Geometry on probability spaces

Constructive Approximation

Volumn 30, Issue 3, 2009, Pages 311-323

  Smale, Steve ^a   Zhou, Ding Xuan ^b  

^a TOYOTA TECHNOLOGICAL INSTITUTE AT CHICAGO   (United States)

^b CITY UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

Dimensionality reduction; Graph Laplacian; Integral operator; Learning theory; Reproducing kernel Hilbert space

Indexed keywords

EID: 70449535860     PISSN: 01764276     EISSN: 14320940     Source Type: Journal    
DOI: 10.1007/s00365-009-9070-2     Document Type: Article

References (25)

1. Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337-404 (1950).
2. Belkin, M., Niyogi, P.: Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput. 15, 1373-1396 (2003).
3. Belkin, M., Niyogi, P.: Convergence of Laplacian eigenmaps. In: Neural Information Processing Systems, vol. 19, pp. 129-136 (2007).
4. Belkin, M., De Vito, E., Rosasco, L.: Random estimates of operators and their spectral properties for learning. Working paper.
5. Bhatia, R.: Matrix Analysis. Graduate Texts in Mathematics, vol. 169. Springer, New York (1997).
6. Blanchard, G., Bousquet, O., Zwald, L.: Statistical properties of kernel principal component analysis. Mach. Learn. 66, 259-294 (2007).
7. Bougleux, S., Elmoataz, A., Melkemi, M.: Discrete Regularization on Weighted Graphs for Image and Mesh Filtering. Lecture Notes in Computer Science, vol. 4485, pp. 128-139. Springer, Berlin (2007).
8. Chung, F. R. K.: Spectral Graph Theory. Regional Conference Series in Mathematics, vol. 92. SIAM, Philadelphia (1997).
9. Coifman, R., Lafon, S., Lee, A., Maggioni, M., Nadler, B., Warner, F., Zucker, S.: Geometric diffusions as a tool for harmonic analysis and structure definition of data: diffusion maps. Proc. Natl. Acad. U. S. A. 102, 7426-7431 (2005).
10. Coifman, R., Maggioni, M.: Diffusion wavelets. Appl. Comput. Harmon. Anal. 21, 53-94 (2006).
11. Cucker, F., Smale, S.: On the mathematical foundations of learning. Bull. Am. Math. Soc. 39, 1-49 (2001).
12. Cucker, F., Zhou, D. X.: Learning Theory: An Approximation Theory Viewpoint. Cambridge University Press, Cambridge (2007).
13. De Vito, E., Caponnetto, A., Rosasco, L.: Model selection for regularized least-squares algorithm in learning theory. Found. Comput. Math. 5, 59-85 (2005).
14. De Vito, E., Rosasco, L., Caponnetto, A., De Giovannini, U., Odone, F.: Learning from examples as an inverse problem. J. Mach. Learn. Res. 6, 883-904 (2005).
15. Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. UCLA CAM Report 07-23, July 2007.
16. Koltchinskii, V., Giné, E.: Random matrix approximation of spectra of integral operators. Bernoulli 6, 113-167 (2000).
17. Pinelis, I.: Optimum bounds for the distributions of martingales in Banach spaces. Ann. Probab. 22, 1679-1706 (1994).
18. Smale, S., Yao, Y.: Online learning algorithms. Found. Comput. Math. 6, 145-170 (2006).
19. Smale, S., Zhou, D. X.: Shannon sampling and function reconstruction from point values. Bull. Am. Math. Soc. 41, 279-305 (2004).
20. Smale, S., Zhou, D. X.: Shannon sampling II. Connections to learning theory. Appl. Comput. Harmonic Anal. 19, 285-302 (2005).
21. Smale, S., Zhou, D. X.: Learning theory estimates via integral operators and their approximations. Constr. Approx. 26, 153-172 (2007).
22. von Luxburg, U., Belkin, M., Bousquet, O.: Consistency of spectral clustering. Ann. Stat. 36, 555-586 (2008).
23. Yao, Y., Rosasco, L., Caponnetto, A.: On early stopping in gradient descent learning. Constr. Approx. 26, 289-315 (2007).
24. Ye, G. B., Zhou, D. X.: Learning and approximation by Gaussians on Riemannian manifolds. Adv. Comput. Math. 29, 291-310 (2008).
25. Zhou, D., Schölkopf, B.: Regularization on discrete spaces. In: Pattern Recognition, Proc. 27th DAGM Symposium, Berlin, pp. 361-368 (2005).