Octave code to approximate kernel eigenfunctions in a Mercer expansion.
Every strictly positive definite and continuous kernel on a bounded set
can be written as
with
This code approximates the unknown eigenfunctions and eigenvalues
using the knowledge of
and
.
The code implements the algorithm of
G. Santin and R. Schaback, Approximation of Eigenfunctions in Kernel-based Spaces, Adv. Comput. Math., Vol. 42 (4), 973–993 (2016) (see also the preprint).
You can start with one of the demos:
- ApproximateEigenbasis1D.m: Example with the Matèrn kernel on an interval.
- ApproximateEigenbasis.m: Example with the Gaussian kernel on the unit disk.
Notice that both demos actually compute the eigenbasis elements as , i.e., with a normalization that makes them orthonormal in the RKHS of the kernel.
If you use this code in your work, please consider citing the paper
@Article{Santin2016,
author = {Santin, Gabriele and Schaback, Robert},
title = {Approximation of eigenfunctions in kernel-based spaces},
journal = {Adv. Comput. Math.},
year = {2016},
volume = {42},
number = {4},
pages = {973--993},
issn = {1572-9044},
doi = {10.1007/s10444-015-9449-5},
}
Latex formulas are rendered using https://jsfiddle.net/8ndx694g/.