Constructive discretization and approximation in reproducing kernel Hilbert spaces

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.