Block Sparse Signal Recovery Research Articles

Block Sparse Signal Recovery in Compressed Sensing: Optimum Active Block Selection and Within-Block Sparsity Order Estimation

In this paper, we develop a new algorithm for recovery of block sparse signals in compressed sensing framework based on orthogonal matching pursuit. Furthermore, we point out that a major issue in conventional sparse signal recovery is the lack of prior knowledge about the order of sparsity. Consequently, block sparse signal recovery algorithms suffer even more from the same problem since two parameters are needed for exact recovery of the signal, order of active blocks and within-block sparsity order. Therefore, we propose a new approach to determining both of the active blocks and within-block sparsity order which is embedded in the proposed block sparse signal recovery algorithm. The simulation results illustrate the improved performance of the proposed method for recovery of block sparse signals compared to the conventional methods which are not aware of the prior information. We also apply our proposed algorithm to ECG signal compression, where the obtained results reveal its efficiency.

A sharp recovery condition for block sparse signals by block orthogonal multi-matching pursuit

We consider the block orthogonal multi-matching pursuit (BOMMP) algorithm for the recovery of block sparse signals. A sharp bound is obtained for the exact reconstruction of block $K$-sparse signals via the BOMMP algorithm in the noiseless case, based on the block restricted isometry constant (block-RIC). Moreover, we show that the sharp bound combining with an extra condition on the minimum $\ell_2$ norm of nonzero blocks of block $K-$sparse signals is sufficient to recover the true support of block $K$-sparse signals by the BOMMP in the noise case. The significance of the results we obtain in this paper lies in the fact that making explicit use of block sparsity of block sparse signals can achieve better recovery performance than ignoring the additional structure in the problem as being in the conventional sense.

Sparse Bayesian learning using correlated hyperparameters for recovery of block sparse signals

Traditional sparse Bayesian learning applies independent hyperparameters to control the sparsity of each coefficient in the sparse signal. For recovery of block-sparse signals with unknown block partitioning structure, pattern coupled sparse Bayesian learning (PCSBL) method considers the sparsity patterns of the neighboring coefficients to be related to each other and uses a predefined parameter to control the relevance between the hyperparameters. However, for most real signals with block sparsity, the degree of relevance between neighboring coefficients is inhomogeneous, and should be data-dependent. This motivates us to modify PCSBL by replacing the single predefined parameter with a set of data-dependent coupling parameters (DDCP) to capture the relevance. In this paper, by performing a linear transformation to the independent hyperparameters, a set of correlated hyperparameters is generated to build the hierarchical Gaussian prior model for recovery of block sparse signals. The coupling parameters which compose the transformation matrix are estimated by the expectation maximization (EM) principle. The relevance between the hyperparameters is data dependent so the sparsity patterns of neighboring coefficients are related in an adaptive way. Compared with existing block sparse recovery methods, our approach is able to encourage the block sparse structure in a more flexible and more reliable way.

Joint Block Sparse Signal Recovery Problem and Applications in LTE Cell Search

We consider the problem of jointly recovering block sparse signals that share the same support set, using multiple measurement vectors (MMVs). We consider the generalized MMV (GMMV) model wherein the different measurement vectors could have been obtained by using different sensing matrices. We study greedy- and convex-programming-based recovery algorithms and theoretically establish their support recovery guarantees. Our results present insights into how the correlation among block sparse signals plays a role in the recovery performance. Next, we consider the problem of cell search in heterogeneous cellular networks (HetNets). With the cell search process, the mobile terminal (MT) acquires synchronization parameters, such as frame timing, residual frequency offset, and the physical-layer identity of a base station (BS), suitable for its connection. In HetNets, due to the increased density of the BS, the MT may receive strong interference from several BSs in its neighborhood. We establish that the problem of cell search in HetNets can be solved by using the GMMV joint block sparse signal recovery framework. We numerically study the performance of the cell search algorithms proposed by using our framework and show that they perform significantly better than the successive interference cancelation algorithm that exists in the literature.

