Title: Some Notes on Error Analysis for Kernel Based Regularized Interpolation
Author(s): Qing Zou
Pages: 689-698
Cite as:
Qing Zou, Some Notes on Error Analysis for Kernel Based Regularized Interpolation, Int. J. Anal. Appl., 18 (5) (2020), 689-698.

Abstract


Kernel based regularized interpolation is one of the most important methods for approximating functions. The theory behind the kernel based regularized interpolation is the well-known Representer Theorem, which shows the form of approximation function in the reproducing kernel Hilbert spaces. Because of the advantages of the kernel based regularized interpolation, it is widely used in many mathematical and engineering applications, for example, dimension reduction and dimension estimation. However, the performance of the approximation is not fully understood from the theoretical perspective. In other word, the error analysis for the kernel based regularized interpolation is lacking. In this paper, some error bounds in terms of the reproducing kernel Hilbert space norm and Sobolev space norm are given to understand the behavior of the approximation function.

Full Text: PDF

 

References


  1. M. Belkin and P. Niyogi, Semi-supervised learning on riemannian manifolds, Mach. Learn. 56 (2004), 209–239. Google Scholar

  2. M. Belkin, P. Niyogi, and V. Sindhwani, Manifold regularization: A geometric framework for learning from labeled and unlabeled examples, J. Mach. Learn. Res. 7 (2006), 2399–2434. Google Scholar

  3. R. Bhatia and C. Davis, A cauchy-schwarz inequality for operators with applications, Linear Algebra Appl. 223 (1995), 119–129. Google Scholar

  4. N. Cressie, The origins of kriging, Math. Geol. 22 (1990), 239–252. Google Scholar

  5. F. R. Deutsch, Best approximation in inner product spaces, Springer Science & Business Media, 2012. Google Scholar

  6. T. Kato, Estimation of iterated matrices, with application to the von neumann condition, Numer. Math. 2 (1960), 22–29. Google Scholar

  7. H.-J. Rong, G.-B. Huang, N. Sundararajan, and P. Saratchandran, Online sequential fuzzy extreme learning machine for function approximation and classification problems, IEEE Trans. Syst. Man Cybern. Part B (Cybernetics), 39 (2009), 1067–1072. Google Scholar

  8. B. Scholkopf, R. Herbrich, and A. J. Smola, A generalized representer theorem, in International conference on computational learning theory, Springer, 2001, 416–426. Google Scholar

  9. D. Shepard, A two-dimensional interpolation function for irregularly-spaced data, in Proceedings of the 1968 23rd ACM national conference, 1968, 517–524. Google Scholar

  10. M. Thakur, B. Samanta, and D. Chakravarty, A non-stationary geostatistical approach to multigaussian kriging for local reserve estimation, Stoch. Environ. Res. Risk Assess. 32 (2018), 2381–2404. Google Scholar

  11. J. J. Thiagarajan, P.-T. Bremer, and K. N. Ramamurthy, Multiple kernel interpolation for inverting non-linear dimensionality reduction and dimension estimation, in 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), IEEE, 2014, 6751–6755. Google Scholar

  12. A. Wannebo, Equivalent norms for the sobolev space $W_{0}^{m,p}(Omega)$, Ark. Mat. 32 (1994), 245–254. Google Scholar


COPYRIGHT INFORMATION

Copyright © 2020 IJAA, unless otherwise stated.