Decentralized and Collaborative Subspace Pursuit: A Communication-Efficient Algorithm for Joint Sparsity Pattern Recovery with Sensor Networks

In this paper, we consider the problem of joint sparsity pattern recovery in a distributed sensor network. The sparse multiple measurement vector signals (MMVs) observed by all the nodes are assumed to have a common (but unknown) sparsity pattern. To accurately recover the common sparsity pattern in a decentralized manner with a low communication overhead of the network, we develop an algorithm named decentralized and collaborative subspace pursuit (DCSP). In DCSP, each node is required to perform three kinds of operations per iteration: 1) estimate the local sparsity pattern by finding the subspace that its measurement vector most probably lies in; 2) share its local sparsity pattern estimate with one-hop neighboring nodes; and 3) update the final sparsity pattern estimate by majority vote based fusion of all the local sparsity pattern estimates obtained in its neighborhood. The convergence of DCSP is proved and its communication overhead is quantitatively analyzed. We also propose another decentralized algorithm named generalized DCSP (GDCSP) by allowing more information exchange among neighboring nodes to further improve the accuracy of sparsity pattern recovery at the cost of increased communication overhead. Experimental results show that, 1) compared with existing decentralized algorithms, DCSP provides much better accuracy of sparsity pattern recovery at a comparable communication cost; and 2) the accuracy of GDCSP is very close to that of centralized processing.Comment: 30 pages, 9 figure

Application of Compressive Sensing Techniques in Distributed Sensor Networks: A Survey

In this survey paper, our goal is to discuss recent advances of compressive sensing (CS) based solutions in wireless sensor networks (WSNs) including the main ongoing/recent research efforts, challenges and research trends in this area. In WSNs, CS based techniques are well motivated by not only the sparsity prior observed in different forms but also by the requirement of efficient in-network processing in terms of transmit power and communication bandwidth even with nonsparse signals. In order to apply CS in a variety of WSN applications efficiently, there are several factors to be considered beyond the standard CS framework. We start the discussion with a brief introduction to the theory of CS and then describe the motivational factors behind the potential use of CS in WSN applications. Then, we identify three main areas along which the standard CS framework is extended so that CS can be efficiently applied to solve a variety of problems specific to WSNs. In particular, we emphasize on the significance of extending the CS framework to (i). take communication constraints into account while designing projection matrices and reconstruction algorithms for signal reconstruction in centralized as well in decentralized settings, (ii) solve a variety of inference problems such as detection, classification and parameter estimation, with compressed data without signal reconstruction and (iii) take practical communication aspects such as measurement quantization, physical layer secrecy constraints, and imperfect channel conditions into account. Finally, open research issues and challenges are discussed in order to provide perspectives for future research directions

Joint Sparse Recovery With Semisupervised MUSIC

Discrete multiple signal classification (MUSIC) with its low computational cost and mild condition requirement becomes a significant noniterative algorithm for joint sparse recovery (JSR). However, it fails in rank defective problem caused by coherent or limited amount of multiple measurement vectors (MMVs). In this letter, we provide a novel sight to address this problem by interpreting JSR as a binary classification problem with respect to atoms. Meanwhile, MUSIC essentially constructs a supervised classifier based on the labeled MMVs so that its performance will heavily depend on the quality and quantity of these training samples. From this viewpoint, we develop a semisupervised MUSIC (SS-MUSIC) in the spirit of machine learning, which declares that the insufficient supervised information in the training samples can be compensated from those unlabeled atoms. Instead of constructing a classifier in a fully supervised manner, we iteratively refine a semisupervised classifier by exploiting the labeled MMVs and some reliable unlabeled atoms simultaneously. Through this way, the required conditions and iterations can be greatly relaxed and reduced. Numerical experimental results demonstrate that SS-MUSIC can achieve much better recovery performances than other MUSIC extended algorithms as well as some typical greedy algorithms for JSR in terms of iterations and recovery probability.Comment: Code is availabl

