientific, SingaporeMATHGoogle Scholar
  23. Van der Kooij AJ (2007) Prediction accuracy and stability of regression with optimal scaling transformations. Unpublished doctoral dissertation, Leiden UniversityGoogle Scholar
  24. Van der Kooij AJ, Meulman JJ, Heiser WJ (2006) Local minima in categorical multiple regression. Comput Stat Data Anal 50: 446–462CrossRefMATHGoogle Scholar
  25. Vapnik VN (2000) The nature of statistical learning theory. Springer, New YorkMATHGoogle Scholar
  26. Young FW (1981) Quantitative analysis of qualitative data. Psychometrika 46: 357–388MATHCrossRefMathSciNetGoogle Scholar
  27. Young FW, De Leeuw J, Takane Y (1976) Additive structure in qualitative data: an alternating least squares method with optimal scaling features. Psychometrika 41: 471–503CrossRefGoogle Scholar
  28. 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: 505–529MATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2008

Authors and Affiliations

  • P. J. F. Groenen
    • 1
  • G. Nalbantov
    • 2
  • J. C. Bioch
    • 1
  1. 1.Econometric InstituteErasmus University RotterdamRotterdamThe Netherlands
  2. 2.ERIM and Econometric InstituteErasmus University RotterdamRotterdamThe Netherlands

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