Abstract
Fixed-Size Least Squares Support Vector Machines (FS-LSSVM) is a powerful tool for solving large scale classification and regression problems. FS-LSSVM solves an over-determined system of M linear equations by using Nyström approximations on a set of prototype vectors (PVs) in the primal. This introduces sparsity in the model along with ability to scale for large datasets. But there exists no formal method for selection of the right value of M. In this paper, we investigate the sparsity-error trade-off by introducing a second level of sparsity after performing one iteration of FS-LSSVM. This helps to overcome the problem of selecting a right number of initial PVs as the final model is highly sparse and dependent on only a few appropriately selected prototype vectors (SV) is a subset of the PVs. The first proposed method performs an iterative approximation of L 0-norm which acts as a regularizer. The second method belongs to the category of threshold methods, where we set a window and select the SV set from correctly classified PVs closer and farther from the decision boundaries in the case of classification. For regression, we obtain the SV set by selecting the PVs with least minimum squared error (mse). Experiments on real world datasets from the UCI repository illustrate that highly sparse models are obtained without significant trade-off in error estimations scalable to large scale datasets.
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References
Hoegaerts, L., Suykens, J.A.K., Vandewalle, J., De Moor, B.: A comparison of pruning algorithms for sparse least squares support vector machines. In: Pal, N.R., Kasabov, N., Mudi, R.K., Pal, S., Parui, S.K. (eds.) ICONIP 2004. LNCS, vol. 3316, pp. 1247–1253. Springer, Heidelberg (2004)
Geebelen, D., Suykens, J.A.K., Vandewalle, J.: Reducing the Number of Support Vectors of SVM classifiers using the Smoothed Seperable Case Approximation. IEEE Transactions on Neural Networks and Learning Systems 23(4), 682–688 (2012)
Suykens, J.A.K., Vandewalle, J.: Least Squares Support Vector Machine Classifiers. Neural Processing Letters 9(3), 293–300 (1999)
Vapnik, V.N.: The Nature of Statistical Learning Theory. Springer (1995)
Suykens, J.A.K., Lukas, L., Vandewalle, J.: Sparse approximation using Least Squares Support Vector Machines. In: Proceedings of IEEE International Symposium on Circuits and Systems (ISCAS 2000), pp. 757–760 (2000)
Li, Y., Lin, C., Zhang, W.: Improved Sparse Least-Squares Support Vector Machine Classifiers. Neurocomputing 69(13), 1655–1658 (2006)
Suykens, J.A.K., Van Gestel, T., De Brabanter, J., De Moor, B., Vandewalle, J.: Least Squares Support Vector Machines. World Scientific Publishing Co., Pte., Ltd., Singapore (2002)
Nyström, E.J.: Über die praktische Auflösung von Integralgleichungen mit Anwendungen auf Randwertaufgaben. Acta Mathematica 54, 185–204 (1930)
Baker, C.T.H.: The Numerical Treatment of Integral Equations. Oxford Claredon Press (1983)
De Brabanter, K., De Brabanter, J., Suykens, J.A.K., De Moor, B.: Optimized Fixed-Size Kernel Models for Large Data Sets. Computational Statistics & Data Analysis 54(6), 1484–1504 (2010)
Karsmakers, P., Pelckmans, K., De Brabanter, K., Hamme, H.V., Suykens, J.A.K.: Sparse conjugate directions pursuit with application to fixed-size kernel methods. Machine Learning, Special Issue on Model Selection and Optimization in Machine Learning 85(1), 109–148 (2011)
Weston, J., Elisseeff, A., Schölkopf, B., Tipping, M.: Use of the Zero Norm with Linear Models and Kernel Methods. Journal of Machine Learning Research 3, 1439–1461 (2003)
Huang, K., Zheng, D., Sun, J., et al.: Sparse Learning for Support Vector Classification. Pattern Recognition Letters 31(13), 1944–1951 (2010)
Lopez, J., De Brabanter, K., Dorronsoro, J.R., Suykens, J.A.K.: Sparse LSSVMs with L 0-norm minimization. In: ESANN 2011, pp. 189–194 (2011)
Blake, C.L., Merz, C.J.: UCI repository of machine learning databases, Irvine, CA (1998), http://archive.ics.uci.edu/ml/datasets.html
Williams, C.K.I., Seeger, M.: Using the Nyström method to speed up kernel machines. In: Advances in Neural Information Processing Systems, vol. 13, pp. 682–688 (2001)
Scott, D.W., Sain, S.R.: Multi-dimensional Density Estimation. Data Mining and Computational Statistics 23, 229–263 (2004)
Xavier de Souza, S., Suykens, J.A.K., Vandewalle, J., Bolle, D.: Coupled Simulated Annealing for Continuous Global Optimization. IEEE Transactions on Systems, Man, and Cybertics - Part B 40(2), 320–335 (2010)
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Mall, R., Suykens, J.A.K. (2013). Sparse Reductions for Fixed-Size Least Squares Support Vector Machines on Large Scale Data. In: Pei, J., Tseng, V.S., Cao, L., Motoda, H., Xu, G. (eds) Advances in Knowledge Discovery and Data Mining. PAKDD 2013. Lecture Notes in Computer Science(), vol 7818. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37453-1_14
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DOI: https://doi.org/10.1007/978-3-642-37453-1_14
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