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Non-uniform Randomized Sampling for Multivariate Approximation by High Order Parzen Windows

Published online by Cambridge University Press:  20 November 2018

Xiang-Jun Zhou ,
Lei Shi  and
Ding-Xuan Zhou
Xiang-Jun Zhou
Affiliation:
Joint Advanced Research Center in Suzhou, University of Science and Technology of China and City University of Hong Kong, Suzhou, Jiangshu, 215123, China e-mail: 50009229@student.cityu.edu.hk sl1983@mail.ustc.edu.cn
Lei Shi
Affiliation:
Joint Advanced Research Center in Suzhou, University of Science and Technology of China and City University of Hong Kong, Suzhou, Jiangshu, 215123, China e-mail: 50009229@student.cityu.edu.hk sl1983@mail.ustc.edu.cn
Ding-Xuan Zhou
Affiliation:
Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, China e-mail: mazhou@cityu.edu.hk
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Abstract

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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.

Keywords

68T0562J02multivariate approximationSobolev spacesnon-uniform randomized samplinghigh order Parzen windowsconvergence rates

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 3 , 01 September 2011 , pp. 566 - 576
Copyright
Copyright © Canadian Mathematical Society 2011

References

[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/S0036144501386986Google Scholar
[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] 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] 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/S0036141097294032Google Scholar
[5] Mukherjee, S. and Zhou, D.-X., Learning coordinate covariances via gradients.. J. Mach. Learn. Res. 7(2006), 519–549.Google Scholar
[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/1176988477Google Scholar
[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-yGoogle Scholar
[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/S0273-0979-04-01025-0Google Scholar
[9] Smale, S. and Zhou, D.-X., Online learning with Markov sampling. Anal. Appl. 7(2009), no. 1, 87–113. doi:10.1142/S0219530509001293Google Scholar
[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-yGoogle Scholar
[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/s10444-008-9073-8Google Scholar