Local sparsity and recovery of fusion frames structured signals

The problem of recovering signals of high complexity from low quality sensing devices is analyzed via a combination of tools from signal processing and harmonic analysis. By using the rich structure offered by the recent development in fusion frames, we introduce a compressed sensing framework in which we split the dense information into sub-channel or local pieces and then fuse the local estimations. Each piece of information is measured by potentially low quality sensors, modeled by linear matrices and recovered via compressed sensing -- when necessary. Finally, by a fusion process within the fusion frames, we are able to recover accurately the original signal. Using our new method, we show, and illustrate on simple numerical examples, that it is possible, and sometimes necessary, to split a signal via local projections and / or filtering for accurate, stable, and robust estimation. In particular, we show that by increasing the size of the fusion frame, a certain robustness to noise can also be achieved. While the computational complexity remains relatively low, we achieve stronger recovery performance compared to usual single-device compressed sensing systems.Comment: 17 figures, 42 page

Robust Recovery of Signals From a Structured Union of Subspaces

Traditional sampling theories consider the problem of reconstructing an unknown signal $x$ from a series of samples. A prevalent assumption which often guarantees recovery from the given measurements is that $x$ lies in a known subspace. Recently, there has been growing interest in nonlinear but structured signal models, in which $x$ lies in a union of subspaces. In this paper we develop a general framework for robust and efficient recovery of such signals from a given set of samples. More specifically, we treat the case in which $x$ lies in a sum of $k$ subspaces, chosen from a larger set of $m$ possibilities. The samples are modelled as inner products with an arbitrary set of sampling functions. To derive an efficient and robust recovery algorithm, we show that our problem can be formulated as that of recovering a block-sparse vector whose non-zero elements appear in fixed blocks. We then propose a mixed $\ell_2/\ell_1$ program for block sparse recovery. Our main result is an equivalence condition under which the proposed convex algorithm is guaranteed to recover the original signal. This result relies on the notion of block restricted isometry property (RIP), which is a generalization of the standard RIP used extensively in the context of compressed sensing. Based on RIP we also prove stability of our approach in the presence of noise and modelling errors. A special case of our framework is that of recovering multiple measurement vectors (MMV) that share a joint sparsity pattern. Adapting our results to this context leads to new MMV recovery methods as well as equivalence conditions under which the entire set can be determined efficiently.Comment: 5 figures. 30 pages. This work has been submitted to the IEEE for possible publicatio

Greedy Subspace Pursuit for Joint Sparse Recovery

In this paper, we address the sparse multiple measurement vector (MMV) problem where the objective is to recover a set of sparse nonzero row vectors or indices of a signal matrix from incomplete measurements. Ideally, regardless of the number of columns in the signal matrix, the sparsity (k) plus one measurements is sufficient for the uniform recovery of signal vectors for almost all signals, i.e., excluding a set of Lebesgue measure zero. To approach the "k+1" lower bound with computational efficiency even when the rank of signal matrix is smaller than k, we propose a greedy algorithm called Two-stage orthogonal Subspace Matching Pursuit (TSMP) whose theoretical results approach the lower bound with less restriction than the Orthogonal Subspace Matching Pursuit (OSMP) and Subspace-Augmented MUltiple SIgnal Classification (SA-MUSIC) algorithms. We provide non-asymptotical performance guarantees of OSMP and TSMP by covering both noiseless and noisy cases. Variants of restricted isometry property and mutual coherence are used to improve the performance guarantees. Numerical simulations demonstrate that the proposed scheme has low complexity and outperforms most existing greedy methods. This shows that the minimum number of measurements for the success of TSMP converges more rapidly to the lower bound than the existing methods as the number of columns of the signal matrix increases.Comment: 55 pages, 8 figures, to be submitted to IEEE Transactions on Information theory, a shorter version was submitted to Proc. IEEE ISIT 201

