Geodesic
Non-uniform Randomized Sampling for Multivariate Approximation by High Order Parzen Windows
Canadian mathematical bulletin, Tome 54 (2011) no. 3, pp. 566-576

Voir la notice de l'article provenant de la source Cambridge University Press

We consider approximation of multivariate functions in Sobolev spaces by high order Parzen windows in a non-uniform sampling setting. Sampling points are neither i.i.d. nor regular, but are noised from regular grids by non-uniform shifts of a probability density function. Sample function values at sampling points are drawn according to probability measures with expected values being values of the approximated function. The approximation orders are estimated by means of regularity of the approximated function, the density function, and the order of the Parzen windows, under suitable choices of the scaling parameter.
DOI : 10.4153/CMB-2011-029-8
Mots-clés : 68T05, 62J02, multivariate approximation, Sobolev spaces, non-uniform randomized sampling, high order Parzen windows, convergence rates 
Zhou, Xiang-Jun; Shi, Lei; Zhou, Ding-Xuan. Non-uniform Randomized Sampling for Multivariate Approximation by High Order Parzen Windows. Canadian mathematical bulletin, Tome 54 (2011) no. 3, pp. 566-576. doi: 10.4153/CMB-2011-029-8
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[1] [1] Aldroubi, A. and Gröchenig, K., Nonuniform sampling and reconstruction in shift-invariant spaces. SIAM Rev. 43(2001), no 4, 585–620. doi:10.1137/S0036144501386986 Google Scholar

[2] [2] Campbell, C. and Ying, Y., Learning coordinate gradients with multi-task kernels. In: Proceedings of the 21st Annual Conference on Learning Theory (COLT), 2008. Google Scholar

[3] [3] de Boor, C. and Jia, R.-Q., Controlled approximation and a characterization of the local approximation order. Proc. Amer. Math. Soc. 95(1985), no. 4, 547–553. Google Scholar

[4] [4] Han, B. and Jia, R.-Q., Multivariate refinement equations and convergence of subdivision schemes. SIAM J. Math. Anal. 29(1998), no. 5, 1177–1999. doi:10.1137/S0036141097294032 Google Scholar

[5] [5] Mukherjee, S. and Zhou, D.-X., Learning coordinate covariances via gradients.. J. Mach. Learn. Res. 7(2006), 519–549. Google Scholar

[6] [6] Pinelis, I., Optimum bounds for the distributions of martingales in Banach spaces. Ann. Probab. 22(1994), no. 4, 1679–1706. doi:10.1214/aop/1176988477 Google Scholar

[7] [7] Smale, S. and Zhou, D.-X., Learning theory estimates via integral operators and their approximations. Constr. Approx. 26(2007), no. 2, 153–172. doi:10.1007/s00365-006-0659-y Google Scholar

[8] [8] Smale, S. and Zhou, D.-X., Shannon sampling and function reconstruction from point values. Bull. Amer. Math. Soc. 41(2004), no. 3, 279–305. doi:10.1090/S0273-0979-04-01025-0 Google Scholar

[9] [9] Smale, S. and Zhou, D.-X., Online learning with Markov sampling. Anal. Appl. 7(2009), no. 1, 87–113. doi:10.1142/S0219530509001293 Google Scholar

[10] [10] Sun, H. and Wu, Q., Regularized least square regression with dependent samples. Adv. Comput. Math. 32(2010), no. 2, 175–189. doi:10.1007/s10444-008-9099-y Google Scholar

[11] [11] Zhou, X.-J. and Zhou, D.-X., High order Parzen windows and randomized sampling. Adv. Comput. Math. 31(2009), no. 4, 349–368. doi:10.1007/s10444-008-9073-8 Google Scholar

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