Abstract
In this paper, we propose a Second Order Cone Programming representable Mahalanobis Ellipsoidal Learning Machine (SOCP-MELM) for One Class Classification (OCC). We propose to utilize the covariance matrix and thus the Mahalanobis distance to replace the Euclidean distance in standard Support Vector Data Description (SVDD). Consequently, we modify and rewrite the SVDD as a standard SOCP problem and then solve it directly in its primal form via interior point methods in polynomial time. By introducing a specified uncertainty model and using the chebyshev inequality, we propose a robust form of SOCP-MELM. Finally, we validate the proposed method using real world benchmark datasets.
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References
Scholkopf, B., Smola, A.: Learning with Kernels, 1st edn. MIT Press, Cambridge, MA (2002)
Tax, D. One-class Classification: Concept-learning in the Absence of Counter-Examples. PhD Thesis, Delft University of Technology (2001)
Scholkopf, B., Platt, J., Shawe-Taylor, J., Smola, A., Williamson, R.: Estimating the Support of a High Dimensional Distribution. Neural Computation 13(7), 1443–1471 (2001)
Tax, D., Duin, R.: Support Vector Domain Description. Pattern Recognition Letters 20(14), 1191–1199 (1999)
Tax, D., Juszczak, P.: Kernel Whitening for One-Class Classification. International Journal of Pattern Recognition and Artificial Intelligence 17(3), 333–347 (2003)
Lanckriet, G., Ghaoui, L.E., Jodan, M.: Robust Novelty Detection with Single-Class MPM. In: Becker, S., Thrun, S., Obermayor, K. (eds.) Advances in Neural Information Processing Systems, vol. 15, MIT Press, Cambridge, MA (2003)
Tsang, I.W., Kwok, J.T., Li, S.: Learning the Kernel in Mahalanobis One-class Support Vector Machines. In: Proceeding of IJCNN 2006 Conference. Vancouver, BC, Canada, pp. 1169–1175 (2006)
Abe, S.: Training of Support Vector Machines with Mahalanobis Kernels. In: Duch, W., Kacprzyk, J., Oja, E., Zadrożny, S. (eds.) ICANN 2005. LNCS, vol. 3697, pp. 571–576. Springer, Heidelberg (2005)
Wang, J., Kwok, J.T., Shen, H.C., Quan, L.: Data-Dependent Kernels for High-Dimensional Data Classification. In: Proceedings of International Joint Conference on Neural Networks, Montreal, Canada, pp. 102–107 (2005)
Wei, X.K., Huang, G.B., Li, Y.H.: Mahalanobis Ellipsoidal Learning Machine for One Class Classification. Manuscript (2007)
Wei, X.K., Huang, G.B., Li, Y.H.: A New One Class Mahalanobis Hyperplane Learning Machine based on QP and SVD. Manuscript (2007)
Chapelle, O.: Training a Support Vector Machine in the Primal. Neural Computation 19, 1155–1178 (2007)
Ruiz, A., Lopez-de-Teruel, P.E.: Nonlinear Kernel-based Statistical Pattern Analysis. IEEE Transactions on Neural Networks 2(1), 16–32 (2001)
Löfberg, J.: YALMIP: A Toolbox for Modeling and Optimization in MATLAB. In: Proceedings of the CACSD Conference, Taipei, Taiwan (2004), Available at: http://control.ee.ethz.ch/~joloef/yalmip.php
Meyer, Carl D.: Matrix Analysis and Applied Linear Algebra. 1st edn. SIAM, Philadelphia, PA (2000)
Alizadeh, F., Goldfarb, D.: Second-order Cone Programming. Mathematical Programming 95, 3–51 (2003)
Lobo, M., Vandenberghe, L., Boyd, S., Lebret, H.: Applications of Second-order Cone Programming. Linear Algebra and its Applications 284, 193–228 (1998)
Blake, C., Keogh, E., Merz, C.J.: UCI Repository of Machine Learning Databases, University of California, Irvine, CA. (1998), Available at http://www.ics.uci.edu/~mlearn/ML-Repository.html
Sturm, J.F.: Using SEDUMI 1.02, A MATLAB toolbox for Optimization over Symmetric Cones. Optimization Methods and Software, 11–12, 625–653 (1999)
Liu, Y., Zheng, Y.-F.: Maximum Enclosing and Maximum Excluding Machine for Pattern Description and Discrimination. In: Proceeding of ICPR 2006 Conference, Hongkong (2006)
Li, Y.-H., Wei, X.-K., Liu, J.-X.: Engineering Applications of Support Vector Machines, 1st edn. China Weapon Industry Press, Beijing (2004)
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Wei, X., Li, Y., Feng, Y., Huang, G. (2007). Solving Mahalanobis Ellipsoidal Learning Machine Via Second Order Cone Programming. In: Huang, DS., Heutte, L., Loog, M. (eds) Advanced Intelligent Computing Theories and Applications. With Aspects of Contemporary Intelligent Computing Techniques. ICIC 2007. Communications in Computer and Information Science, vol 2. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74282-1_133
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DOI: https://doi.org/10.1007/978-3-540-74282-1_133
Publisher Name: Springer, Berlin, Heidelberg
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