REQUIRED NUMBER OF ITERATIONS FOR SPARSE SIGNAL RECOVERY VIA ORTHOGONAL LEAST SQUARES

In countless applications,we need to reconstruct a K-sparse signal x∈R n from noisy measurements y=Φx+v,whereΦ∈R^(m×n)is a sensing matrix and v∈R m is a noise vector.Orthogonal least squares(OLS),which selects at each step the column that results in the most significant decrease in the residual power,is one of the most popular sparse recovery algorithms.In this paper,we investigate the number of iterations required for recovering x with the OLS algorithm.We show that OLS provides a stable reconstruction of all K-sparse signals x in[2.8K]iterations provided thatΦsatisfies the restricted isometry property(RIP).Our result provides a better recovery bound and fewer number of required iterations than those proposed by Foucart in 2013.

DOA estimation in unknown colored noise using covariance differencing and sparse signal recovery

A direction-of-arrival （DOA） estimation algorithm is presented based on covariance differencing and sparse signal recovery, in which the desired signal is embedded in noise with unknown covariance. The key point of the algorithm is to eliminate the noise component by forming the difference of original and transformed covariance matrix, as well as cast the DOA estimation considered as a sparse signal recovery problem. Concerning accuracy and complexity of estimation, the authors take a vectorization operation on difference matrix, and further enforce sparsity by reweighted l1-norm penalty. We utilize data-validation to select the regularization parameter properly. Meanwhile, a kind of symmetric grid division and refinement strategy is introduced to make the proposed algorithm effective and also to mitigate the effects of limiting estimates to a grid of spatial locations. Compared with the covariance-differencing-based multiple signal classification （MUSIC） method, the proposed is of salient features, including increased resolution, improved robustness to colored noise, distinguishing the false peaks easily, but with no requiring of prior knowledge of the number of sources.

A Relaxed-PPA Contraction Method for Sparse Signal Recovery

Sparse signal recovery is a topic of considerable interest,and the literature in this field is already quite immense.Many problems that arise in sparse signal recovery can be generalized as a convex programming with linear conic constraints.In this paper,we present a new proximal point algorithm(PPA) termed as relaxed-PPA(RPPA) contraction method,for solving this common convex programming.More precisely,we first reformulate the convex programming into an equivalent variational inequality(VI),and then efficiently explore its inner structure.In each step,our method relaxes the VI-subproblem to a tractable one,which can be solved much more efficiently than the original VI.Under mild conditions,the convergence of the proposed method is proved.Experiments with l1 analysis show that RPPA is a computationally efficient algorithm and compares favorably with the recently proposed state-of-the-art algorithms.

Pulse Signal Recovery Method Based on Sparse Representation

Pulse signal recovery is to extract useful amplitude and time information from the pulse signal contaminated by noise. It is a great challenge to precisely recover the pulse signal in loud background noise. The conventional approaches,which are mostly based on the distribution of the pulse energy spectrum,do not well determine the locations and shapes of the pulses. In this paper,we propose a time domain method to reconstruct pulse signals. In the proposed approach,a sparse representation model is established to deal with the issue of the pulse signal recovery under noise conditions. The corresponding problem based on the sparse optimization model is solved by a matching pursuit algorithm. Simulations and experiments validate the effectiveness of the proposed approach on pulse signal recovery.

Robust signal recovery algorithm for structured perturbation compressive sensing

It is understood that the sparse signal recovery with a standard compressive sensing（CS） strategy requires the measurement matrix known as a priori. The measurement matrix is, however, often perturbed in a practical application.In order to handle such a case, an optimization problem by exploiting the sparsity characteristics of both the perturbations and signals is formulated. An algorithm named as the sparse perturbation signal recovery algorithm（SPSRA） is then proposed to solve the formulated optimization problem. The analytical results show that our SPSRA can simultaneously recover the signal and perturbation vectors by an alternative iteration way, while the convergence of the SPSRA is also analytically given and guaranteed. Moreover, the support patterns of the sparse signal and structured perturbation shown are the same and can be exploited to improve the estimation accuracy and reduce the computation complexity of the algorithm. The numerical simulation results verify the effectiveness of analytical ones.

A NEW SUFFICIENT CONDITION FOR SPARSE RECOVERY WITH MULTIPLE ORTHOGONAL LEAST SQUARES

A greedy algorithm used for the recovery of sparse signals,multiple orthogonal least squares(MOLS)have recently attracted quite a big of attention.In this paper,we consider the number of iterations required for the MOLS algorithm for recovery of a K-sparse signal x∈R^(n).We show that MOLS provides stable reconstruction of all K-sparse signals x from y=Ax+w in|6K/ M|iterations when the matrix A satisfies the restricted isometry property(RIP)with isometry constantδ_(7K)≤0.094.Compared with the existing results,our sufficient condition is not related to the sparsity level K.