Localized Low-Rank Promoting for Recovery of Block-Sparse Signals With Intrablock Correlation

Localized Low-Rank Promoting for Recovery of Block-Sparse Signals With Intrablock Correlation

A perturbation analysis of block-sparse compressed sensing via mixed ℓ2/ℓ1 minimization

In this paper, the recovery of block-sparse signals is considered by the completely perturbed mixed [Formula: see text] minimization method and a sufficient condition is established to guarantee the robust recovery. The obtained result generalizes the existing result on complete perturbation to the block setting. Specially, we not only improve the condition related to block-restricted isometry property, but also better the error upper bound if the result degenerates to the general case. In addition, some numerical experiments are also carried out to demonstrate the block structure which is an important factor in the process of recovering block-sparse signals, and present outperformance of the mixed [Formula: see text] minimization method comparing with the [Formula: see text] minimization method in the completely perturbed model.

Iterative Bayesian Reconstruction of Non-IID Block-Sparse Signals

This paper presents a novel Block Iterative Bayesian Algorithm (Block-IBA) for reconstructing block-sparse signals with unknown block structures. Unlike the existing algorithms for block sparse signal recovery which assume the clustered nonzero elements of the unknown signal to be independent and identically distributed (i.i.d.), we use a more adequate Bernoulli-Gaussian hidden Markov model (BGHMM) to characterize the non-i.i.d. block-sparse signals commonly encountered in practice. The Block-IBA iteratively estimates the amplitudes and positions of the block-sparse signal using the steepest-ascent based Expectation-Maximization, and effectively selects the nonzero elements of the block-sparse signal by a diminishing threshold. The global convergence of Block-IBA is analyzed and proved based on the non-i.i.d. property of BGHMM and error vector method. The effectiveness of Block-IBA is demonstrated by simulations on synthetic and real-life data.

On the Feedback Reduction of Multiuser Relay Networks Using Compressive Sensing

This paper presents a comprehensive performance analysis of full-duplex multiuser relay networks employing opportunistic scheduling with noisy and compressive feedback. Specifically, two feedback techniques based on compressive sensing (CS) theory are introduced and their effect on the system performance is analyzed. The problem of joint user identity and signal-to-noise ratio (SNR) estimation at the base-station is casted as a block sparse signal recovery problem in CS. Using existing CS block recovery algorithms, the identity of the strong users is obtained and their corresponding SNRs are estimated using the best linear unbiased estimator (BLUE). To minimize the effect of feedback noise on the estimated SNRs, a backoff strategy that optimally backsoff on the noisy estimated SNRs is introduced, and the error covariance matrix of the noise after CS recovery is derived. Finally, closed-form expressions for the end-to-end SNRs of the system are derived. Numerical results show that the proposed techniques drastically reduce the feedback air-time and achieve a rate close to that obtained by scheduling techniques that require dedicated error-free feedback from all network users. Key findings of this paper suggest that the choice of half-duplex or full-duplex SNR feedback is dependent on the channel coherence interval, and on low coherence intervals, full-duplex feedback is superior to the interference-free half-duplex feedback.

Bayesian Hypothesis Testing for Block Sparse Signal Recovery

A novel block Bayesian hypothesis testing algorithm (BBHTA) is presented for reconstructing block-sparse signals with unknown block structures. The BBHTA detects and recovers the supports and then estimates the amplitudes of block sparse signal. The support detection and recovery are performed by a Bayesian hypothesis testing. Using the detected and reconstructed supports, the nonzero amplitudes are then estimated by linear minimum mean-square error estimation. Numerical experiments demonstrate the effectiveness of BBHTA.

Downlink Synchronization Techniques for Heterogeneous Cellular Networks

Base stations (BS) in a cellular network send unique synchronization signals based on their physical layer cell-identity, in every frame. To synchronize with a BS, a mobile terminal (MT) needs to identify the following synchronization parameters—starting time of the frame, relative frequency offset and the cell-identity of the corresponding BS. In a heterogeneous cellular network, due to the increased density of low-powered base stations, the MT receives signals from multiple BS in its vicinity. We consider the downlink synchronization problem where the MT tries to recover the synchronization parameters of all the BS in its neighborhood, using which, MT can subsequently choose the suitable BS for its connection. In any given scenario, the number of BS in the neighborhood of MT is relatively small compared to the total number of BS in the entire network. Exploiting this sparseness, we present a two-stage synchronization approach where the first stage identifies the frame timing using an approximation to the maximum likelihood detector and the second stage recovers the cell identities and the corresponding frequency offsets of all the neighborhood BS using block sparse signal recovery framework of compressive sensing. We analytically and numerically study the recovery guarantees of our proposed techniques and establish their superior performance over existing successive interference cancellation and matched filtering approaches.

