Induced Voronoï Kernels for Principal Manifolds Approximation

  • Michaël Aupetit
  • Pierre Couturier
  • Pierre Massotte
Conference paper


We present a new interpolation technique allowing to build an approximation of a priori unknown principal manifolds of a data set which may be non-linear, non-connected and of various intrinsic dimension. This technique is based onto the Induced Delaunay Triangulation of the data built by a Topology Representing Network. It opens the way to a new field of research and applications in data analysis, data modeling and forecasting.


Intrinsic Dimension Vector Quantization Delaunay Triangulation Interpolation Technique Discrete Equivalent 
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  1. [1]
    Hastie T, Stuetzle W. Principal curves. Journal of the American Statistical Association 1989, vol. 84, no. 406, pp. 502–516MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    Mulier F, Cherkassky V. Self-organization as an iterative kernel smoothing process. Neural Computation 1995, vol. 7, pp. 1165–1177CrossRefGoogle Scholar
  3. [3]
    Ritter H. Parametrized self-organizing maps. In S. Gielen and B. Kappen Eds., Proc. of the Int. Conf. on Art. Neural Networks 1993, pp. 568–575, Springer Verlag, Berlin.Google Scholar
  4. [4]
    Walter J. Rapid learning in robotics. Cuvillier Verlag, Göttingen, Germany. Url:
  5. [5]
    Göppert J, Rosenstiel W. Interpolation in SOM: Improved generalization by iterative methods. In EC2&Cie eds., Proc. of the Int. Conf. on Artificial Neural Networks 1995, vol. 10, Paris, France.Google Scholar
  6. [6]
    Aupetit M, Couturier P, Massotte P. Function approximation with continuous self-organizing maps using neighboring influence interpolation. Proc. of NC’2000, Berlin.Google Scholar
  7. [7]
    Martinetz T, Schulten K. Topology Representing Networks. Neural Networks 1994, vol. 7, no. 3, pp. 507–522, Elsevier Science.CrossRefGoogle Scholar
  8. [8]
    Martinetz T, Berkovich S, Schulten K. “Neural-Gas” network for vector quantization and its application to time-series prediction. IEEE Trans, on Neural Networks 1993, vol. 4, no. 4, pp. 558–569CrossRefGoogle Scholar
  9. [9]
    Moody J, Darken C. Fast learning in networks of locally-tuned processing units. Neural Computation 1989, vol. 1, pp. 281–294, MIT.CrossRefGoogle Scholar
  10. [10]
    Bartels R, Beaty J, Barsky B. B-splines. Mathematiques et CAO, vol.4, Hermès 1988.Google Scholar
  11. [11]
    Sibson R. A brief description of natural neighbour interpolation. Interpreting Multivariate Data, pp. 21-36, V. Barnet eds., Wiley, Chichester 1981.Google Scholar
  12. [12]
    Aupetit M, Couturier P, Massotte P. Vector Quantization with γ-Observable Neighbors. Workshop on Self-Organizing Maps (WSOM01), Lincoln, UK, June 2001.Google Scholar
  13. [13]
    Marquardt D. J. Soc Appl. Math 1963, vol. 11, pp.431–441MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2001

Authors and Affiliations

  • Michaël Aupetit
    • 1
  • Pierre Couturier
    • 1
  • Pierre Massotte
    • 1
  1. 1.Parc Scientifique Georges BesseLGI2P - EMA - Site EERIENîmesFrance

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