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K-Means-Type Algorithms: A Generalized Convergence Theorem and Characterization of Local Optimality
January 1984 (vol. 6 no. 1)
pp. 81-87
Shokri Z. Selim, Department of Systems Engineering, University of Petroleum and Minerals, Dhahran, Saudi Arabia.
M. A. Ismail, Department of Computer Science, University of Windsor, Windsor, Canada.
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.
Citation:
Shokri Z. Selim, M. A. Ismail, "K-Means-Type Algorithms: A Generalized Convergence Theorem and Characterization of Local Optimality," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 6, no. 1, pp. 81-87, Jan. 1984, doi:10.1109/TPAMI.1984.4767478
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