Better bases for kernel spaces

TL;DR

This paper develops stable, local bases for kernel spaces that are computationally efficient and have desirable stability and decay properties, applicable to a broad class of kernels including Sobolev and polyharmonic types.

Contribution

It introduces a method to construct localized, sparse basis functions for kernel spaces with provable stability and decay, improving computational efficiency.

Findings

Basis functions can be constructed using about O((log N)^d) kernels.

The basis construction has a computational cost of O(N(log N)^d).

The basis is proven to be L_p stable with polynomial decay estimates.

Abstract

In this article we investigate the feasibility of constructing stable, local bases for computing with kernels. In particular, we are interested in constructing families $(b_{\xi})_{\xi\in\Xi}$ that function as bases for kernel spaces $S(k,\Xi)$ so that each basis function is constructed using very few kernels. In other words, each function $b_{\zeta}(x) = \sum_{\xi\in\Xi} A_{\zeta,\xi} k(x,\xi)$ is a linear combination of samples of the kernel with few nonzero coefficients $A_{\zeta,\xi}$. This is reminiscent of the construction of the B-spline basis from the family of truncated power functions. We demonstrate that for a large class of kernels (the Sobolev kernels as well as many kernels of polyharmonic and related type) such bases exist. In fact, the basis elements can be constructed using a combination of roughly $O(\log N)^d$ kernels, where $d$ is the local dimension of the manifold and $N$ is the dimension of the kernel space (i.e. $N=#\Xi$). Viewing this as a preprocessing step -- the construction of the basis has computational cost $O(N(\log N)^d)$. Furthermore, we prove that the new basis is $L_p$ stable and satisfies polynomial decay estimates that are stationary with respect to the density of $\Xi$.