Bases for kernel-based spaces

Since it is well-known (De Marchi and Schaback (2001) [4]) that standard bases of kernel translates are badly conditioned while the interpolation itself is not unstable in function space, this paper surveys the choices of other bases. All data-dependent bases turn out to be defined via a factorizati...

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Bibliographic Details
Published in:Journal of computational and applied mathematics 2011-09, Vol.236 (4), p.575-588
Main Authors: Pazouki, Maryam, Schaback, Robert
Format: Article
Language:English
Subjects:
Adaptive algorithms
Adaptivity
Cholesky factorization
Computation
Computational efficiency
Duality
Factorization
Interpolation
Kernels
Matching pursuit
Mathematical analysis
Scattered data
Stability
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Summary:Since it is well-known (De Marchi and Schaback (2001) [4]) that standard bases of kernel translates are badly conditioned while the interpolation itself is not unstable in function space, this paper surveys the choices of other bases. All data-dependent bases turn out to be defined via a factorization of the kernel matrix defined by these data, and a discussion of various matrix factorizations (e.g. Cholesky, QR, SVD) provides a variety of different bases with different properties. Special attention is given to duality, stability, orthogonality, adaptivity, and computational efficiency. The “Newton” basis arising from a pivoted Cholesky factorization turns out to be stable and computationally cheap while being orthonormal in the “native” Hilbert space of the kernel. Efficient adaptive algorithms for calculating the Newton basis along the lines of orthogonal matching pursuit conclude the paper.
ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2011.05.021