On relevant dimensions in kernel feature spaces

Journal of Machine Learning Research

Volumn 9, Issue , 2008, Pages 1875-1908

  Braun, Mikio L ^a   Buhmann, Joachim M ^b   Müller, Klaus Robert ^a,c  

^a TECHNISCHE UNIVERSITÄT BERLIN   (Germany)

^b ETH ZURICH   (Switzerland)

^c FRAUNHOFER FIRST   (Germany)

Author keywords

Dimension reduction; Effective dimensionality; Feature space; Kernel methods

Indexed keywords

CLASSIFICATION (OF INFORMATION); SUPERVISED LEARNING;

DATA SETS; DIMENSION REDUCTION; EFFECTIVE DIMENSIONALITY; FEATURE SPACE; KERNEL METHODS;

SET THEORY;

EID: 50949127418     PISSN: 15324435     EISSN: 15337928     Source Type: Journal    
DOI: None     Document Type: Article

References (27)

1. Gilles Blanchard, Olivier Bousquet, and Laurent Zwald. Statistical properties of kernel principal component analysis. Machine Learning, 66(2-3):259-294, 2007.
2. Mikio L. Braun. Accurate error bounds for the eigenvalues of the kernel matrix. Journal of Machine Learning Research, 7:2303-2328, Nov 2006.
3. Mikio L. Braun. Spectral Properties of the Kernel Matrix and Their Application to Kernel Methods in Machine Learning. PhD thesis, University of Bonn, 2005. Available electronically at http://hss.ulb.uni-bonn.de/diss_online/ math_nat_fak/2005/braun_mikio.
4. Chris J. C. Burges. A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2(2):121-167, 1998.
5. Chandler Davis and William M. Kahan. The rotation of eigenvectors by a perturbation, iii. SIAM Journal of Numerical Analysis, 7:1-46, 1970.
6. γ dimension for regression in reproducing kernel hilbert spaces. In Proceedings of Algorithmic Learning Theory, 1999.
7. Edwin T. Jaynes. Information theory and statistical mechanics. Physical Review, 160(620-630), 1957.
8. Vladimir Koltchinskii and Evariste Giné. Random matrix approximation of spectra of integral operators. Bernoulli, 6(1):113-167, 2000.
9. Vladimir I. Koltchinskii. Asymptotics of spectral projections of some random matrices approximating integral operators. Progress in Probability, 43:191-227, 1998.
10. Sebastian Mika. Kernel Fisher Discriminants. PhD thesis, Technische Universität Berlin, December 2002.
11. Klaus-Robert Müller, Sebastian Mika, Gunnar Rätsch, Koji Tsuda, and Bernhard Schölkopf. An introduction to kernel-based learning algorithms. IEEE Transaction on Neural Networks, 12(2): 181-201, May 2001.
12. Gunnar Rätsch, Takashi Onoda, and Klaus-Robert Müller. Soft margins for AdaBoost. Machine Learning, 42(3):287-320, March 2001.
13. Bernhard Schölkopf and Alexander J. Smola. Learning with Kernels. MIT Press, 2002.
14. Bernhard Schölkopf, Alexander J. Smola, and Klaus-Robert Müller. Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation, 10:1299-1319, 1998.
15. Bernhard Schölkopf, Sebastian Mika, Christopher J. C. Burges, Philipp Knirsch, Klaus-Robert Müller, Gunnar Rätsch, and Alex J. Smola. Input space vs. feature space in kernel-based methods. IEEE Transactions on Neural Networks, 10(5):1000-1017, 1999.
16. John Shawe-Taylor, Peter L. Bartlett, Robert C. Williamson, and Martin Anthony. Structural risk minimization over date-dependent hierarchies. IEEE Transactions on Information Theory, 44(5): 1926-1940, 1998.
17. John Shawe-Taylor, Christopher K. I. Williams, Nello Christianini, and Jaz Kandola. On the eigen-spectrum of the Gram matrix and the generalization error of kernel-PCA. IEEE Transactions on Information Theory, 51(7):2510-2522, July 2005.
18. Alex J. Smola, Bernhard Schölkopf, and Klaus-Robert Müller. The connection between regularization operators and support vector kernels. Neural Networks, 11(4):637-649, 1998.
19. Sören Sonnenburg, Gunnar Rätsch, and Bernhard Schölkopf. Large scale genomic sequence SVM classifiers. In Proceedings of the 22nd International Machine Learning Conference, pages 848-855. ACM Press, 2005.
20. Vladimir Vapnik. The Nature of Statistical Learning Theory. Springer, 1995.
21. Vladimir Vapnik. Statistical Learning Theory. Wiley, 1998.
22. Régis Vert, Laurent Zwald, Gilles Blanchard, and Pascal Massart. Kernel projection machine: a new tool for pattern recognition. In Advances in Neural Information Processing Systems (NIPS 2004), pages 1649-1656. 2005, 2005.
23. Ulrike von Luxburg. Statistical Learning with Similarity and Dissimilarity Functions. PhD thesis, Technische Universität Berlin, 2004.
24. Grace Wahba. Spline Models For Observational Data. Society for Industrial and Applied Mathematics, 1990.
25. Robert C. Williamson, Alex J. Smola, and Bernhard Schölkopf. Generalization bounds for regularization networks and support vector machines via entropy numbers of compact operators. IEEE Transaction on Information Theory, 47(6):2516-2532, 2001.
26. Tong Zhang. Learning bounds for kernel regression using effective data dimensionality. Neural Computation, 17:2077-2098, 2005.
27. Laurent Zwald and Gilles Blanchard. On the convergence of eigenspaces in kernel principal components analysis. In Advances in Neural Information Processing Systems (NIPS 2005), volume 18, 2006.