Convergence theory for fuzzy c-means: Counterexamples and repairs
- 1 September 1987
- journal article
- Published by Institute of Electrical and Electronics Engineers (IEEE) in IEEE Transactions on Systems, Man, and Cybernetics
- Vol. 17 (5) , 873-877
- https://doi.org/10.1109/tsmc.1987.6499296
Abstract
A counterexample to the original incorrect convergence theorem for the fuzzy c-means (FCM) clustering algorithms (see J.C. Bezdak, IEEE Trans. Pattern Anal. and Math. Intell., vol.PAMI-2, no.1, pp.1-8, 1980) is provided. This counterexample establishes the existence of saddle points of the FCM objective function at locations other than the geometric centroid of fuzzy c-partition space. Counterexamples previously discussed by W.T. Tucker (1987) are summarized. The correct theorem is stated without proof: every FCM iterate sequence converges, at least along a subsequence, to either a local minimum or saddle point of the FCM objective function. Although Tucker's counterexamples and the corrected theory appear elsewhere, they are restated as a caution not to further propagate the original incorrect convergence statement.Keywords
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