Constructive discretization and approximation in reproducing kernel Hilbert spaces

TL;DR

This paper extends a sparsification algorithm to infinite-dimensional spaces, providing new discretization inequalities and improved bounds for least-squares approximation errors in reproducing kernel Hilbert spaces.

Contribution

It generalizes a sparsification algorithm to infinite dimensions, yielding more constructive approximation bounds and improved constants in reproducing kernel Hilbert spaces.

Findings

Dimension-independent discretization inequalities in $L_2$ and sup-norms.

A new infinite-dimensional variant of the sparsification algorithm.

Enhanced constants and oversampling factors in approximation bounds.

Abstract

We generalize the sparsification algorithm of Batson, Spielman and Srivastava, making one part of the result dimension-independent. In particular, we recover discretization inequalities in $L_2$- and sup-norms on general finite-dimensional subspaces, prove a suitable infinite-dimensional variant, and discuss the implications for the error of least-squares approximation based on samples. This gives a more constructive version of several recently established approximation bounds, some of which relied on the stronger and less constructive result of Marcus, Spielman and Srivastava. We also improve the constants and oversampling factors in these results.