Damping Noise-Folding and Enhanced Support Recovery in Compressed Sensing

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 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 l1-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 l1-minimization or iteratively re-weighted l1-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 methods based on l1-minimization.