New Conditions on Stable Recovery of Weighted Sparse Signals via Weighted $$l_1$$ l 1 Minimization

Abstract

The goal of phaseless compressed sensing is to recover an unknown sparse or approximately sparse signal from the magnitude of its measurements. However, it does not take advantage of any support information of the original signal. Therefore, our main contribution in this paper is to extend the theoretical framework for phaseless compressed sensing to incorporate with prior knowledge of the support structure of the signal. Specifically, we investigate two conditions that guarantee stable recovery of a weighted k-sparse signal via weighted l 1 minimization without any phase information. We first prove that the weighted null space property (WNSP) is a sufficient and necessary condition for the success of weighted l 1 minimization for weighted k-sparse phase retrievable. Moreover, we show that if a measurement matrix satisfies the strong weighted restricted isometry property (SWRIP), then the original signal can be stably recovered from the phaseless measurements.

Stable recovery of weighted sparse signals from phaseless measurements via weighted l1 minimization

“…Proposition 20, Theorem 3.4 Suppose that satisfies the WRIP of order 2 k with constant for . Let with ‖ e ‖ 2 ≤ ϵ , and be the solution to ().…”

“…By exploiting such partial support information, there has been a large amount of research on the recovery of a sparse signal via the weighted l 1 minimization 11–17 . Moreover, based on the weighted l 1 minimization, a weighted sparse recovery problem has been intensively studied in previous works 18–21 as well, in which a weight function is considered into the sparsity structure. Specifically, given a weight function with w i ≥ 1, a signal is called a weighted k ‐sparse signal, if …”

“…Then, the recovery of a weighted k ‐sparse signal by using the weighted l 1 minimization is referred to as the weighted sparse recovery problem. For more details, we refer the reader to previous works 18–21 …”

“…By exploiting such partial support information, there has been a large amount of research on the recovery of a sparse signal via the weighted l 1 minimization. [11][12][13][14][15][16][17] Moreover, based on the weighted l 1 minimization, a weighted sparse recovery problem has been intensively studied in previous works [18][19][20][21] as well, in which a weight function is considered into the sparsity structure. Specifically, given a weight function w…”

“…For more details, we refer the reader to previous works. [18][19][20][21] Phase retrieval is a fundamental problem of recovering a signal only from the magnitude of its linear measurements. It can be found in several areas, such as radar signal processing, 22 quantum mechanics, 23 and optics.…”