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 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 mdimensional 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.

On efficient algorithms for computing near-best polynomial approximations to high-dimensional, Hilbert-valued functions from limited samples

“…See, e.g., [46,69]. More recently, several works have also introduced sampling schemes that achieve linear in s sample complexity -i.e., optimal up to a constant [20,45,84,90,130]. Unfortunately, it is unknown whether or not linear or log-linear sample complexity possible in the compressed sensing setting, where the target subspace is unknown.…”

A sharp upper bound for sampling numbers in $L_{2}$

“…This earlier paper is applied to approximation theory in e.g. [18,24,34] and more recently in [2]. It presents a slightly less powerful method, requiring additional weights, but comes with an almost linear algorithmic complexity, see [17], and much smaller constants, which could make the bound presented here sharp also in terms of numerical values.…”