Nonlinear Projection Using Geodesic Distances and the Neural Gas Network

  • Pablo A. Estévez
  • Andrés M. Chong
  • Claudio M. Held
  • Claudio A. Perez
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4131)


A nonlinear projection method that uses geodesic distances and the neural gas network is proposed. First, the neural gas algorithm is used to obtain codebook vectors, and a connectivity graph is concurrently created by using competitive Hebbian rule. A procedure is added to tear or break non-contractible cycles in the connectivity graph, in order to project efficiently ‘circular’ manifolds such as cylinder or torus. In the second step, the nonlinear projection is created by applying an adaptation rule for codebook positions in the projection space. The mapping quality obtained with the proposed method outperforms CDA and Isotop, in terms of the trustworthiness, continuity, and topology preservation measures.


Input Space Geodesic Distance Output Space Adaptation Rule Short Path Tree 


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  1. 1.
    Cox, T.F., Cox, M.A.A.: Multidimensional Scaling, 2nd edn. Chapman & Hall/CRC (2001)Google Scholar
  2. 2.
    Demartines, P., Hérault, J.: Curvilinear component analysis: A self-organizing neural network for nonlinear mapping of data sets. IEEE Trans. on Neural Networks 8, 148–154 (1997)CrossRefGoogle Scholar
  3. 3.
    Dijkstra, F.W.: A note on two problems in connection with graphs. Num. Math. 1, 269–271 (1959)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Estévez, P.A., Chong, A.M.: Geodesic Nonlinear Projection using the Neural Gas Network. In: Int. Joint Conference on Neural Networks, Vancouver, Canada (2006) (to appear)Google Scholar
  5. 5.
    Estévez, P.A., Figueroa, C.J.: Online data visualization using the neural gas network. Neural Networks (2006) (in press)Google Scholar
  6. 6.
    Estévez, P.A., Figueroa, C.J.: Online nonlinear mapping using the neural gas network. In: Proceedings of the Workshop on Self-Organizing Maps (WSOM 2005), pp. 299–306 (2005)Google Scholar
  7. 7.
    Kaski, S., Nikkilä, J., Oja, M., Venna, J., Törönen, P., Castrén, E.: Trustworthiness and metrics in visualizing similarity of gene expression. BMC Bioinformatics 4, 48 (2003)CrossRefGoogle Scholar
  8. 8.
    Kohonen, T.: Self–Organizing Maps. Springer, Berlin (1995)Google Scholar
  9. 9.
    König, A.: Interactive visualization and analysis of hierarchical neural projections for data mining. IEEE Trans. on Neural Networks 11, 615–624 (2000)CrossRefGoogle Scholar
  10. 10.
    Lee, J.A., Verleysen, M.: Nonlinear dimensionality reduction of data manifolds with essential loops. Neurocomputing 67, 29–53 (2005)CrossRefGoogle Scholar
  11. 11.
    Lee, J.A., Lendasse, A., Verleysen, M.: Nonlinear projection with curvilinear distances: Isomap versus curvilinear distance analysis. Neurocomputing 57, 49–76 (2004)CrossRefGoogle Scholar
  12. 12.
    Lee, J.A., Verleysen, M.: Nonlinear projection with the Isotop method. In: Proceedings of the International Conference on Artificial Neural Networks, pp. 933–938 (2002)Google Scholar
  13. 13.
    Lee, J.A., Lendasse, A., Donckers, N., Verleysen, M.: A robust nonlinear projection method. In: Proceedings of European Symposium on Artificial Neural Networks (ESSAN 2000), pp. 13–20 (2000)Google Scholar
  14. 14.
    Martinetz, T.M., Schulten, K.J.: Topology representing networks. Neural Networks 7, 507–522 (1994)CrossRefGoogle Scholar
  15. 15.
    Martinetz, T.M., Berkovich, S.G., Schulten, K.J.: Neural gas network for vector quantization and its application to time-series prediction. IEEE Trans. on Neural Networks 4, 558–569 (1993)CrossRefGoogle Scholar
  16. 16.
    Martinetz, T.M., Schulten, K.J.: A neural gas network learns topologies. Artificial Neural Networks, pp. 397–402. Elsevier, Amsterdam (1991)Google Scholar
  17. 17.
    Prim, R.C.: Shortest connection networks and some generalizations. Bell System Tech. J. 36, 1389–1401 (1957)Google Scholar
  18. 18.
    Venna, J., Kaski, S.: Local multidimensional scaling with controlled tradeoff between trustworthiness and continuity. In: Proceedings of the Workshop on Self-Organizing Maps (WSOM 2005), pp. 695–702 (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Pablo A. Estévez
    • 1
  • Andrés M. Chong
    • 1
  • Claudio M. Held
    • 1
  • Claudio A. Perez
    • 1
  1. 1.Dept. Electrical EngineeringUniversity of ChileSantiagoChile

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