A theoretical perspective of solving phaseless compressive sensing via its nonconvex relaxation

You, Guowei; Huang, Zheng-Hai; Wang, Yong

doi:10.1016/j.ins.2017.06.020

Cited by 7 publications

(11 citation statements)

References 37 publications

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“…The signal recovery, as an important process of compressed sensing, has an essential impact on reconstruction accuracy. Therefore, researchers have proposed many improvement strategies to improve the accuracy of signal recovery, which include the following four categories: (1) Greedy algorithm based on local search strategy [48,49]; (2) convex optimization algorithm based on linear programming problem [50,51]; (3) non-convex optimization algorithm based on linear programming problem [52,53]; and (4) reconstruction algorithm based on natural heuristic algorithm [54,55]. In the process of reconstruction [56], the signals can be compressed and sampled in real-time; moreover, the original signals also can be recovered by some specific reconstruction algorithms.…”

Section: Related Workmentioning

confidence: 99%

Adaptive Bat Algorithm Optimization Strategy for Observation Matrix

et al. 2019

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Bat algorithm, as an optimization strategy of the observation matrix, has been widely used. Observation matrix has a direct impact on the reconstructed signal accuracy as a projection transformation matrix, and it has been widely used in various algorithms. However, for the traditional experimental process, randomly generated observation matrices often result in a larger reconstruction error and unstable reconstruction results. Therefore, it is a challenge to retain more feature information of the original signal and reduce reconstruction error. To obtain a more accurate reconstruction signal and less memory space, it is important to select an effective compression and reconstruction strategy. To solve this problem, an adaptive bat algorithm is proposed to optimize the observation matrix in this paper. For the adaptive bat algorithm, we design a dynamic adjustment strategy of the optimal radius to improve its global convergence ability. The results of our simulation experiments verify that, compared with other algorithms, it can effectively reduce the reconstruction error and has stronger robustness.

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Section: Related Workmentioning

confidence: 99%

Adaptive Bat Algorithm Optimization Strategy for Observation Matrix

et al. 2019

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“…Recently, Fung and Mangasarian (2011) discussed the relationship between the 0 and p minimal solutions to the system of linear equalities and inequalities with box constraints; and they showed that there is a (problem dependent) constant smaller or equal to one, denoted by p * , such that any optimal solution of p -minimization with p ∈ [0, p * ) also solves the concerned 0 -minimization. Such an important property has also been investigated for the system of linear equations (Peng et al 2015), the linear complementarity problem (Chen and Xiang 2016) and the phaseless compressive sensing problem (You et al 2017).…”

Section: Introductionmentioning

confidence: 99%

“…Inspired by the studies mentioned above, we investigate the partially sparsest solution of the union of finite polytopes. This problem covers a wide range which contains the problems discussed in Fung and Mangasarian (2011), Peng et al (2015) and You et al (2017) as special cases. We consider this problem via its nonconvex relaxation by using a class of nonconvex approximation functions which contains Schattenp quasi-norm as a special case.…”

Section: Introductionmentioning

confidence: 99%

The sparsest solution of the union of finite polytopes via its nonconvex relaxation

2019

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Sparse optimization problems have gained much attention since 2004. Many approaches have been developed, where nonconvex relaxation methods have been a hot topic in recent years. In this paper, we study a partially sparse optimization problem, which finds a partially sparsest solution of a union of finite polytopes. We discuss the relationship between its solution set and the solution set of its nonconvex relaxation. In details, by using geometrical properties of polytopes and properties of a family of well-defined nonconvex functions, we show that there exists a positive constant p * ∈ (0, 1] such that for every p ∈ [0, p * ), all optimal solutions to the nonconvex relaxation with the parameter p are also optimal solutions to the original sparse optimization problem. This provides a theoretical basis for solving the underlying problem via its nonconvex relaxation. Moreover, we show that the problem we concerned covers a wide range of problems so that several important sparse optimization problems are its subclasses. Finally, by an example we illustrate our theoretical findings.

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“…In these areas of application, the original signal to be recovered is often sparse. Hence, it is very natural to combine phase retrieval with compressed sensing, which is called phaseless compressed sensing [12,16,27,30]. The goal of phaseless compressed sensing is to reconstruct an unknown sparse signal x 0 ∈ C N from the magnitude of its noisy measurements y = |Ax 0 | + e, where A ∈ C m×N is the measurement matrix, e is the noise vector, and |•| denotes the element-wise absolute value.…”

Section: Introductionmentioning

confidence: 99%

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

Huo

Sun

Xiao

2017

Circuits Syst Signal Process

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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.

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A theoretical perspective of solving phaseless compressive sensing via its nonconvex relaxation

Cited by 7 publications

References 37 publications

Adaptive Bat Algorithm Optimization Strategy for Observation Matrix

Adaptive Bat Algorithm Optimization Strategy for Observation Matrix

The sparsest solution of the union of finite polytopes via its nonconvex relaxation

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

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