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A new efficient algorithm based on DC programming and DCA for clustering. (English) Zbl 1198.90327
Summary: In this paper, a version of K-median problem, one of the most popular and best studied clustering measures, is discussed. The model using squared Euclidean distances terms to which the K-means algorithm has been successfully applied is considered. A fast and robust algorithm based on DC (Difference of Convex functions) programming and DC Algorithms (DCA) is investigated. Preliminary numerical solutions on real-world databases show the efficiency and the superiority of the appropriate DCA with respect to the standard K-means algorithm.

90C26 Nonconvex programming, global optimization
65K10 Numerical optimization and variational techniques
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[1] Alon N., Spencer J.H. (1991): The Probabilistic Method. Wiley, New York, NY
[2] Arora, S., Kannan, R.: Learning mixtures of arbitrary Gaussians. In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, July 6–8, pp. 247–257. Heraklion, Crete, Greece (2001) · Zbl 1323.68440
[3] Al-Sultan K. (1995): A Tabu search approach to the clustering problem. Pattern Recogn. 28(9): 443–1453 · Zbl 05478426
[4] Bradley, P.S., Mangasarian, O.L., Street, W.N.: Clustering via concave minimization, Technical Report 96-03, Computer Sciences Department, University of Wisconsin, Madison, Wisconsin May 1996. Advances In: Mozer, M.C., Jordan, M.I., Petsche, T. (eds.) Neural Information processing Systems 9, pp. 368–374. MIT Press, Cambridge, MA Available by ftp://ftp.cs.wisc.edu/math-prog/tech-trports/96-03.ps.Z.
[5] Bradley, B.S., Mangasarian, O.L.: Feature selection via concave minimization and support vector machines. In: Shavlik, J. (eds.) Machine Learning Proceedings of the Fifteenth International Conferences(ICML’98), pp. 82–90. San Francisco, CA 1998, Morgan Kaufmann.
[6] Charikar, M., Guha, S.: Improved combinatorial algorithms for facility location and k-median problems. In: Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science, 17–18 October, pp. 378–388. New York, NY, USA (1999)
[7] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k-median problem. In: Proceedings of the 31st Annual ACM Symposium on Theory of Computing pp. 1–10 (1999) · Zbl 1023.90037
[8] De Leeuw J. (1997): Applications of convex analysis to multidimensional scaling, Recent developments. In: Barra J.R., et al. (eds). Statistics. North-Holland Publishing company, Amsterdam, pp. 133–145
[9] De Leeuw J. (1988): Convergence of the majorization method for multidimensional scaling. J. Classi. 5, 163–180 · Zbl 0692.62056
[10] Dhilon I.S., Korgan J., Nicholas C. (2003): Feature Selection and Document Clustering. In: Berry M.W. (eds). A Comprehensive Survey of Text Mining. Springer-Verlag, Berlin, pp. 73–100
[11] Duda R.O., Hart P.E. (1972): Pattern Classification and Scene Analysis. Wiley, New York · Zbl 0277.68056
[12] Feder, T., Greene, D.: Optimal algorithms for approximate clustering. In: Proceedings of the 20th Annual ACM Symposium on Theory of Computing (STOC) May 2–4, Chicago, Illinois, USA (1988)
[13] Fisher D. (1988): Knowledge acquisition via incremental conceptual clusterin. Mach. Learning, 2, 139–172
[14] Fukunaga K. (1990): Statistical Pattern Recognition. Academic Press, NY · Zbl 0711.62052
[15] Hiriart Urruty J.B., Lemarechal C. (1993): Convex Analysis and Minimization Algorithms. Springer Verlag, Berlin, Heidelberg
[16] Jain A.K., Dubes R.C. (1988): Algorithms for Clustering Data. Prentice-Hall Inc, Englewood Cliffs, NJ · Zbl 0665.62061
[17] Hartigan J.A. (1975): Clustering Algorithms. Wiley, New York · Zbl 0372.62040
[18] Le Thi Hoai An: Contribution à l’optimisation non convexe et l’optimisation globale: Théorie, Algoritmes et Applications. Habilitation à Diriger des Recherches, Université de Rouen, Juin (1997)
[19] Le Thi Hoai An, Pham Dinh Tao (1997): Solving a class of linearly constrained indefinite quadratic problems by DC algorithms. J. Global Optim. 11(3): 253–285 · Zbl 0905.90131
[20] Le Thi Hoai An, Pham Dinh Tao: DC programming approach for large-scale molecular optimization via the general distance geometry problem. Nonconvex Optimization and Its Applications, Special Issue ”Optimization in Computational Chemistry and Molecular Biology: Local and Global Approaches”, pp. 301–339. Kluwer Academic Publishers, Dordrecht (2000) (This special issue contains refereed invited papers submitted at the conference on optimization in computational chemistry and molecular biology: local and global approaches held at Princeton University, 7–9 May 1999)