Iterative Reweighted <formula formulatype="inline"><tex Notation="TeX">$\ell_{2}/\ell_{1}$</tex> </formula> Recovery Algorithms for Compressed Sensing of Block Sparse Signals

In many applications of compressed sensing the signal is block sparse, i.e., the non-zero elements of the sparse signal are clustered in blocks. Here, we propose a family of iterative algorithms for the recovery of block sparse signals. These algorithms, referred to as iterative reweighted l 2 /l 1 minimization algorithms (IR-l 2 /l 1 ), solve a weighted l 2 /l 1 minimization in each iteration. Our simulation and analytical results on the recovery of both ideally and approximately block sparse signals show that the proposed iterative algorithms have significant advantages in terms of accuracy and the number of required measurements over non-iterative approaches as well as existing iterative methods. In particular, we demonstrate that, by increasing the block length, the performance of the proposed algorithms approaches the Wu-Verdu theoretical limit. The improvement in performance comes at a rather small cost in complexity increase. Further improvement in performance is achieved by using a priori information about the location of non-zero blocks, even if such a priori information is not perfectly reliable.

Block-sparsity-based localization in wireless sensor networks

In this paper, we deal with the localization problem in wireless sensor networks, where a target sensor location must be estimated starting from few measurements of the power present in a radio signal received from sensors with known locations. Inspired by the recent advances in sparse approximation, the localization problem is recast as a block-sparse signal recovery problem in the discrete spatial domain. In this paper, we develop different RSS-fingerprinting localization algorithms and propose a dictionary optimization based on the notion of the coherence to improve the reconstruction efficiency. The proposed protocols are then compared with traditional fingerprinting methods both via simulation and on-field experiments. The results prove that our methods outperform the existing ones in terms of the achieved localization accuracy.

Synthesized multitask compressive sensing for block-sparse signal recovery

The paper considers the problem of reconstructing blocks-sparse signals. A new algorithm, called synthesized multitask compressive sensing (SMCS), is proposed. In contrast to existing methods that rely on the availability of the sparsity structure information, the SMCS algorithm resorts to the multitask compressive sensing (MCS) technique for signal recovery. The SMCS algorithm synthesizes new compressive sensing (CS) tasks via circular-shifting operations and utilizes the minimum description length (MDL) principle to determine the proper set of the synthesized CS tasks for signal reconstruction. An outstanding advantage of SMCS is that it can achieve good signal reconstruction performance without using prior information on the block-sparsity structure. Simulations corroborate the theoretical developments.

Pattern-Coupled Sparse Bayesian Learning for Recovery of Block-Sparse Signals

We consider the problem of recovering block-sparse signals whose cluster patterns are unknown a priori. Block-sparse signals with nonzero coefficients occurring in clusters arise naturally in many practical scenarios. However, the knowledge of the block partition is usually unavailable in practice. In this paper, we develop a new sparse Bayesian learning method for recovery of block-sparse signals with unknown cluster patterns. A pattern-coupled hierarchical Gaussian prior is introduced to characterize the pattern dependencies among neighboring coefficients, where a set of hyperparameters are employed to control the sparsity of signal coefficients. The proposed hierarchical model is similar to that for the conventional sparse Bayesian learning. However, unlike the conventional sparse Bayesian learning framework in which each individual hyperparameter is associated independently with each coefficient, in this paper, the prior for each coefficient not only involves its own hyperparameter, but also its immediate neighbor hyperparameters. In doing this way, the sparsity patterns of neighboring coefficients are related to each other and the hierarchical model has the potential to encourage structured-sparse solutions. The hyperparameters are learned by maximizing their posterior probability. We exploit an expectation-maximization (EM) formulation to develop an iterative algorithm that treats the signal as hidden variables and iteratively maximizes a lower bound on the posterior probability. In the M-step, a simple suboptimal solution is employed to replace a gradient-based search to maximize the lower bound. Numerical results are provided to illustrate the effectiveness of the proposed algorithm.

Recovery of block sparse signals by a block version of StOMP

It is in general NP-hard to pursue the sparsest solution of an underdetermined system of linear equations. The Stagewise Orthogonal Matching Pursuit (StOMP) algorithm has been proposed in Donoho et al. [1] to recover sparse signals from compressed measurements, which is a greedy algorithm with low computational complexity and has some particularly interesting theoretical properties. On the other hand, problems of block sparse signal recovery have arisen in fields like signal processing and systems biology, under which a block sparse signal recovery algorithm with low computational complexity and theoretical guarantees is needed. In this paper, a block version of StOMP is proposed, termed Block Stagewise Orthogonal Matching Pursuit (Block-StOMP). Block-StOMP combines advantages of StOMP and block structure of signals for a high-efficiency recovery of the block sparse signal. Moreover, a rigorous theoretical analysis for Block-StOMP is given in this paper. Compared with StOMP and other algorithms, Block-StOMP has excellent recovery performance when the measurement noise level is moderate.

