Abstract
The problem of clustering a set of data is a textbook machine learning problem, but at the same time, at heart, a typical optimization problem. Given an objective function, such as minimizing the intra-cluster distances or maximizing the inter-cluster distances, the task is to find an assignment of data points to clusters that achieves this objective. In this paper, we present a constraint programming model for a centroid based clustering and one for a density based clustering. In particular, as a key contribution, we show how the expressivity introduced by the formulation of the problem by constraint programming makes the standard problem easy to be extended with other constraints that permit to generate interesting variants of the problem. We show this important aspect in two different ways: first, we show how the formulation of the density-based clustering by constraint programming makes it very similar to the label propagation problem and then, we propose a variant of the standard label propagation approach.
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References
Babaki, B., Guns, T., Nijssen, S.: Constrained clustering using column generation. In: Simonis, H. (ed.) CPAIOR 2014. LNCS, vol. 8451, pp. 438–454. Springer, Heidelberg (2014)
Bar-Hillel, A., Hertz, T., Shental, N., Weinshall, D.: Learning distance functions using equivalence relations. In: ICML, pp. 11–18 (2003)
Basu, S., Banerjee, A., Mooney, R.J.: Active semi-supervision for pairwise constrained clustering. In: SDM (2004)
Basu, S., Bilenko, M., Mooney, R.J.: A probabilistic framework for semi-supervised clustering. In: KDD, pp. 59–68 (2004)
Berthold, M.R., Borgelt, C., Hppner, F., Klawonn, F.: Guide to Intelligent Data Analysis: How to Intelligently Make Sense of Real Data, 1st edn. Springer, London (2010)
Bezdek, J.C.: Pattern Recognition with Fuzzy Objective Function Algorithms. Kluwer Academic Publishers, Norwell (1981)
Bilenko, M., Basu, S., Mooney, R.J.: Integrating constraints and metric learning in semi-supervised clustering. In: ICML, ACM (2004)
Coscia, M., Giannotti, F., Pedreschi, D.: A classification for community discovery methods in complex networks. Stat. Anal. Data Min. 4(5), 512–546 (2011)
Dao, T.-B.-H., Duong, K.-C., Vrain, C.: A declarative framework for constrained clustering. In: Blockeel, H., Kersting, K., Nijssen, S., Železný, F. (eds.) ECML PKDD 2013, Part III. LNCS, vol. 8190, pp. 419–434. Springer, Heidelberg (2013)
Davidson, I., Ravi, S.S.: Clustering with constraints: feasibility issues and the k-means algorithm. In: SDM (2005)
Davidson, I., Ravi, S.S.: Identifying and generating easy sets of constraints for clustering. In: Proceedings, The Twenty-First National Conference on Artificial Intelligence and the Eighteenth Innovative Applications of Artificial Intelligence Conference (AAAI), pp. 336–341 (2006)
Davidson, I., Ravi, S.S.: The complexity of non-hierarchical clustering with instance and cluster level constraints. DMKD 14(1), 25–61 (2007)
Davidson, I., Ravi, S.S.: Using instance-level constraints in agglomerative hierarchical clustering: theoretical and empirical results. Data Min. Knowl. Discov. 18(2), 257–282 (2009)
Dunn, J.C.: A fuzzy relative of the isodata process and its use in detecting compact well-separated clusters. J. Cybern. 3(3), 32–57 (1974)
Ester, M., Kriegel, H.-P., Sander, J., Xu, X.: A density-based algorithm for discovering clusters in large spatial databases with noise. In: Simoudis, E., Han, J., Fayyad, U.M. (eds.) KDD, pp. 226–231. AAAI Press (1996)
Guns, T., Nijssen, S., Raedt, L.D.: k-pattern set mining under constraints. IEEE Trans. Knowl. Data Eng. 25(2), 402–418 (2013)
Hansen, P., Aloise, D.: A survey on exact methods for minimum sum-of-squares clustering. http://www.math.iit.edu/Buck65files/msscStLouis.pdf, pp. 1–2, January 2009
Hartigan, J.A., Wong, M.A.: Algorithm AS 136: a k-means clustering algorithm. J. Roy. Stat. Soc. Ser. C (Appl. Stat.) 28(1), 100–108 (1979)
Merle, O.D., Hansen, P., Jaumard, B., Mladenović, N.: An interior point algorithm for minimum sum of squares clustering. SIAM J. Sci. Comput. 21, 1485–1505 (1997)
Mueller, M., Kramer, S.: Integer linear programming models for constrained clustering. In: Pfahringer, B., Holmes, G., Hoffmann, A. (eds.) DS 2010. LNCS, vol. 6332, pp. 159–173. Springer, Heidelberg (2010)
Negrevergne, B., Guns, T.: Constraint-based sequence mining using constraint programming. In: Michel, L. (ed.) CPAIOR 2015. LNCS, vol. 9075, pp. 288–305. Springer, Heidelberg (2015)
Okabe, M., Yamada, S.: Clustering by learning constraints priorities. In: ICDM, pp. 1050–1055 (2012)
Raghavan, U.N., Albert, R., Kumara, S.: Near linear time algorithm to detect community structures in large-scale networks. Phys. Rev. E 76(2), 036106+ (2007)
Ruiz, C., Spiliopoulou, M., Menasalvas, E.: C-DBSCAN: density-based clustering with constraints. In: An, A., Stefanowski, J., Ramanna, S., Butz, C.J., Pedrycz, W., Wang, G. (eds.) RSFDGrC 2007. LNCS (LNAI), vol. 4482, pp. 216–223. Springer, Heidelberg (2007)
Wagstaff, K., Cardie, C.: Clustering with instance-level constraints. In: ICML, pp. 1103–1110 (2000)
Wagstaff, K., Cardie, C.: Clustering with instance-level constraints. In: AAAI/IAAI, p. 1097 (2000)
Wagstaff, K., Cardie, C., Rogers, S., Schrödl, S.: Constrained k-means clustering with background knowledge. In: ICML, pp. 577–584 (2001)
Xing, E.P., Ng, A.Y., Jordan, M.I., Russell, S.: Distance metric learning, with application to clustering with side-information. In: Advances in Neural Information Processing Systems, vol. 15, pp. 505–512. MIT Press (2002)
Acknowledgements
This work was supported by the European Commission under the project Inductive Constraint Programming (ICON) contract number FP7-284715.
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Grossi, V., Monreale, A., Nanni, M., Pedreschi, D., Turini, F. (2015). Clustering Formulation Using Constraint Optimization. In: Bianculli, D., Calinescu, R., Rumpe, B. (eds) Software Engineering and Formal Methods. SEFM 2015. Lecture Notes in Computer Science(), vol 9509. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49224-6_9
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DOI: https://doi.org/10.1007/978-3-662-49224-6_9
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