A Newton basis for Kernel spaces
Journal of Approximation Theory
Stability of kernel-based interpolation
Advances in Computational Mathematics
Matching pursuits with time-frequency dictionaries
IEEE Transactions on Signal Processing
Bases for conditionally positive definite kernels
Journal of Computational and Applied Mathematics
A new stable basis for radial basis function interpolation
Journal of Computational and Applied Mathematics
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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.