K-Means-Type Algorithms: A Generalized Convergence Theorem and Characterization of Local Optimality

Authors:
Shokri Z. Selim;M. A. Ismail
Affiliations:
Department of Systems Engineering, University of Petroleum and Minerals, Dhahran, Saudi Arabia.;Department of Computer Science, University of Windsor, Windsor, Canada.
Venue:
IEEE Transactions on Pattern Analysis and Machine Intelligence
Year:
1984

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Abstract

The K-means algorithm is a commonly used technique in cluster analysis. In this paper, several questions about the algorithm are addressed. The clustering problem is first cast as a nonconvex mathematical program. Then, a rigorous proof of the finite convergence of the K-means-type algorithm is given for any metric. It is shown that under certain conditions the algorithm may fail to converge to a local minimum, and that it converges under differentiability conditions to a Kuhn-Tucker point. Finally, a method for obtaining a local-minimum solution is given.