PerTurbo: A New Classification Algorithm Based on the Spectrum Perturbations of the Laplace-Beltrami Operator

  • Nicolas Courty
  • Thomas Burger
  • Johann Laurent
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6911)


PerTurbo, an original, non-parametric and efficient classification method is presented here. In our framework, the manifold of each class is characterized by its Laplace-Beltrami operator, which is evaluated with classical methods involving the graph Laplacian. The classification criterion is established thanks to a measure of the magnitude of the spectrum perturbation of this operator. The first experiments show good performances against classical algorithms of the state-of-the-art. Moreover, from this measure is derived an efficient policy to design sampling queries in a context of active learning. Performances collected over toy examples and real world datasets assess the qualities of this strategy.


Support Vector Machine Active Learning Dimensionality Reduction Heat Kernel Gaussian Mixture Model 
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.
    Duda, R.O., Hart, P.E., Stork, D.G.: Pattern Classification. Wiley, Chichester (2001)zbMATHGoogle Scholar
  2. 2.
    Chavel, I.: Eigenvalues in Riemannian geometry. Academic Press, Orlando (1984)zbMATHGoogle Scholar
  3. 3.
    Lafon, S., Lee, A.B.: Diffusion maps and coarse-graining: A unified framework for dimensionality reduction, graph partitioning, and data set parameterization. IEEE Transactions on Pattern Analysis and Machine Intelligence 28, 1393–1403 (2006)CrossRefGoogle Scholar
  4. 4.
    Nadler, B., Lafon, S., Coifman, R., Kevrekidis, I.: Diffusion maps, spectral clustering and eigenfunctions of fokker-planck operators. In: NIPS (2005)Google Scholar
  5. 5.
    Reuter, M., Wolter, F.-E., Peinecke, N.: Laplace-beltrami spectra as ”shape-dna” of surfaces and solids. Computer-Aided Design 38(4), 342–366 (2006)CrossRefGoogle Scholar
  6. 6.
    Rustamov, R.: Laplace-beltrami eigenfunctions for deformation invariant shape representation. In: Proc. of the Fifth Eurographics Symp. on Geometry Processing, pp. 225–233 (2007)Google Scholar
  7. 7.
    Knossow, D., Sharma, A., Mateus, D., Horaud, R.: Inexact matching of large and sparse graphs using laplacian eigenvectors. In: Torsello, A., Escolano, F., Brun, L. (eds.) GbRPR 2009. LNCS, vol. 5534, pp. 144–153. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  8. 8.
    Öztireli, C., Alexa, M., Gross, M.: Spectral sampling of manifolds. ACM, New York (2010)CrossRefGoogle Scholar
  9. 9.
    Coifman, R.R., Lafon, S.: Diffusion maps. Applied and Computational Harmonic Analysis 21(1), 5–30 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ham, J., Lee, D., Mika, S., Schölkopf, B.: A kernel view of the dimensionality reduction of manifolds. In: Proc. of the International Conference on Machine learning, ICML 2004, pp. 47–57 (2004)Google Scholar
  11. 11.
    Belkin, M., Sun, J., Wang, Y.: Constructing laplace operator from point clouds in rd. In: Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2009, pp. 1031–1040. Society for Industrial and Applied Mathematics, Philadelphia (2000)Google Scholar
  12. 12.
    Dey, T., Ranjan, P., Wang, Y.: Convergence, stability, and discrete approximation of laplace spectra. In: Proc. of ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, pp. 650–663 (2010)Google Scholar
  13. 13.
    Luxburg, U.: A tutorial on spectral clustering. Statistics and Computing 17(4), 395–416 (2007)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Belkin, M., Niyogi, P.: Laplacian eigenmaps for dimensionality reduction and data representation. Neural computation 15(6), 1373–1396 (2003)CrossRefzbMATHGoogle Scholar
  15. 15.
    Neumaier, A.: Solving ill-conditioned and singular linear systems: A tutorial on regularization. Siam Review 40(3), 636–666 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lee, J.A., Verleysen, M.: Nonlinear dimensionality reduction. Springer, Heidelberg (2007)CrossRefzbMATHGoogle Scholar
  17. 17.
    Schölkopf, B., Smola, A., Müller, K.-R.: Nonlinear component analysis as a kernel eigenvalue problem. Neural Computing 10(5), 1299–1319 (1998)CrossRefGoogle Scholar
  18. 18.
    Aizerman, M.A., Braverman, E.M., Rozonoèr, L.: Theoretical foundations of the potential function method in pattern recognition learning. Automation and remote control 25(6), 821–837 (1964)zbMATHGoogle Scholar
  19. 19.
    Meyer, C.: Matrix Analysis and Applied Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia (2000)CrossRefGoogle Scholar
  20. 20.
    Haasdonk, B., Pȩkalska, E.: Classification with Kernel Mahalanobis Distance Classifiers. In: Advances in Data Analysis, Data Handling and Business Intelligence, Studies in Classification, Data Analysis, and Knowledge Organization, pp. 351–361 (2008)Google Scholar
  21. 21.
    Settles, B.: Active learning literature survey. Computer Sciences Technical Report 1648, University of Wisconsin–Madison (2009)Google Scholar
  22. 22.
    Frank, A., Asuncion, A.: UCI machine learning repository (2010)Google Scholar
  23. 23.
    Karatzoglou, A., Smola, A., Hornik, K., Zeileis, A.: kernlab–an S4 package for kernel methods in R. Journal of Statistical Software 11(9) (2004)Google Scholar
  24. 24.
    Wolpert, D.H., Macready, W.G.: No free lunch theorems for optimization. IEEE Transactions on Evolutionary Computation 1(1), 67–82 (1997)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Nicolas Courty
    • 1
  • Thomas Burger
    • 2
  • Johann Laurent
    • 2
  1. 1.Université Européenne de BretagneUniversité de Bretagne SudValoriaFrance
  2. 2.Université Européenne de Bretagne, CNRS, Lab-STICCUniversité de Bretagne SudFrance

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