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How the initialization affects the stability of the қ-meansalgorithm∗

Published online by Cambridge University Press:  04 September 2012

Sébastien Bubeck ,
Marina Meilă  and
Ulrike von Luxburg
Sébastien Bubeck
Affiliation:
Centre de Recerca Matemàtica, Barcelona, Spain. sbubeck@crm.cat
Marina Meilă
Affiliation:
University of Washington, Department of Statistics, Seattle, U.S.A.; mmp@stat.washington.edu
Ulrike von Luxburg
Affiliation:
Max Planck Institute for Biological Cybernetics, Tübingen, Germany; ulrike.luxburg@tuebingen.mpg.de
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Abstract

We investigate the role of the initialization for the stability of the қ-means clusteringalgorithm. As opposed to other papers, we consider the actual қ-means algorithm (also knownas Lloyd algorithm). In particular we leverage on the property that this algorithm can getstuck in local optima of the қ-means objective function. We are interested in the actualclustering, not only in the costs of the solution. We analyze when differentinitializations lead to the same local optimum, and when they lead to different localoptima. This enables us to prove that it is reasonable to select the number of clustersbased on stability scores.

Keywords

Clusteringқ-meansstabilitymodel selection
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 16 , 2012 , pp. 436 - 452
Copyright
© EDP Sciences, SMAI, 2012

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References

Références

D. Arthur and S. Vassilvitskii, қ-means++ : the advantages of careful seeding, in Proc. of SODA (2007).
S. Ben-David and U. von Luxburg, Relating clustering stability to properties of cluster boundaries, in Proc. of COLT (2008).
S. Ben-David, U. von Luxburg and D. Pál, A sober look on clustering stability, in Proc. of COLT (2006).
S. Ben-David, D. Pál and H.-U. Simon, Stability of қ-means clustering, in Proc. of COLT (2007).
L. Bottou and Y. Bengio, Convergence properties of the қ-means algorithm, in Proc. of NIPS (1995).
Dasgupta, S. and Schulman, L., A probabilistic analysis of EM for mixtures of separated, spherical Gaussians. J. Mach. Learn. Res. 8 (2007) 203–226. Google Scholar
S. Graf and H. Luschgy, Foundations of Quantization for Probability Distributions. Springer (2000).
Hochbaum, D. and Shmoys, D., A best possible heuristic for the -center problem. Math. Operat. Res. 10 (1985) 180–184. Google Scholar
Lange, T., Roth, V., Braun, M. and Buhmann, J., Stability-based validation of clustering solutions. Neural Comput. 16 (2004) 1299–1323. Google ScholarPubMed
R. Ostrovsky, Y. Rabani, L.J. Schulman and C. Swamy, The effectiveness of Lloyd-type methods for the қ-means problem, in Proc. of FOCS (2006).
O. Shamir and N. Tishby, Cluster stability for finite samples, in Proc. of NIPS (2008).
O. Shamir and N. Tishby, Model selection and stability in қ-means clustering, in Proc. of COLT (2008).
O. Shamir and N. Tishby, On the reliability of clustering stability in the large sample regime, in Proc. of NIPS (2008).
N. Srebro, G. Shakhnarovich and S. Roweis, An investigation of computational and informational limits in Gaussian mixture clustering, in Proc. of ICML (2006).
Z. Zhang, B. Dai and A. Tung, Estimating local optimums in EM algorithm over Gaussian mixture model, in Proc. of ICML (2008).