Constructive subsampling of finite frames with applications in optimal function recovery

Abstract

In this paper we present new constructive methods, random and deterministic, for the efficient subsampling of finite frames in ${\mathbb{C}}^m$. Based on a suitable random subsampling strategy, we are able to extract from any given frame with bounds $0<A\le B<\infty$ (and condition $B/A$) a similarly conditioned reweighted subframe consisting of merely $\mathcal{O}(m\log m)$ elements. Further, utilizing a deterministic subsampling method based on principles developed by Batson, Spielman, and Srivastava to control the spectrum of sums of Hermitian rank-1 matrices, we are able to reduce the number of elements to $\mathcal{O}(m)$ (with a constant close to one). By controlling the weights via a preconditioning step, we can, in addition, preserve the lower frame bound in the unweighted case. This permits the derivation of new quasi-optimal unweighted (left) Marcinkiewicz-Zygmund inequalities for $L_2(D,\nu )$ with constructible node sets of size $\mathcal{O}(m)$ for $m$-dimensional subspaces of bounded functions. Those can be applied e.g. for (plain) least-squares sampling reconstruction of functions, where we obtain new quasi-optimal results avoiding the Kadison-Singer theorem. Numerical experiments indicate the applicability of our results.