Generation of point sets by convex optimization for interpolation in reproducing kernel Hilbert spaces

TL;DR

This paper introduces convex optimization algorithms for generating point sets in reproducing kernel Hilbert spaces, enabling efficient kernel-based interpolation with competitive performance to existing greedy methods.

Contribution

It presents novel second-order cone programming algorithms for selecting points in RKHS interpolation, including a one-shot and a sequential variant.

Findings

Algorithms produce point sets comparable to $P$-greedy method.

Sequential algorithm allows adding multiple points per iteration.

Numerical results demonstrate competitive performance.

Abstract

We propose algorithms to take point sets for kernel-based interpolation of functions in reproducing kernel Hilbert spaces (RKHSs) by convex optimization. We consider the case of kernels with the Mercer expansion and propose an algorithm by deriving a second-order cone programming (SOCP) problem that yields $n$ points at one sitting for a given integer $n$. In addition, by modifying the SOCP problem slightly, we propose another sequential algorithm that adds an arbitrary number of new points in each step. Numerical experiments show that in several cases the proposed algorithms compete with the $P$-greedy algorithm, which is known to provide nearly optimal points.