# Damping Noise-Folding and Enhanced Support Recovery in Compressed Sensing

Massimo Fornasier^{*}, Marco Artina and Steffen Peter

The practice of compressive sensing suffers importantly in terms of the efficien\-cy/accuracy trade-off when acquiring noisy signals prior to measurement. It is rather common to find results treating the noise affecting the measurements, avoiding in this way to face the so-called \textit{noise-folding} phenomenon, related to the noise in the signal, eventually amplified by the measurement procedure. In this paper we present a new decoding procedure, combining $\ell_1$-minimization followed by a selective least $p$-powers, which not only is able to reduce this component of the original noise, but also has enhanced properties in terms of support identification with respect to the sole $\ell_1$-minimization. We prove such features, providing relatively simple and precise theoretical guarantees. We additionally confirm and support the theoretical estimates by extensive numerical simulations, which give a statistics of the robustness of the new decoding procedure with respect to more classical $\ell_1$-minimization.

Mathematics Subject Classification: 94A12 65F22

Keywords: Noise folding in compressed sensing, support identification, $\ell_1$-minimization, selective least p-powers, iterative thresholding, phase transitions.

Minisymposion: Noise Estimation, Model Selection and Bilevel Optimization