Constructive subsampling of finite frames with applications in optimal function recovery

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.