Greedy Recombination Interpolation Method (GRIM)

TL;DR

GRIM is a novel method combining greedy growth and recombination techniques to find sparse function approximations, extending previous methods and demonstrating competitive performance in kernel quadrature tasks.

Contribution

Introduces GRIM, a new interpolation method that integrates growth and thinning techniques, and applies it to kernel quadrature with promising results.

Findings

GRIM achieves sparse function approximations effectively.

Performance of GRIM matches other kernel quadrature methods.

First application of recombination outside measure support reduction.

Abstract

In this paper we develop the Greedy Recombination Interpolation Method (GRIM) for finding sparse approximations of functions initially given as linear combinations of some (large) number of simpler functions. In a similar spirit to the CoSaMP algorithm, GRIM combines dynamic growth-based interpolation techniques and thinning-based reduction techniques. The dynamic growth-based aspect is a modification of the greedy growth utilised in the Generalised Empirical Interpolation Method (GEIM). A consequence of the modification is that our growth is not restricted to being one-per-step as it is in GEIM. The thinning-based aspect is carried out by recombination, which is the crucial component of the recent ground-breaking convex kernel quadrature method. GRIM provides the first use of recombination outside the setting of reducing the support of a measure. The sparsity of the approximation found by GRIM is controlled by the geometric concentration of the data in a sense that is related to a particular packing number of the data. We apply GRIM to a kernel quadrature task for the radial basis function kernel, and verify that its performance matches that of other contemporary kernel quadrature techniques.