Block-sparse Signals Research Articles

Robust Reconstruction of Block Sparse Signals from Adaptively One‐Bit Measurements

Though various theoretical results and algorithms have been proposed in one-bit Compressed sensing (1-bit CS), there are few studies on more structured signals, such as block sparse signals. We address the problem of recovering block sparse signals from one-bit measurements. We first propose two recovery schemes, one based on second-order cone programming and the other based on hard thresholding, for common non-adaptively thresholded one-bit measurements. Note that the worst-case error in recovering sparse signals from non-adaptively thresholded one-bit measurements is bounded below by a polynomial of oversampling factor. To break the limit, we introduce a recursive strategy that allows the thresholds in quantization to be adaptive to previous measurements at each iteration. Using the scheme, we propose two iterative algorithms and show that corresponding recovery errors are both exponential functions of the oversampling factor. Several simulations are conducted to reveal the superiority of our methods to existing approaches.

ISAR Imaging Based on Multiple Measurement Vector Model Sparse Signal Recovery Algorithm

A multiple measurement vector (MMV) model blocks sparse signal recovery. ISAR imaging algorithm is proposed to improve ISAR imaging quality. Firstly, the sparse imaging model is built, and block sparse signal recovery algorithm-based MMV model is applied to ISAR imaging. Then, a negative exponential function is proposed to approximately block L0 norm. The optimization solution of smoothed function is obtained by constructing a decreasing sequence. Finally, the correction steps are added to ensure the optimal solution of the block sparse signal along the fastest descent direction. Several simulations and real data simulation experiments verify the proposed algorithm has advantages in imaging time and quality.

Weighted lp − l1 minimization methods for block sparse recovery and rank minimization

This paper considers block sparse recovery and rank minimization problems from incomplete linear measurements. We study the weighted [Formula: see text] [Formula: see text] norms as a nonconvex metric for recovering block sparse signals and low-rank matrices. Based on the block [Formula: see text]-restricted isometry property (abbreviated as block [Formula: see text]-RIP) and matrix [Formula: see text]-RIP, we prove that the weighted [Formula: see text] minimization can guarantee the exact recovery for block sparse signals and low-rank matrices. We also give the stable recovery results for approximately block sparse signals and approximately low-rank matrices in noisy measurements cases. Our results give the theoretical support for block sparse recovery and rank minimization problems.

An Extended Block Restricted Isometry Property for Sparse Recovery with Non-Gaussian Noise

We study the recovery conditions of weighted mixed l2/lp minimization for block sparse signal reconstruction from compressed measurements when partial block supportinformation is available. We show ...

Statistical inference for the block sparsity of complex‐valued signals

Block sparsity is an important parameter in many algorithms to successfully recover block-sparse signals under the framework of compressive sensing. However, it is often unknown and needs to be estimated. Recently there emerges a few research works about how to estimate block sparsity of real-valued signals, while there is, to the best of our knowledge, no research that has been done on complex-valued signals. In this study, the authors propose a method to estimate the block sparsity of complex-valued signal. Its statistical properties are obtained and verified by simulations. In addition, the authors demonstrate the importance of accurately estimating the block sparsity through a sensitivity analysis.

Orthogonal Projection-Based Channel Estimation for Multi-Panel Millimeter Wave MIMO

Multi-panel MIMO is a promising technology in millimeter wave communications. Due to its partially hybrid structure and non-uniform antenna array, existing channel estimation cannot be directly applied to multi-panel MIMO. In this paper, we study channel estimation for multi-panel mmWave MIMO. We firstly model channel estimation of multi-panel MIMO as a block-sparse signal recovery problem. Then, according to maximum likelihood criterion, we propose an orthogonal projection method to firstly detect the support of the sparse channel response vector and then perform least square estimation, based on which we also propose a low-complexity greedy support detection in order to reduce computational complexity. Finally, through analyzing pairwise error probability of support detection, we verify the performance gain of joint multi-panel channel estimation over single panel channel estimation, and we further find that joint multi-panel channel estimation with independently generated random combining matrices outperforms that with identically generated combining matrices. Numerical results verify the superiority of our proposed joint multi-panel orthogonal projection based channel estimation over existing schemes.

Block-sparse two-dimensional off-grid beamforming with arbitrary planar array geometry.

For a sound field observed on a planar sensor array, compressive sensing (CS) reconstructs the two-dimensional (2D) direction-of-arrival (DOA) of multiple sources using a sparsity constraint. Conventional compressive beamforming methods suffer from grid mismatch, where true DOAs do not fall on the discretized angular search grid. This paper adopts a CS-based model, which can reconstruct block-sparse signals, and the model treats DOAs and the off-grid DOA compensation parts as blocks to deal with the off-grid 2D beamforming. The method is illustrated by numerical simulations and shows high estimation accuracy. Also, the approach does not require a specific array configuration and is suitable for arbitrary planar array geometry, which is practically useful. Since propeller tip vortex cavitation induces noise sources located sparsely near the propeller tip, the high-resolution of the method is demonstrated with experimental data from cavitation tunnel experiments.

