Convergence theory for fuzzy c-means: Counterexamples and repairs

1987/09/01 by James C. Bezdek, Richard J. Hathaway, Michael J. Sabin +1 · 9 citations
Computer Science · #Advanced Algebra and Logic #Rough Sets and Fuzzy Logic #Advanced Clustering Algorithms Research

paper · doi:10.1109/tsmc.1987.6499296

Abstract

A counterexample to the original incorrect convergence theorem for the fuzzy <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</i> -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 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</i> -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.

Cited by

• Lagrangian based methods for coherent structure detection
• Neighborhood-Adaptive Accelerated Fuzzy C-Means for Three-Dimensional Millimeter-Wave Image Segmentation
• On the convergence of the sparse possibilistic c-means algorithm
• Unsupervised Multi Class Segmentation of 3D Images with Intensity\n Inhomogeneities
• Review: Metaheuristic Search-Based Fuzzy Clustering Algorithms
• From Large Deviations to Semidistances of Transport and Mixing: Coherence Analysis for Finite Lagrangian Data. [europepmc]
• Morphologies in-between: The impact of the first steps on the human talus. [europepmc]
• Performance evaluation of spatial fuzzy C-means clustering algorithm on GPU for image segmentation. [europepmc]
• Human talar ontogeny: Insights from morphological and trabecular changes during postnatal growth [europepmc]

Related

• On Coreset Constructions for the Fuzzy K-Means Problem
• Impact of Exponent Parameter Value for the Partition Matrix on the Performance of Fuzzy C Means Algorithm
• Accelerated Fuzzy C-Means Clustering Based on New Affinity Filtering and Membership Scaling
• Hybrid Fuzzy-Crisp Clustering Algorithm: Theory and Experiments
• Complexity and Approximation of the Fuzzy K-Means Problem