Restricted p-isometry properties of nonconvex block-sparse compressed sensing

In this paper, by generalizing the notion of restricted p-isometry constant (0<p≤1) defined by Chartrand and Staneva [1] to the setting of block-sparse signal recovery, we establish a general restricted p-isometry property (p-RIP) condition for recovery of (nearly) block-sparse signals via mixed l2/lp-minimization. Moreover, we derive a lower bound on the necessary number of Gaussian measurements for the p-RIP condition to hold with high probability, which shows clearly that fewer measurements with smaller p are needed for exact recovery of block-sparse signals via mixed l2/lp-minimization than when p=1.

A Simple Gaussian Measurement Bound for Exact Recovery of Block-Sparse Signals

We present a probabilistic analysis on conditions of the exact recovery of block-sparse signals whose nonzero elements appear in fixed blocks. We mainly derive a simple lower bound on the necessary number of Gaussian measurements for exact recovery of such block-sparse signals via the mixedl2/lq (0&lt;q≤1)norm minimization method. In addition, we present numerical examples to partially support the correctness of the theoretical results. The obtained results extend those known for the standardlqminimization and the mixedl2/l1minimization methods to the mixedl2/lq (0&lt;q≤1)minimization method in the context of block-sparse signal recovery.

On recovery of block-sparse signals via mixed l 2 /l q (0 &lt; q ≤ 1) norm minimization

Compressed sensing (CS) states that a sparse signal can exactly be recovered from very few linear measurements. While in many applications, real-world signals also exhibit additional structures aside from standard sparsity. The typical example is the so-called block-sparse signals whose non-zero coefficients occur in a few blocks. In this article, we investigate the mixed l 2/l q (0 < q ≤ 1) norm minimization method for the exact and robust recovery of such block-sparse signals. We mainly show that the non-convex l 2/l q (0 < q < 1) minimization method has stronger sparsity promoting ability than the commonly used l 2/l 1 minimization method both practically and theoretically. In terms of a block variant of the restricted isometry property of measurement matrix, we present weaker sufficient conditions for exact and robust block-sparse signal recovery than those known for l 2/l 1 minimization. We also propose an efficient Iteratively Reweighted Least-Squares (IRLS) algorithm for the induced non-convex optimization problem. The obtained weaker conditions and the proposed IRLS algorithm are tested and compared with the mixed l 2/l 1 minimization method and the standard l q minimization method on a series of noiseless and noisy block-sparse signals. All the comparisons demonstrate the outperformance of the mixed l 2/l q (0 < q < 1) method for block-sparse signal recovery applications, and meaningfulness in the development of new CS technology.

Extension of SBL Algorithms for the Recovery of Block Sparse Signals With Intra-Block Correlation

We examine the recovery of block sparse signals and extend the recovery framework in two important directions; one by exploiting the signals' intra-block correlation and the other by generalizing the signals' block structure. We propose two families of algorithms based on the framework of block sparse Bayesian learning (BSBL). One family, directly derived from the BSBL framework, require knowledge of the block structure. Another family, derived from an expanded BSBL framework, are based on a weaker assumption on the block structure, and can be used when the block structure is completely unknown. Using these algorithms, we show that exploiting intra-block correlation is very helpful in improving recovery performance. These algorithms also shed light on how to modify existing algorithms or design new ones to exploit such correlation and improve performance.

Block sparse recovery via mixed l 2/l 1 minimization

We consider efficient methods for the recovery of block sparse signals from underdetermined system of linear equations. We show that if the measurement matrix satisfies the block RIP with δ2s < 0.4931, then every block s-sparse signal can be recovered through the proposed mixed l2/l1-minimization approach in the noiseless case and is stably recovered in the presence of noise and mismodeling error. This improves the result of Eldar and Mishali (in IEEE Trans. Inform. Theory 55: 5302–5316, 2009). We also give another sufficient condition on block RIP for such recovery method: δs < 0.307.