Reliable Recovery of Hierarchically Sparse Signals for Gaussian and Kronecker Product Measurements

We propose and analyze a solution to the problem of recovering a block sparse signal with sparse blocks from linear measurements. Such problems naturally emerge inter alia in the context of mobile communication, in order to meet the scalability and low complexity requirements of massive antenna systems and massive machine-type communication. We introduce a new variant of the Hard Thresholding Pursuit (HTP) algorithm referred to as HiHTP. We provide both a proof of convergence and a recovery guarantee for noisy Gaussian measurements that exhibit an improved asymptotic scaling in terms of the sampling complexity in comparison with the usual HTP algorithm. Furthermore, hierarchically sparse signals and Kronecker product structured measurements naturally arise together in a variety of applications. We establish the efficient reconstruction of hierarchically sparse signals from Kronecker product measurements using the HiHTP algorithm. Additionally, we provide analytical results that connect our recovery conditions to generalized coherence measures. Again, our recovery results exhibit substantial improvement in the asymptotic sampling complexity scaling over the standard setting. Finally, we validate in numerical experiments that for hierarchically sparse signals, HiHTP performs significantly better compared to HTP.

Robust Weighted Subspace Fitting for DOA Estimation via Block Sparse Recovery

In this letter, a novel robust block sparse recovery algorithm by using the weighted subspace fitting (WSF) is proposed to deal with the direction-of-arrival (DOA) problem under the condition of unknown mutual coupling. Firstly, a novel block sparse representation signal model based on the WSF is established to settle the effect of unknown mutual coupling. Then, the sparse constraint problem is investigated, and a regularization criterion between the sparsity penalty and subspace fitting error is given. Finally, the DOA estimation problem can be converted into a block sparse recovery problem. Some experimental results are carried out to prove the performance of proposed method in the case of unknown mutual coupling.

Error bounds of block sparse signal recovery based on q-ratio block constrained minimal singular values

In this paper, we introduce the q-ratio block constrained minimal singular values (BCMSV) as a new measure of measurement matrix in compressive sensing of block sparse/compressive signals and present an algorithm for computing this new measure. Both the mixed ℓ2/ℓq and the mixed ℓ2/ℓ1 norms of the reconstruction errors for stable and robust recovery using block basis pursuit (BBP), the block Dantzig selector (BDS), and the group lasso in terms of the q-ratio BCMSV are investigated. We establish a sufficient condition based on the q-ratio block sparsity for the exact recovery from the noise-free BBP and developed a convex-concave procedure to solve the corresponding non-convex problem in the condition. Furthermore, we prove that for sub-Gaussian random matrices, the q-ratio BCMSV is bounded away from zero with high probability when the number of measurements is reasonably large. Numerical experiments are implemented to illustrate the theoretical results. In addition, we demonstrate that the q-ratio BCMSV-based error bounds are tighter than the block-restricted isotropic constant-based bounds.

Variance State Propagation for Structured Sparse Bayesian Learning

We propose a compressed sensing algorithm termed variance state propagation (VSP) for block-sparse signals, i.e., sparse signals that have nonzero coefficients occurring in clusters. The VSP algorithm is developed under the Bayesian framework. A hierarchical Gaussian prior is introduced to depict the clustered patterns in the sparse signal. Markov random field (MRF) is introduced to characterize the state of the variances of the Gaussian priors. Such a hierarchical prior has the potential to encourage clustered patterns and suppress isolated coefficients whose patterns are different from their respective neighbors. The core idea of our algorithm is to iteratively update the variances in the prior Gaussian distribution. The message passing technique is employed in the design of the algorithm. For messages that are difficult to calculate, we correspondingly design reasonable methods to achieve approximate calculations. The hyperparameters can be updated within the iteration process. Simulation results demonstrate that the VSP algorithm is able to handle a variety of block-sparse signal recovery tasks and presents a significant advantage over the existing methods.

A Novel Block Sparse Reconstruction Method for DOA Estimation With Unknown Mutual Coupling

In this letter, we consider the direction-of-arrival (DOA) estimation in the presence of unknown mutual coupling in application to uniform linear arrays (ULAs). A novel method is proposed that treats the DOA estimation as a block sparse signal reconstruction problem with a modified array manifold matrix which utilizes the information of entire array output. A block smoothed $\ell _{0}$ norm approximation technique is introduced to solve the problem. Finally, an iterative proximal algorithm is proposed. Simulation results are presented that show the superiority of the proposed method against other known techniques.

