Eignets for function approximation on manifolds

Applied and Computational Harmonic Analysis

Volumn 29, Issue 1, 2010, Pages 63-87

  Mhaskar, H N ^a  

^a CALIFORNIA STATE UNIVERSITY   (United States)

Author keywords

Data dependent manifolds; Direct and converse theorems of approximation; Kernel based approximation; RBF networks; Simultaneous approximation; Stability estimates

Indexed keywords

CONVERSE THEOREMS; DATA DEPENDENT; KERNEL-BASED APPROXIMATION; RBF NETWORK; SIMULTANEOUS APPROXIMATION; STABILITY ESTIMATES;

LYAPUNOV METHODS; MATHEMATICAL OPERATORS; OPTIMIZATION;

RADIAL BASIS FUNCTION NETWORKS;

EID: 77955273256     PISSN: 10635203     EISSN: 1096603X     Source Type: Journal    
DOI: 10.1016/j.acha.2009.08.006     Document Type: Article

References (35)

1. M. Belkin, I. Matveeva, and P. Niyogi Regularization and regression on large graphs Proc. of Computational Learning Theory 2004 Banff Canada
2. M. Belkin, P. Niyogi, Semi-supervised learning on Riemannian manifolds, Machine Learning J., in press; also, Tech. Report TR-2001-30, Univ. of Chicago, Computer Science Dept
3. M. Belkin, and P. Niyogi Towards a theoretical foundation for Laplacian-based manifold methods J. Comput. System Sci. 74 8 2008 1289 1308
4. M. Belkin, and P. Niyogi Convergence of Laplacian eigenmaps http://www.cse.ohio-state.edu/~mbelkin/papers/CLEM-08.pdf
5. J. Bergh, and J. Löfström Interpolation Spaces, an Introduction 1976 Springer-Verlag Berlin
6. C.K. Chui, and D.L. Donoho Special Issue: Diffusion Maps and Wavelets Appl. Comput. Harmon. Anal. 21 1 2006
7. R.R. Coifman, and M. Maggioni Diffusion wavelets Appl. Comput. Harmon. Anal. 21 2006 53 94
8. R.A. DeVore, and G.G. Lorentz Constructive Approximation 1993 Springer-Verlag Berlin
9. S.B. Damelin On bounds for diffusion, discrepancy and fill distance metrics. Principal manifolds for data visualization and dimension reduction Lect. Notes Comput. Sci. Eng. vol. 58 2008 Springer Berlin 261 270
10. S.B. Damelin A walk through energy, discrepancy, numerical integration and group invariant measures on measurable subsets of Euclidean space Numer. Algorithms 48 1-3 2008 213 235
11. S.B. Damelin, and A.J. Devaney Local Paley Wiener theorems for analytic functions on the unit sphere Inverse Problems 23 2007 1 12
12. D.L. Donoho, and C. Grimes Image manifolds which are isometric to Euclidean space http://www-stat.stanford.edu/~donoho/Reports/2002/WhenIsometry. pdf
13. D.L. Donoho, O. Levi, J.-L. Starck, and V.J. Martinez Multiscale Geometric analysis for 3-D catalogues http://www-stat.stanford.edu/~donoho/ Reports/2002/MGA3D.pdf
14. F. Filbir, H.N. Mhaskar, A Bernstein inequality for diffusion polynomials corresponding to a generalized heat kernel, manuscript
15. F. Filbir, and W. Themistoclakis Polynomial approximation on the sphere using scattered data Math. Nachr. 281 5 2008 650 668
16. A. Grigorýan Estimates of heat kernels on Riemannian manifolds E.B. Davies, Y. Safarov, Spectral Theory and Geometry Edinburgh, 1998 London Math. Soc. Lecture Note Ser. vol. 273 1999 Cambridge University Press Cambridge 140 225
17. A. Grigorýan, Heat kernels and function theory on metric measure spaces, in: L. Ji, P. Li, R. Schoen, L. Simon (Eds.), Handbook of Geometric Analysis, vol. 2, in: Advanced Lectures in Math., 2008, in press
18. X. He, S. Yan, Y. Hu, P. Niyogi, and H.-J. Zhang Face recognition using Laplacian faces IEEE Trans. Pattern Anal. Mach. Intell. 27 3 2005 328 340
19. L. Hörmander The spectral function of an elliptic operator Acta Math. 121 1968 193 218
20. P.W. Jones, M. Maggioni, and R. Schul Universal local parametrizations via heat kernels and eigenfunctions of the Laplacian arXiv:0709.1975
21. J. Keiner, S. Kunis, and D. Potts Efficient reconstruction of functions on the sphere from scattered data J. Fourier Anal. Appl. 13 2007 435 458
22. Yu.A. Kordyukov Lp-theory of elliptic differential operators on manifolds of bounded geometry Acta Appl. Math. 23 3 1991 223 260
23. S. Lafon, Diffusion maps and geometric harmonics, PhD thesis, Yale University, Dept. of Mathematics & Applied Mathematics, 2004
24. Q.T. Le Gia, and H.N. Mhaskar Localized linear polynomial operators and quadrature formulas on the sphere SIAM J. Numer. Anal. 47 1 2008/2009 440 466
25. M. Maggioni, and H.N. Mhaskar Diffusion polynomial frames on metric measure spaces Appl. Comput. Harmon. Anal. 24 3 2008 329 353
26. H.N. Mhaskar Introduction to the Theory of Weighted Polynomial Approximation 1996 World Scientific Singapore
27. H.N. Mhaskar When is approximation by Gaussian networks necessarily a linear process? Neural Networks 17 2004 989 1001
28. H.N. Mhaskar, and J. Prestin Polynomial frames: A fast tour C.K. Chui, M. Neamtu, L. Schumaker, Approximation Theory XI Gatlinburg, 2004 2005 Nashboro Press Brentwood 287 318
29. S. Minakshisundaram, and A. Pleijel Some properties of eigenfunctions of the Laplace operator on Riemannian manifolds Canad. J. Math. 1 1949 242 256
30. N. Saito Data analysis and representation on a general domain using eigenfunctions of Laplacian Appl. Comput. Harmon. Anal. 25 1 2008 68 97
31. A. Singer From graph to manifold Laplacian: The convergence rate Appl. Comput. Harmon. Anal. 21 1 2006 128 134
32. B. Xu Derivatives of the spectral function and Sobolev norms of eigenfunctions on a closed Riemannian manifold Ann. Global Anal. Geom. 26 2004 231 252
33. R. Xu, S.B. Damelin, and D. Wunsch Clustering of cancer tissues using diffusion maps and fuzzy ART with gene expression data BME 2008 42 49
34. R. Xu, S.B. Damelin, D.C. Wunsch II, Applications of diffusion maps in gene expression data-based cancer diagnosis analysis, in: Proceedings of the 29th Annual International Conference of IEEE Engineering in Medicine and Biology Society, Lyon, France, August 2007, pp. 4613-4616
35. A. Zygmund Trigonometric Series 1977 Cambridge University Press Cambridge