On the Convergence of the Sparse Possibilistic C-Means Algorithm

In this paper, a convergence proof for the recently proposed cost function optimization sparse possibilistic c-means (SPCM) algorithm is provided. Specifically, it is shown that the algorithm will converge to one of the local minima of its associated cost function. It is also shown that similar convergence results can be derived for the well-known possibilistic c-means (PCM) algorithm proposed by Krishnapuram and Keller, 1996, if we view it as a special case of SPCM. Note that the convergence results for PCM are stronger than those established in previous works.

[1]  Frank Klawonn,et al.  A contribution to convergence theory of fuzzy c-means and derivatives , 2003, IEEE Trans. Fuzzy Syst..

[2]  James M. Keller,et al.  A possibilistic fuzzy c-means clustering algorithm , 2005, IEEE Transactions on Fuzzy Systems.

[3]  Athanasios A. Rontogiannis,et al.  Sparsity-Aware Possibilistic Clustering Algorithms , 2015, IEEE Transactions on Fuzzy Systems.

[4]  George F. Corliss.  Which Root Does the Bisection Algorithm Find , 1977 .

[5]  James C. Bezdek,et al.  A Convergence Theorem for the Fuzzy ISODATA Clustering Algorithms , 1980, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[6]  Chuleerat Jaruskulchai,et al.  Possibilistic Exponential Fuzzy Clustering , 2013, Journal of Computer Science and Technology.

[7]  James M. Keller,et al.  A possibilistic approach to clustering , 1993, IEEE Trans. Fuzzy Syst..

[8]  J. Jäkel,et al.  A New Convergence Proof of Fuzzy c-Means , 2005, IEEE Transactions on Fuzzy Systems.

[9]  E. M. L. Beale,et al.  Nonlinear Programming: A Unified Approach. , 1970 .

[10]  James C. Bezdek,et al.  Pattern Recognition with Fuzzy Objective Function Algorithms , 1981, Advanced Applications in Pattern Recognition.

[11]  Junjie Wu,et al.  Advances in K-means clustering: a data mining thinking , 2012 .

[12]  Kuo-Lung Wu,et al.  Unsupervised possibilistic clustering , 2006, Pattern Recognit..

[13]  Nan Yang,et al.  On the convergence of some possibilistic clustering algorithms , 2013, Fuzzy Optim. Decis. Mak..

[14]  W. T. Tucker,et al.  Convergence theory for fuzzy c-means: Counterexamples and repairs , 1987, IEEE Transactions on Systems, Man, and Cybernetics.

[15]  B. Ripley,et al.  Pattern Recognition , 1968, Nature.

[16]  John N. Tsitsiklis,et al.  Parallel and distributed computation , 1989 .

[17]  Robert R. Meyer,et al.  Sufficient Conditions for the Convergence of Monotonic Mathematical Programming Algorithms , 1976, J. Comput. Syst. Sci..

[18]  J. A. Hartigan,et al.  A k-means clustering algorithm , 1979 .

[19]  James M. Keller,et al.  The possibilistic C-means algorithm: insights and recommendations , 1996, IEEE Trans. Fuzzy Syst..