Approximation of eigenfunctions in kernel-based spaces

By Gabriele Santin, Robert Schaback. 2014

64 citations recorded by the source corpus.

Kernel-based methods in Numerical Analysis have the advantage of yielding optimal recovery processes in the “native” Hilbert space ℋ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {H}$\end{document} in which they are reproducing. Continuous kernels on compact domains have an expansion into eigenfunctions that are both L2-orthonormal and orthogonal in ℋ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {H}$\end{document} (Mercer expansion). This paper examines the corresponding eigenspaces and proves that they have optimality properties among all other subspaces of ℋ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {H}$\end{document}. These results have strong connections to n-widths in Approximation Theory, and they establish that errors of optimal approximations are closely related to the decay of the eigenvalues. Though the eigenspaces and eigenvalues are not readily available, they can be well approximated using the standard n-dimensional subspaces spanned by translates of the kernel with respect to n nodes or centers. We give error bounds for the numerical approximation of the eigensystem via such subspaces. A series of examples shows that our numerical technique via a greedy point selection strategy allows to calculate the eigensystems with good accuracy.

Subfields: Computational Mechanics, Numerical Analysis, Statistical and Nonlinear Physics

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

In Numerical Analysis

In Statistical and Nonlinear Physics

Citations in the corpus

Control functionals for Monte Carlo integration (2014, methodology): “rmulae Though the eigenbasis of T is not readily available, it can be well approximated using the standard m-dimensional subspaces spanned by translates of the kernel with respect to m design points (Santin and Schaback, 2015). Details are provided below. 2.4.1 Regularised least-squares As a numeric”
Diving into the shallows: a computational perspective on large-scale shallow learning (2017, background): “The rate for the practically important smooth radial kernels, including Gaussian kernels, Cauchy kernel and a number of other kernel families, is given in in [SS16], Theorem 6.”