Using Directional Curvatures to Visualize Folding Patterns of the GTM Projection Manifolds

  • Peter Tino
  • Ian Nabney
  • Yi Sun
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2130)


In data visualization, characterizing local geometric properties of non-linear projection manifolds provides the user with valuable additional information that can influence further steps in the data analysis. We take advantage of the smooth character of GTM projection manifold and analytically calculate its local directional curvatures. Curvature plots are useful for detecting regions where geometry is distorted, for changing the amount of regularization in non-linear projection manifolds, and for choosing regions of interest when constructing detailed lower-level visualization plots.


Latent Space Data Space Rubber Sheet Curvature Plot Directional Curvature 
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  1. 1.
    Bates, D.M., Watts, D.G.: Relative curvature measures of nonlinearity (with Discussion). J. R. Stat. Soc. B 42 (1980) 1–25zbMATHMathSciNetGoogle Scholar
  2. 2.
    Bauer, H.U., Pawelzik, K.: Quantifying the neighborhood preservation of self-organizing feature maps. IEEE Transactions on Neural Networks 3 (1992) 570–579CrossRefGoogle Scholar
  3. 3.
    Bishop, C.M.: Neural Networks for Pattern Recognition. Oxford University Press, Oxford, UK (1995)Google Scholar
  4. 4.
    Bishop, C.M. Svensén, M., Williams, C.K.I.: Magnification Factors for the SOM and GTM Algorithms. In: Proceedings 1997 Workshop on Self-Organizing Maps, Helsinki, Finland. (1997)Google Scholar
  5. 5.
    Bishop, C.M. Svensén, M., Williams, C.K.I.: Magnification Factors for the GTM Algorithm. In: Proceedings IEE Fifth International Conference on Artificial Neural Networks. IEE, London (1997) 64–69CrossRefGoogle Scholar
  6. 6.
    Bishop, C.M., Svensén, M., Williams, C.K.I.: GTM: The Generative Topographic Mapping. Neural Computation 1 (1998) 215–235CrossRefGoogle Scholar
  7. 7.
    Kohonen, T.: The self-organizing map. Proceedings of the IEEE 9 (1990) 1464–1479CrossRefGoogle Scholar
  8. 8.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge, England (1985)zbMATHGoogle Scholar
  9. 9.
    Seber, G.A.F., Wild, C.J.: Nonlinear Regression. John Wiley and Sons, New York, NY (1989)zbMATHCrossRefGoogle Scholar
  10. 10.
    Tino, P., Nabney, I.: Constructing localized non-linear projection manifolds in a principled way: hierarchical Generative Topographic Mapping. Technical Report NCRG/2000/011, Neural Computation Research Group, Aston University, UK. (2000)Google Scholar
  11. 11.
    Willmann, T., Der, R., Martinez, T.: A new quantitative measure of topology preservation in Kohonen’s feature maps. In: ICNN’94 Proceedings. IEEE Service Center (1994) 645–648Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Peter Tino
    • 1
  • Ian Nabney
    • 1
  • Yi Sun
    • 1
  1. 1.Neural Computing Research GroupAston UniversityBirminghamUK

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