Constructive subsampling of finite frames with applications in optimal function recovery
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In this paper we present new constructive methods, random and deterministic, for the efficient subsampling of finite frames in ℂ^m. Based on a suitable random subsampling strategy, we are able to extract from any given frame with bounds 0 < A ≤ B < ∞ (and condition B/A) a similarly conditioned reweighted subframe consisting of merely 𝒪(m log m) elements. Further, utilizing a deterministic subsampling method based on principles developed in [1, Sec. 3], we are able to reduce the number of elements to 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 allows to derive new quasi-optimal unweighted (left) Marcinkiewicz-Zygmund inequalities for L_2(D, ν) with constructible node sets of size 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.
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