Damping Noise-Folding and Enhanced Support Recovery in Compressed Sensing

TL;DR

This paper introduces two new decoding methods in compressed sensing that effectively reduce noise amplification and improve support recovery, supported by theoretical guarantees and extensive numerical simulations.

Contribution

The paper proposes novel decoding procedures combining $ extit{ ext{l}_1}$-minimization with regularized least $p$-powers or iterative hard thresholding, enhancing noise reduction and support identification.

Findings

Reduced noise-folding effects in decoding

Improved support recovery over classical methods

Validated robustness through extensive simulations

Abstract

The practice of compressed sensing suffers importantly in terms of the efficiency/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 two new decoding procedures, combining $\ell_1$-minimization followed by either a regularized selective least $p$-powers or an iterative hard thresholding, which not only are able to reduce this component of the original noise, but also have enhanced properties in terms of support identification with respect to the sole $\ell_1$-minimization or iteratively re-weighted $\ell_1$-minimization. We prove such features, providing relatively simple and precise theoretical guarantees. We additionally confirm and support the theoretical results by extensive numerical simulations, which give a statistics of the robustness of the new decoding procedures with respect to more classical $\ell_1$-minimization and iteratively re-weighted $\ell_1$-minimization.