A new stable basis for RBF approximation

TL;DR

This paper introduces a new orthonormal basis for Radial Basis Function approximation that improves stability and conditioning, based on a weighted SVD of the kernel matrix, with proven convergence and stability bounds.

Contribution

It presents a novel orthonormal basis derived from a weighted SVD of the kernel matrix, enhancing RBF approximation stability and linking to the eigenbasis of a related operator.

Findings

Provides convergence estimates for the new basis

Establishes stability bounds for interpolation and least-squares approximation

Connects the basis to the eigenbasis of a compact operator

Abstract

It's well know that Radial Basis Function approximants suffers of bad conditioning if the simple basis of translates is used. A recent work of M.Pazouki and R.Schaback gives a quite general way to build stable, orthonormal bases for the native space based on a factorization of the kernel matrix A. Starting from that setting we describe a particular orthonormal basis that arises from a weighted singular value decomposition of A. This basis is related to a discretization of the compact operator which leads to the so-called eigenbasis, and provides a connection with it. We give convergence estimates and stability bound for the interpolation and the discrete least-squares approximation based on this basis, which involves the eigenvalues of such an operator.