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Neural Processing Letters

, Volume 8, Issue 2, pp 181–192 | Cite as

The Softmap Algorithm

  • Steven Raekelboom
  • Marc M. Van Hulle
Article
  • 28 Downloads

Abstract

A new unsupervised competitive learning rule is introduced, called the Self-organizing free-topology map (Softmap) algorithm, for nonparametric density estimation. The receptive fields of the formal neurons are overlapping, radially-symmetric kernels, the radii of which are adapted to the local input density together with the weight vectors which define the kernel centers. A fuzzy code membership function is introduced in order to encompass, in a novel way, the presence of overlapping receptive fields in the competitive learning scheme. Furthermore, a computationally simple heuristic is introduced for determining the overall degree of smoothness of the resulting density estimate. Finally, the density estimation performance is compared to that of the variable kernel method, VBAR and Kohonen's SOM algorithm.

nonparametric density estimation vector quantization unsupervised competitive learning variable kernel method 

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References

  1. 1.
    T. Kohonen, “Self-organized formation of topologically correct feature maps”, Biol. Cybern., Vol. 43, pp. 59–69, 1982.Google Scholar
  2. 2.
    T. Kohonen, Self-organizing maps, Springer: Heidelberg, 1995.Google Scholar
  3. 3.
    H. Ritter and K. Schulten, “On the stationary state of Kohonen's self-organizing sensory mapping”, Biol. Cybern., Vol. 54, pp. 99–106, 1986.Google Scholar
  4. 4.
    S. Grossberg, “Adaptive pattern classification and universal recoding: I. Parallel development sand coding of neural feature detectors”, Biol. Cybern., Vol. 23, pp. 121–134, 1976.Google Scholar
  5. 5.
    M.M. Van Hulle and D. Martinez, “On an unsupervised learning rule for scalar quantization following the maximum entropy principle”, Neural Computation, Vol. 5, pp. 939–953, 1993.Google Scholar
  6. 6.
    M.M. Van Hulle, “Nonparametric Density Estimation and-regression Achieved with a Learning Rule for Equiprobabilistic Topographic Map Formation”, in Proc. IEEE NNSP96, pp. 33–41, Seika, Kyoto, 1996.Google Scholar
  7. 7.
    M.M. Van Hulle, “Topographic Map Formation by Maximizing Unconditional Entropy: A Plausible Strategy for 'On-line' Unsupervised Competitive Learning and Non-Parametric Density Estimation”, IEEE Transactions on Neural Networks, Vol. 7(5), pp. 1299–1305, 1996.Google Scholar
  8. 8.
    M.M. Van Hulle, “Topology-preserving map formation achieved with a purely local unsupervised competitive learning rule”, Neural Networks, Vol. 10(3), pp. 431–446, 1997.Google Scholar
  9. 9.
    E. Parzen, “On estimation of a probability density function and mode”, Ann. Math. Statist., Vol. 33, pp. 1065–1076, 1962.Google Scholar
  10. 10.
    B.W. Silverman, Density Estimation for Statistics and Data Analysis, Chapman and Hall: London, 1992.Google Scholar
  11. 11.
    L.A. Zadeh, “Fuzzy sets”, Information and control, Vol. 8, pp. 338–353, 1965.Google Scholar
  12. 12.
    L. Breiman, W. Meisel and E. Purcell, “Variable kernel estimates of multivariate densities”, Technometrics, Vol. 19, pp. 135–144, 1977.Google Scholar
  13. 13.
    I.S. Abramson, “On bandwidth variation of kernel estimates - a square root law”, Ann. Statist., Vol. 10, pp. 1217–1223, 1982.Google Scholar
  14. 14.
    R. Der and M. Herrmann, “Instabilities in Self-Organized Feature Maps with Short Neighborhood Range,” in M. Verleysen (ed.) Proc. of the European Symposium on Artificial Neural Networks - ESANN'94, pp. 271–276, Brussels, Belgium, 1994.Google Scholar
  15. 15.
    F. Mulier and V. Cherkassky, “Self-organization as an iterative kernel smoothing process”, Neural Computation, Vol. 7, pp. 1165–1177, 1995.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Steven Raekelboom
    • 1
  • Marc M. Van Hulle
    • 1
  1. 1.Laboratorium voor Neuro- en PsychofysiologieK.U. Leuven, Campus GasthuisbergLeuvenBelgium

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