Statistical properties of kernel principal component analysis

Lecture Notes in Artificial Intelligence (Subseries of Lecture Notes in Computer Science)

Volumn 3120, Issue , 2004, Pages 594-608

  Zwald, Laurent ^a   Bousquet, Olivier ^b   Blanchard, Gilles ^c  

^a UNIVERSITÉ PARIS SACLAY   (France)

^b MAX PLANCK INSTITUTE FOR BIOLOGICAL CYBERNETICS   (Germany)

^c NONE   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; COMPUTATIONAL COMPLEXITY; DATA REDUCTION; EIGENVALUES AND EIGENFUNCTIONS; MATRIX ALGEBRA; STATISTICAL MECHANICS; RISK ASSESSMENT; SUPPORT VECTOR MACHINES;

HILBERT SPACES; KERNEL METHODS; RADEMACHER AVERAGES; SUPPORT VECTOR MACHINE (SVM); INFINITE DIMENSIONS; KERNEL ALGORITHMS; KERNEL OPERATOR; KERNEL PRINCIPAL COMPONENT ANALYSES (KPCA); NON-ASYMPTOTIC SETTINGS; RECONSTRUCTION ERROR; REPRODUCING KERNEL HILBERT SPACES; STATISTICAL PROPERTIES;

PRINCIPAL COMPONENT ANALYSIS;

EID: 9444287119     PISSN: 03029743     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1007/978-3-540-27819-1_41     Document Type: Conference Paper

References (20)

1. T. W. Anderson. Asymptotic theory for principal component analysis. Ann. Math. Stat., 34:122-148, 1963.
2. P. Bartlett, O. Bousquet, and S. Mendelson. Localized Rademacher complexities, 2003. Submitted, available at http://www.kyb.mpg.de/publications/ pss/ps2000.ps.
3. P.L. Bartlett and S. Mendelson. Rademacher and Gaussian complexities: risk bounds and structural results. Journal of Machine Learning Research, 3:463-482, 2002.
4. P. Baxendale. Gaussian measures on function spaces. Amer. J. Math., 98:891-952, 1976.
5. P. Besse. Etude descriptive d'un pocessus; approximation, interpolation. PhD thesis, Université de Toulouse, 1979.
6. P. Besse. Approximation spline de l'analyse en composantes principales d'une variable aléatoire hilbertienne. Ann. Fac. Sci. Toulouse (Math.), 12(5):329-349, 1991.
7. S. Boucheron, G. Lugosi, and P. Massart. A sharp concentration inequality with applications. Random Structures and Algorithms, 16:277-292, 2000.
8. J. Dauxois and A. Pousse. Les analyses factorielles en calcul des probabilités et en statistique: essai d'étude synthétique. PhD thesis.
9. N. Dunford and J. T. Schwartz. Linear Operators Part II: Spectral Theory, Self Adjoint Operators in Hilbert Space. Number VII in Pure and Applied Mathematics. John Wiley & Sons, New York, 1963.
10. W. Hoeffding. Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association, 58:13-30, 1963.
11. V. Koltchinskii. Asymptotics of spectral projections of some random matrices approximating integral operators. Progress in Probability, 43:191-227, 1998.
12. V. Koltchinskii and E. Giné. Random matrix approximation of spectra of integral operators. Bernoulli, 6(1):113-167, 2000.
13. P. Massart. Some applications of concentration inequalities to statistics. Annales de la Faculté des Sciencies de Toulouse, IX:245-303, 2000.
14. S. Mendelson. Estimating the performance of kernel classes. Journal of Machine Learning Research, 4:759-771, 2003.
15. J. O. Ramsay and C. J. Dalzell. Some tools for functional data analysis. Journal of the Royal Statistical Society, Series B, 53(3):539-572, 1991.
16. B. Schölkopf, A. J. Smola, and K.-R. Müller. Kernel principal component analysis. In B. Schölkopf, C. J. C. Burges, and A. J. Smola, editors, Advances in Kernel Methods - Support Vector Learning, pages 327-352. MIT Press, Cambridge, MA, 1999. Short version appeared in Neural Computation 10:1299-1319, 1998.
17. B. Schölkopf, A. J. Smola, and K.-R. Müller. Kernel principal component analysis. In B. Schölkopf, C. J. C. Burges, and A. J. Smola, editors, Advances in Kernel Methods - Support Vector Learning, pages 327-352. MIT Press, Cambridge, MA, 1999. Short version appeared in Neural Computation 10:1299-1319, 1998.
18. J. Shawe-Taylor, C. Williams, N. Cristianini, and J. Kandola. Eigenspectrum of the gram matrix and its relationship to the operator eigenspectrum. In Algorithmic Learning Theory : 13th International Conference, ALT 2002, volume 2533 of Lecture Notes in Computer Science, pages 23-40. Springer-Verlag, 2002. Extended version available at http://www.support-vactor. net/papers/eigenspectrum.pdf.
19. M. Torki. Etude de la sensibilité de toutes les valeurs propres non nulles d'un opérateur compact autoadjoint. Technical Report LA097-05, Université Paul Sabatier, 1997. Available at http://mip.ups-tlse.fr/ publi/rappLAO/97.05.ps.gz.
20. R. G. Williamson, J. Shawe-Taylor, B. Schölkopf, and A. J. Smola. Sample-based generalization bounds. IEEE Transactions on Information Theory, 1999. Submitted. Also: NeuroCOLT Technical Report NC-TR-99-055.