Generalized Residual Ratio Thresholding

Simultaneous orthogonal matching pursuit (SOMP) and block OMP (BOMP) are two widely used techniques for sparse support recovery in multiple measurement vector (MMV) and block sparse (BS) models respectively. For optimal performance, both SOMP and BOMP require \textit{a priori} knowledge of signal sparsity or noise variance. However, sparsity and noise variance are unavailable in most practical applications. This letter presents a novel technique called generalized residual ratio thresholding (GRRT) for operating SOMP and BOMP without the \textit{a priori} knowledge of signal sparsity and noise variance and derive finite sample and finite signal to noise ratio (SNR) guarantees for exact support recovery. Numerical simulations indicate that GRRT performs similar to BOMP and SOMP with \textit{a priori} knowledge of signal and noise statistics.Comment: 13 pages, 8 figure

Improving M-SBL for Joint Sparse Recovery using a Subspace Penalty

The multiple measurement vector problem (MMV) is a generalization of the compressed sensing problem that addresses the recovery of a set of jointly sparse signal vectors. One of the important contributions of this paper is to reveal that the seemingly least related state-of-art MMV joint sparse recovery algorithms - M-SBL (multiple sparse Bayesian learning) and subspace-based hybrid greedy algorithms - have a very important link. More specifically, we show that replacing the $\log\det(\cdot)$ term in M-SBL by a rank proxy that exploits the spark reduction property discovered in subspace-based joint sparse recovery algorithms, provides significant improvements. In particular, if we use the Schatten-$p$ quasi-norm as the corresponding rank proxy, the global minimiser of the proposed algorithm becomes identical to the true solution as $p \rightarrow 0$. Furthermore, under the same regularity conditions, we show that the convergence to a local minimiser is guaranteed using an alternating minimization algorithm that has closed form expressions for each of the minimization steps, which are convex. Numerical simulations under a variety of scenarios in terms of SNR, and condition number of the signal amplitude matrix demonstrate that the proposed algorithm consistently outperforms M-SBL and other state-of-the art algorithms

Efficient iterative thresholding algorithms with functional feedbacks and convergence analysis

An accelerated class of adaptive scheme of iterative thresholding algorithms is studied analytically and empirically. They are based on the feedback mechanism of the null space tuning techniques (NST+HT+FB). The main contribution of this article is the accelerated convergence analysis and proofs with a variable/adaptive index selection and different feedback principles at each iteration. These convergence analysis require no longer a priori sparsity information $s$ of a signal. %key theory in this paper is the concept that the number of indices selected at each iteration should be considered in order to speed up the convergence. It is shown that uniform recovery of all $s$-sparse signals from given linear measurements can be achieved under reasonable (preconditioned) restricted isometry conditions. Accelerated convergence rate and improved convergence conditions are obtained by selecting an appropriate size of the index support per iteration. The theoretical findings are sufficiently demonstrated and confirmed by extensive numerical experiments. It is also observed that the proposed algorithms have a clearly advantageous balance of efficiency, adaptivity and accuracy compared with all other state-of-the-art greedy iterative algorithms

Compressed Sensing for Wireless Communications : Useful Tips and Tricks

As a paradigm to recover the sparse signal from a small set of linear measurements, compressed sensing (CS) has stimulated a great deal of interest in recent years. In order to apply the CS techniques to wireless communication systems, there are a number of things to know and also several issues to be considered. However, it is not easy to come up with simple and easy answers to the issues raised while carrying out research on CS. The main purpose of this paper is to provide essential knowledge and useful tips that wireless communication researchers need to know when designing CS-based wireless systems. First, we present an overview of the CS technique, including basic setup, sparse recovery algorithm, and performance guarantee. Then, we describe three distinct subproblems of CS, viz., sparse estimation, support identification, and sparse detection, with various wireless communication applications. We also address main issues encountered in the design of CS-based wireless communication systems. These include potentials and limitations of CS techniques, useful tips that one should be aware of, subtle points that one should pay attention to, and some prior knowledge to achieve better performance. Our hope is that this article will be a useful guide for wireless communication researchers and even non-experts to grasp the gist of CS techniques