BSBL-Based DOA and Polarization Estimation with Linear Spatially Separated Polarization Sensitive Array

The problem of multiple incident signals’ direction of arrival (DOA) and polarization estimation with linear spatially separated polarization sensitive array (SS-PSA) is investigated in sparse Bayesian learning (SBL) framework. SS-PSA is widely studied due to its low mutual coupling compared with the spatially collocated polarization sensitive array. On the one hand, the sparse representation of data model of proposed array can be expressed skillfully as a block-sparse representation. On the other hand, block sparse Bayesian learning (BSBL) algorithm has excellent performance in recovering the block-sparse signals. Therefore, in this paper, BSBL algorithm is extended to SS-PSA for DOA and polarization estimation. Firstly, a sparse representation model without parameters coupling is established, where the sparse coefficient vector is a block-sparse signal. Secondly, the expectation–maximization method in BSBL framework, i.e., BSBL-EM algorithm, is proposed to recover the block-sparse signal. Lastly, angles estimation can be obtained from the recovered sparse signal according to the intra-block correlation. Simulation results demonstrate that the BSBL-EM algorithm used in DOA and polarization estimation with SS-PSA exhibits better performance compared with Block Orthogonal Matching Pursuit algorithm and Group Basis Pursuit.

Block normalised iterative hard thresholding algorithm for compressed sensing

In this letter, the authors propose block normalised iterative hard thresholding (BNIHT) algorithm for the recovery of block sparse signal, in which the non-zero elements are presented in clusters. Based on block restricted isometry property, the sufficient conditions to guarantee the convergence of BNIHT are derived. In addition, the number of required iterations is obtained. The simulation experiment shows that BNIHT algorithm is superior to the block IHT (BIHT) algorithm when the step size satisfies μ < 1 .

Compressed channel estimation for massive MIMO-OFDM systems over doubly selective channels

Doubly selective (DS) channel estimation for the downlink massive multiple-input-multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) systems is a challenging problem, due to the requirements on high pilots overhead and prohibitive complexity. In this paper, by exploiting the highly correlated spatial structure of the obtained array response vectors and sparsity of the multipath signal components of the massive MIMO-OFDM channels, a modified spatial basis expansion model (modified-SBEM) is introduced. Thus, using complex exponential (CE-) modified-SBEM (i.e., modified CE-SBEM) can improve the resolution of the angles of departures (AoDs) information to represent the downlink with far fewer parameter dimensions, since the AoDs are much slower than path gains. Subsequently, we jointly design the effective pilot power and pilot placement for sparse channel estimation by means of an extended model. Our design is based on the block-coherence and sub-coherence simultaneous minimization of the measurement matrix associated with the massive MIMO-OFDM system pilot subcarriers. Furthermore, we leverage the sparse nature of the massive MIMO-OFDM system to formulate the quantized AoDs estimation into a block-sparse signal recovery problem, where the measurement matrix is designed based on the estimated virtual AoD. Thus, a new algorithm namely, generalized quasi-block simultaneous orthogonal matching pursuit (gQBSO), is introduced to solve the problem by providing sparse signal reconstruction solution. Simulation results demonstrate that the proposed scheme can effectively estimate the DS channel for massive MIMO-OFDM systems compared with other existing algorithms. For example, at SNR=20 dB for K=4 users, Doppler shift=0.093 with NT=32 antenna size, the adaptive-QBSO algorithm with G-SBEM and the proposed gQBSO with modified-SBEM can realize approximately 75.44%and 85.14% of the NMSE achieved by the oracle estimator with modified-SBEM.

Block-sparse signal recovery based on orthogonal matching pursuit via stage-wise weak selection

For recovering the block-sparse signal with unknown block structures, this paper presents a Block Stage-wise Weak Orthogonal Matching Pursuit (BSWOMP) algorithm. Which partitions the original signal into different groups or segments and conducts the selection for these groups separately using weak selection method to update reconstructing support set. And it proves that the Restricted Isometry Property (RIP) of BSWOMP algorithm is slack than Stage-wise Weak Orthogonal Matching Pursuit (SWOMP) algorithm and Block Orthogonal Matching Pursuit (BOMP) algorithm, this means BSWOMP algorithm generates better convergence than traditional sparse signal recovery algorithm. To further analysis of the performance of BSWOMP, the effectiveness of this algorithm is demonstrated by simulations. Experiments results show that the proposed algorithm effectively selects the nonzero elements of the block-sparse signal by a threshold, and compared with the traditional algorithms, BSWOMP algorithm has higher recovering stability, expedites the approximate conjugate gradient update step and brings an evident performance improvement and application prospect.

