Learning and approximation by Gaussians on Riemannian manifolds

Advances in Computational Mathematics

Volumn 29, Issue 3, 2008, Pages 291-310

  Ye, Gui Bo ^a   Zhou, Ding Xuan ^b  

^a FUDAN UNIVERSITY   (China)

^b CITY UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

Approximation; Gaussian kernels; Learning theory; Multi kernel least square regularization scheme; Reproducing kernel Hilbert spaces; Riemannian manifolds

Indexed keywords

EID: 55049127622     PISSN: 10197168     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10444-007-9049-0     Document Type: Article

References (27)

1. N. Aronszajn 1950 Theory of reproducing kernels Trans. Amer. Math. Soc. 68 337 404
2. Belkin, M., Niyogi, P.: Towards a theoretical foundation for Laplacian-based manifold methods. In: Auer, P., Meir, R. (eds.) COLT 2005, pp. 486-500 (2005)
3. M. Belkin P. Niyogi 2004 Semi-supervised learning on Riemannian manifolds Mach. Learn. 56 209 239
4. M. Belkin P. Niyogi 2003 Laplacian eigenmaps for dimensionality reduction and data representation Neural Comput. 15 1373 1396
5. Boothby, W.M.: An Introduction to Differentiable Manifolds and Riemannian Geometry. Academic, New York (1986)
6. Bousquet, O., Chapelle, O., Hein, M.: Measure based regularization. In: NIPS (2003)
7. do Carmo, M.: Riemannian Geometry. Birkhäuser, Boston (1992)
8. D.R. Chen Q. Wu Y. Ying D.X. Zhou 2004 Support vector machine soft margin classifiers: error analysis J. Mach. Learn. Res. 5 1143 1175
9. Cucker, F., Zhou, D.X.: Learning Theory: An Approximation Theory Viewpoint. Cambridge University Press, Cambridge (2007)
10. E. De Vito A. Caponnetto L. Rosasco 2005 Model selection for regularized least-squares algorithm in learning theory Found. Comput. Math. 5 59 85
11. Ditzian, Z., Totik, V.: Moduli of Smoothness. Springer, New York (1987)
12. D. Donoho C. Grimes 2003 Hessian eigenmaps: locally linear embedding techniques for high-dimensional data Proc. Natl. Acad. Sci. USA 100 5591 5596
13. T. Evgeniou M. Pontil T. Poggio 2000 Regularization networks and suport vector machines Adv. Comput. Math. 13 1 50
14. E. Gine V. Koltchinskii 2006 Empirical graph Laplacian approximation of Laplace-Beltrami operators: large sample results IMS Lecture Notes-monograph Series High Dimensional Probability 51 238 259
15. Hardin, D., Tsamardinos, I., Aliferis, C.F.: A theoretical characterization of linear SVM-based feature selection. Proc. of the 21st Int. Conf. on Machine Learning, Banff, Canada (2004)
16. Lee, J.M.: Riemannian Manifolds, Springer, New York (1997)
17. Micchelli, C.A., Pontil, M., Wu, Q. Zhou, D.X.: Error bounds for learning the kernel. In: Pattern Analysis, Statistical Modelling and Computational Learning (PASCAL), Research note no. 05/09, pp. 1-14 (2005)
18. Mukherjee, S., Wu, Q., Zhou, D.X.: Learning gradients and feature selection on manifolds. Preprint (2007)
19. S. Smale D.X. Zhou 2003 Estimating the approximation error in learning theory Anal. Appl. 1 17 41
20. Smale, S., Zhou, D.X.: Learning theory estimates via integral operators and their applications. Constr. Approx. (2007) doi: 10.1007/s00365-006-0659-y
21. von Luxburg, U., Belkin, M., Bousquet, O.: Consistency of spectral clustering. Ann. Statist. (2007) (in press)
22. Q. Wu Y. Ying D.X. Zhou 2007 Multi-kernel regularized classifiers J. Complexity 23 108 134
23. Q. Wu D.X. Zhou 2005 SVM soft margin classifiers: linear programming versus quadratic programming Neural Comput. 17 1160 1187
24. Ye, G.B., Zhou, D.X.: Fully online classification by regularization. Appl. Comput. Harmon. Anal. (2007) doi: 10.1016/j.acha.2006.12.001
25. Y. Ying D.X. Zhou 2007 Learnability of Gaussians with flexible variances J. Mach. Learn. Res. 8 249 276
26. D.X. Zhou 2002 The covering number in learning theory J. Complexity 18 739 767
27. D.X. Zhou 2003 Capacity of reproducing kernel spaces in learning theory IEEE Trans. Inform Theory 49 1743 1752