Simulated Annealing C-Means Clustering Algorithm Convergence Proof

  • Piotr Boguś
  • Anna Maria Massone
  • Francesco Masulli
Conference paper
Part of the Advances in Soft Computing book series (AINSC, volume 19)


The paper presents a simple proof of convergence for the simulated annealing c-means (SACM) algorithm. This proof supports the excellent experimental performances shown by this algorithm that are also due to an accurate modeling of clusters making use of the Mahalanobis distance.


Simulated Annealing Mahalanobis Distance Simulated Annealing Algorithm Cluster Problem Deterministic Annealing 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    J. C. Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithms, second edition, Plenum Press, 1987Google Scholar
  2. 2.
    P. Bogus, A.M. Massone, F. Masulli, A Simulated Annealing C-Means Clustering Algorithm, R. Parenti, F. Masulli (eds.), Proceedings of the third International ICSC Symposia on Intelligent Industrial Automation IAA’99 and Soft Computing SOCO’99, June 1–4 1999, Genova, Italy, ICSC Academic Press, pp. 534–540, 1999.Google Scholar
  3. 3.
    P. Bogus, A.M. Massone, F. Masulli, A. Schenone, Interactive graphical system for segmentation of multimodal medical volumes using fuzzy clustering, Machine GRAPHICS & VISION, vol. 7, no. 4, pp. 781–791, 1998.Google Scholar
  4. 4.
    P. Bogus, Simulated Annealing in Clustering. Colloquia in Artificial Intelligence, Proceedings of CAI 2000 Third Polish Conference on Theory and Applications of Artificial Intelligence, Lódz, October 5–7, pp. 67–84, 2000.Google Scholar
  5. 5.
    R. Duda, P. Hart, Pattern Classification and Scene Analysis, New York, Wiley Interscience, 1973.MATHGoogle Scholar
  6. 6.
    R. A. Fisher, The use of multiple measurements in taxonomic problems, Annual Eugenics, vol. 7, no. 2, pp. 179–188, 1936.CrossRefGoogle Scholar
  7. 7.
    S. Geman, D. Geman, Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. PAMI-6, no. 6, pp. 721–741, 1984.MATHCrossRefGoogle Scholar
  8. 8.
    R. Krishnapuram, J. M. Keller, A Possibilistic Approach to Clustering, IEEE Transactions on Fuzzy Systems, vol. 1, no. 2, pp. 98–110, 1993.CrossRefGoogle Scholar
  9. 9.
    P. J. M. van Laarhoven, E. H. L. Aarts, Simulated Annealing: Theory and Applications, D. Reidel Publishing Co., Dordrecht, Holland, 1987.CrossRefGoogle Scholar
  10. 10.
    N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, and E. Teller, Equation of state calculations for fast computing machines, Journal of Chemical Physics, vol. 21, pp. 1087–1092, 1953.CrossRefGoogle Scholar
  11. 11.
    S. Kirkpatric, C.D. Gelatt, Jr., M.P. Vecchi, Optimization by simulated annealing, Science, vol. 220, pp. 671–680, 1983.MathSciNetCrossRefGoogle Scholar
  12. 12.
    F. Masulli, P. Bogus, M. Artuso, A. Schenone, The Maximum Entropy Principle and Its Application to the Segmentation of Multivariate Medical Images, In R. Tadeusiewicz, L. Rutkowski, J. Chojcan (eds.), Proceedings of the Third Conference „Neural Networks And Their Application“, Kule (Poland), 14 X -18 X 97, Polish Neural Networks Society, Cz@stochowa, pp. 285–291, 1997.Google Scholar
  13. 13.
    F. Masulli, M. Artuso, P. Bogus, A. Schenone, Application of Possibilistic Clustering to the Segmentation of Multivariate Medical Images, In: P. Blonda, M. Castellano, A. Petrosino (eds.), Proceedings of the Second Italian Workshop on Fuzzy Logic, Bari, Italy, 19–20 March 1997, World Scientific, pp. 105–112, 1998.Google Scholar
  14. 14.
    Rose, E. Gurewitz, G. C. Fox, Statistical Mechanics and Phase Transitions in Clustering, Physical Review Letters, vol. 65, no. 8, pp. 945–948, 1990.CrossRefGoogle Scholar
  15. 15.
    K. Rose, E. Gurewitz, G. C. Fox, A deterministic annealing approach to clustering, Pattern Recognition Letters, vol. 11, pp. 589–594, 1990.MATHCrossRefGoogle Scholar
  16. 16.
    S. S. Selim, K. Alsultan, A simulated annealing algorithm for the clustering problem, Pattern Recognition, vol. 24, no. 19, pp. 1003–1008, 1991.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Piotr Boguś
    • 1
  • Anna Maria Massone
    • 2
    • 3
  • Francesco Masulli
    • 2
    • 4
  1. 1.Department of Physics and BiophysicsMedical University of GdańskGdańskPoland
  2. 2.INFM National Institute for Physics of MatterGenovaItaly
  3. 3.Institut of High Energy PhysicsUniversité de Lausanne Bâtiment des SciencesSwitzerland
  4. 4.Department of Computer SciencesUniversity of PisaPisaItaly

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