Sharp sufficient condition of block signal recovery via l 2 / l 1 ‐minimisation

This work gains a sharp sufficient condition on the block restricted isometry property for the recovery of sparse signal and corresponding upper bound estimate of error. Under the certain assumption, the signal with block structure can be stably recovered in the presence of noisy case and the block sparse signal can be exactly reconstructed in the noise-free case. Besides, an example is proposed to exhibit the condition is sharp. Numerical simulations are carried out to demonstrate that authors’ results are verifiable and l 2/l 1 minimisation method is robust and stable for the recovery of block sparse signals.

Sample-Efficient Algorithms for Recovering Structured Signals From Magnitude-Only Measurements

We consider the problem of recovering a signal $ \mathbf {x^{*}}\in \mathbb {R}^{n}$ , from magnitude-only measurements, $y_{i} = \left |{ \left \langle{ { \mathbf {a}_{i}}, \mathbf {x^{*}} }\right \rangle }\right | $ for $i=\{1,2,\ldots ,m\}$ . This is a stylized version of the classical phase retrieval problem , and is a fundamental challenge in nano- and bio-imaging systems, astronomical imaging, and speech processing. It is well-known that the above problem is ill-posed, and therefore some additional assumptions on the signal and/or the measurements are necessary. In this paper, we consider the case where the underlying signal $ \mathbf {x^{*}}$ is $s$ -sparse. For this case, we develop a novel recovery algorithm that we call Compressive Phase Retrieval with Alternating Minimization , or CoPRAM . Our algorithm is simple and is obtained via a natural combination of the classical alternating minimization approach for phase retrieval with the CoSaMP algorithm for sparse recovery. Despite its simplicity, we prove that our algorithm achieves a sample complexity of $ \mathcal {O}\left ({{s^{2} \log n}}\right )$ with Gaussian measurements $ { \mathbf {a}_{i}}$ , which matches the best known existing results; moreover, it also demonstrates linear convergence in theory and practice. An appealing feature of our algorithm is that it requires no extra tuning parameters other than the signal sparsity level $s$ . Moreover, we show that our algorithm is robust to noise. The quadratic dependence of sample complexity on the sparsity level is sub-optimal, and we demonstrate how to alleviate this via additional assumptions beyond sparsity. First, we study the (practically) relevant case where the sorted coefficients of the underlying sparse signal exhibit a power law decay. In this scenario, we show that the CoPRAM algorithm achieves a sample complexity of $ \mathcal {O}\left ({{s \log n}}\right )$ , which is close to the information-theoretic limit. We then consider the case where the underlying signal $ \mathbf {x^{*}}$ arises from structured sparsity models. We specifically examine the case of block-sparse signals with uniform block size of $b$ and block sparsity $k=s/b$ . For this problem, we design a recovery algorithm that we call Block CoPRAM that further reduces the sample complexity to $ \mathcal {O}\left ({{ks \log n}}\right )$ . For sufficiently large block lengths of $b=\Theta (s)$ , this bound equates to $ \mathcal {O}\left ({{s \log n}}\right )$ . To our knowledge, our approach constitutes the first family of linearly convergent algorithms for signal recovery from magnitude-only Gaussian measurements that exhibit a sub-quadratic dependence on the signal sparsity level.

The High Order Block RIP Condition for Signal Recovery

In this paper, we consider the recovery of block sparse signals, whose nonzero entries appear in blocks (or clusters) rather than spread arbitrarily throughout the signal, from incomplete linear measurement. A high order sufficient condition based on block RIP is obtained to guarantee the stable recovery of all block sparse signals in the presence of noise, and robust recovery when signals are not exactly block sparse via mixed $l_{2}/l_{1}$ minimization. Moreover, a concrete example is established to ensure the condition is sharp. The significance of the results presented in this paper lies in the fact that recovery may be possible under more general conditions by exploiting the block structure of the sparsity pattern instead of the conventional sparsity pattern.

ISAR Imaging Based on Block Sparse Smoothed L0 Norm Recovery Algorithm

In order to obtain high resolution inverse synthetic aperture radar (ISAR) sparse images, a block sparse signal recovery ISAR imaging algorithm is proposed by considering the cluster characteristics of target in this paper. Firstly, the ISAR sparse imaging model is established, the imaging is converted to L0 norm optimization problem. Secondly, one negative exponential function sequence is used as smoothed function to approach the block L0 norm. Finally, the revised step is added to ensure solving the optimization problem along the steepest descent gradient direction and the cost function is updated for the next loop. Simulation results verify the proposed algorithm is effective.