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In this paper we present new constructive methods, random and deterministic,for the efficient subsampling of finite frames in $\mathbb C^m$. Based on asuitable random subsampling strategy, we are able to extract from any givenframe with bounds $0<A\le B<\infty$ (and condition $B/A$) a similarlyconditioned reweighted subframe consisting of merely $\mathcal{O}(m\log m)$elements. Further, utilizing a deterministic subsampling method based onprinciples developed by Batson, Spielman, and Srivastava to control thespectrum of sums of Hermitian rank-1 matrices, we are able to reduce the numberof elements to $\mathcal{O}(m)$ (with a constant close to one). By controllingthe weights via a preconditioning step, we can, in addition, preserve the lowerframe bound in the unweighted case. This permits the derivation of newquasi-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 newquasi-optimal results avoiding the Kadison-Singer theorem. Numericalexperiments indicate the applicability of our results.

Constructive subsampling of finite frames with applications in optimal function recovery