Abstract
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.
Chapter PDF
Similar content being viewed by others
Keywords
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.
References
Duda, R.O., Hart, P.E., Stork, D.G.: Pattern Classification. Wiley, Chichester (2001)
Chavel, I.: Eigenvalues in Riemannian geometry. Academic Press, Orlando (1984)
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)
Nadler, B., Lafon, S., Coifman, R., Kevrekidis, I.: Diffusion maps, spectral clustering and eigenfunctions of fokker-planck operators. In: NIPS (2005)
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)
Rustamov, R.: Laplace-beltrami eigenfunctions for deformation invariant shape representation. In: Proc. of the Fifth Eurographics Symp. on Geometry Processing, pp. 225–233 (2007)
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)
Öztireli, C., Alexa, M., Gross, M.: Spectral sampling of manifolds. ACM, New York (2010)
Coifman, R.R., Lafon, S.: Diffusion maps. Applied and Computational Harmonic Analysis 21(1), 5–30 (2006)
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)
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)
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)
Luxburg, U.: A tutorial on spectral clustering. Statistics and Computing 17(4), 395–416 (2007)
Belkin, M., Niyogi, P.: Laplacian eigenmaps for dimensionality reduction and data representation. Neural computation 15(6), 1373–1396 (2003)
Neumaier, A.: Solving ill-conditioned and singular linear systems: A tutorial on regularization. Siam Review 40(3), 636–666 (1998)
Lee, J.A., Verleysen, M.: Nonlinear dimensionality reduction. Springer, Heidelberg (2007)
Schölkopf, B., Smola, A., Müller, K.-R.: Nonlinear component analysis as a kernel eigenvalue problem. Neural Computing 10(5), 1299–1319 (1998)
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)
Meyer, C.: Matrix Analysis and Applied Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia (2000)
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)
Settles, B.: Active learning literature survey. Computer Sciences Technical Report 1648, University of Wisconsin–Madison (2009)
Frank, A., Asuncion, A.: UCI machine learning repository (2010)
Karatzoglou, A., Smola, A., Hornik, K., Zeileis, A.: kernlab–an S4 package for kernel methods in R. Journal of Statistical Software 11(9) (2004)
Wolpert, D.H., Macready, W.G.: No free lunch theorems for optimization. IEEE Transactions on Evolutionary Computation 1(1), 67–82 (1997)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Courty, N., Burger, T., Laurent, J. (2011). PerTurbo: A New Classification Algorithm Based on the Spectrum Perturbations of the Laplace-Beltrami Operator. In: Gunopulos, D., Hofmann, T., Malerba, D., Vazirgiannis, M. (eds) Machine Learning and Knowledge Discovery in Databases. ECML PKDD 2011. Lecture Notes in Computer Science(), vol 6911. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23780-5_33
Download citation
DOI: https://doi.org/10.1007/978-3-642-23780-5_33
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-23779-9
Online ISBN: 978-3-642-23780-5
eBook Packages: Computer ScienceComputer Science (R0)