[21] Le Thi Hoai An, Pham Dinh Tao (2001): DC programming approach for solving the multidimensional scaling problem. Nonconvex optimizations and its applications: special issue ”From local to global optimization”, pp. 231–276. Kluwer Academic Publishers, Dordrecht
[22] Le Thi Hoai An, Pham Dinh Tao: DC programming: theory, algorithms and applications. The state of the art. In Proceedings of The First International Workshop on Global Constrained Optimization and Constraint Satisfaction (Cocos’ 02), 28 p. Valbonne-Sophia Antipolis, France, 2–4 October (2002)
[23] Le Thi Hoai An, Pham Dinh Tao (2003): Large Scale Molecular Optimization from distances matrices by a DC optimization approach. SIAM J. Optim. 14(1): 77–116 · Zbl 1075.90071
[24] Le Thi Hoai An, Pham Dinh Tao (2005): The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems. Ann. Oper. Res. 133, 23–46 · Zbl 1116.90122
[25] Liu, Y., Shen, X., Doss, H.: Multicategory {\(\psi\)}–learning and support vector machine (29 p.). Conference on Machine Learning, Statistics and Discovery, June 22–26, Department of Statistics, the Ohio State University (2003)
[26] Mangasarian O.L. (1997): Mathematical programming in data mining. Data Mining and Knowl. discov. 1, 183–201
[27] MacQueen J.B. (1967): Some Methods for classification and analysis of multivariate observations. In: Proceedings of 5th Berkeley Symposium on Mathematical Statistics and Probability”, Berkeley, University of California Press,1, 281–297 · Zbl 0214.46201
[28] Meyerson A., O’Callaghan L., Plotkin S. (2004): A k-Median algorithm with running time independent of data size. Machine Learn. 56, 61–87 · Zbl 1093.68635
[29] Neumann, J., Schnörr, C., Steidl, G.: SVM-based feature selection by direct objective minimisation, Pattern Recognition. In: Proceeding of 26th DAGM Symposium, vol. 3175, pp. 212–219, LNCS (2004)
[30] Pham Dinh Tao. (1984): Convergence of subgradient method for computing the bound-norm of matrices, Linear Algebra Appl. 62, 163–182 · Zbl 0563.65029
[31] Pham Dinh Tao. (1984): Algorithmes de calcul d’une forme quadratique sur la boule unité de la norme du maximum. Numer. Math. 45, 377–440 · Zbl 0531.65022
[32] Pham Dinh Tao, Le Thi Hoai An.(1997): Convex analysis approach to d.c. programming: Theory, Algorithms and Applications. Acta Mathematica Vietnamica, (dedicated to Professor Hoang Tuy on the occasion of his 70th birthday), 22(1): 289–355 · Zbl 0895.90152
[33] Pham Dinh Tao, Le Thi Hoai An. (1998): DC optimization algorithms for solving the trust region subproblem. SIAM J. Optim. 8, 476–505 · Zbl 0913.65054
[34] Rao M.R. (1971): Cluster analysis and mathematical programming. J. Amer. Stat. Associ., 66, 622–626 · Zbl 0238.90042
[35] Rockafellar R.T. (1970): Convex Analysis. Princeton University, Princeton
[36] Selim S.Z., Ismail M.A. (1984): K-means-Type algorithms: a generalized convergence theorem and characterization of local optimilaty. (Book Series) IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-6, 81–87 · Zbl 0546.62037
[37] Schüle T., Schnörr C., Weber S., Hornegger J. (2005): Discrete tomography by convex-concave regularization and d.c. programming. Discr. Appl. Math. 151, 229–243 · Zbl 1131.68571
[38] Weber S., Schüle T., Schnörr C. (2005): Prior learning and convex-concave regularization of binary tomography. Electr. Notes in Discr. Math. 20, 313–327 · Zbl 1179.68192
[39] Weber S., Schnörr C., Schüle T., Hornegger J., (2005): Binary Tomography by Iterating Linear Programs, Klette, R., Kozera, R., Noakes, L., and Weickert, J., (eds.) Computational Imaging and Vision – Geometric Properties from Incomplete Data, Kluwer Academic Press, Dordrecht
[40] Wong, T., Katz, R., McCanne, S.: A preference clustering protocol for large-Scale Multicast Applications, Proceedings of the First International COST264 Workshop on Networked Group Communication, November 17–20, LNCS, pp. 1–18. Pisa, Italy (1999)
[41] Wolberg W.H., Street W.N., Mangasarian O.L. (1995): Image analysis and machine learning applied to breast cancer diagnosis and prognosis. Anal. Quant. Cytol. Histol. 17(2): 77–87 · Zbl 0857.90073
[42] Wolberg W.H., Street W.N., Heisey D.M., Mangasarian O.L. (1995):Computerized breast cancer diagnosis and prognosis from fine-needle aspirates. Arch. Surg. 130, 511–516
[43] Wolberg W.H., Street W.N., Heisey D.M., Mangasarian O.L. (1995): Computer-derived nuclear features distinguish malignant from benign breast cytology. Hum. Patholo., 26, 792–796
[44] Yuille A.L., Rangarajan A. (2002): The Concave-Convex Procedure (CCCP). Avances in Neural Information Processing Systems 14, pp. 1033–1040. MIT Press, Cambridge, MA
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