Modified Iterative Method for Recovery of Sparse Multiple Measurement Problems

We consider the problem of constructing one sparse signal from a few measurements. This problem has been extensively addressed in the literature, providing many sub-optimal methods that assure convergence to a locally optimal solution under specific conditions. There are a few measurements associated with every signal, where the size of each measurement vector is less than the sparse signal＇s size. All of the sparse signals have the same unknown support. We generalize an existing algorithm for the recovery of one sparse signal from a single measurement to this problem and analyze its performances through simulations. We also compare the construction performance with other existing algorithms. Finally, the proposed method also shows advantages over the OMP （Orthogonal Matching Pursuit） algorithm in terms of the computational complexity.

THE EXACT RECOVERY OF SPARSE SIGNALS VIA ORTHOGONAL MATCHING PURSUIT

This paper aims to investigate sufficient conditions for the recovery of sparse signals via the orthogonal matching pursuit （OMP） algorithm. In the noiseless case, we present a novel sufficient condition for the exact recovery of all k-sparse signals by the OMP algorithm, and demonstrate that this condition is sharp. In the noisy case, a sufficient condition for recovering the support of k-sparse signal is also presented. Generally, the computation for the restricted isometry constant （RIC） in these sufficient conditions is typically difficult, therefore we provide a new condition which is not only computable but also sufficient for the exact recovery of all k-sparse signals.

Sparse recovery method for far-field and near-field sources localization using oblique projection

In this paper, we propose a novel source localization method to estimate parameters of arbitrary field sources, which may lie in near-field region or far-field region of array aperture. The proposed method primarily constructs two special spatial-temporal covariance matrixes which can avoid the array aperture loss, and then estimates the frequencies of signals to obtain the oblique projection matrixes. By using the oblique projection technique, the covariance matrixes can be transformed into several data matrixes which only contain single source information, respectively. At last, based on the sparse signal recovery method, these data matrixes are utilized to solve the source localization problem. Compared with the existing typical source localization algorithms, the proposed method improves the estimation accuracy, and provides higher angle resolution for closely spaced sources scenario. Simulation results are given to demonstrate the performance of the proposed algorithm.

Coherence-based performance analysis of the generalized orthogonal matching pursuit algorithm

The performance guarantees of generalized orthogonal matching pursuit（ gOMP） are considered in the framework of mutual coherence. The gOMP algorithmis an extension of the well-known OMP greed algorithmfor compressed sensing. It identifies multiple N indices per iteration to reconstruct sparse signals.The gOMP with N≥2 can perfectly reconstruct any K-sparse signals frommeasurement y = Φx if K 〈1/N（1/μ-1） ＋1,where μ is coherence parameter of measurement matrix Φ. Furthermore,the performance of the gOMP in the case of y = Φx ＋ e with bounded noise ‖e‖2≤ε is analyzed and the sufficient condition ensuring identification of correct indices of sparse signals via the gOMP is derived,i. e.,K 〈1/N（1/μ-1）＋1-（2ε/Nμxmin） ,where x min denotes the minimummagnitude of the nonzero elements of x. Similarly,the sufficient condition in the case of G aussian noise is also given.

1-Bit compressive sensing: Reformulation and RRSP-based sign recovery theory

Recently, the 1-bit compressive sensing （1-bit CS） has been studied in the field of sparse signal recovery. Since the amplitude information of sparse signals in 1-bit CS is not available, it is often the support or the sign of a signal that can be exactly recovered with a decoding method. We first show that a necessary assumption （that has been overlooked in the literature） should be made for some existing theories and discussions for 1-bit CS. Without such an assumption, the found solution by some existing decoding algorithms might be inconsistent with 1-bit measurements. This motivates us to pursue a new direction to develop uniform and nonuniform recovery theories for 1-bit CS with a new decoding method which always generates a solution consistent with 1-bit measurements. We focus on an extreme case of 1-bit CS, in which the measurements capture only the sign of the product of a sensing matrix and a signal. We show that the 1-bit CS model can be reformulated equivalently as an t0-minimization problem with linear constraints. This reformulation naturally leads to a new linear-program-based decoding method, referred to as the 1-bit basis pursuit, which is remarkably different from existing formulations. It turns out that the uniqueness condition for the solution of the 1-bit basis pursuit yields the so-called restricted range space property （RRSP） of the transposed sensing matrix. This concept provides a basis to develop sign recovery conditions for sparse signals through 1-bit measurements. We prove that if the sign of a sparse signal can be exactly recovered from 1-bit measurements with 1-bit basis pursuit, then the sensing matrix must admit a certain RRSP, and that if the sensing matrix admits a slightly enhanced RRSP, then the sign of a k-sparse signal can be exactly recovered with 1-bit basis pursuit.