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

1 Application of Compressi v e Sensing T echniques in Distrib uted Sensor Networks: A Surv ey Thakshila W imalajeewa, Senior Member , IEEE and Pramod K V arshney , Life F ellow , IEEE Abstract —In this survey paper , our goal is to discuss recent advances of compressiv e sensing (CS) based solutions in wireless sensor networks (WSNs) including the main ongoing/r ecent resear ch efforts, challenges and resear ch 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 requir ement 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 se veral factors to be considered beyond the standard CS framework. W e 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 prob- lems such as detection, classiﬁcation and parameter estimation, with compressed data without signal reconstruction and (iii) take practical communication aspects such as measur ement 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. Index T erms —Wir eless sensor networks, Data gathering, Dis- tributed inference, Data compression, Compressive sensing (CS), Distributed/decentralized CS, Fading channels, Physical layer secrecy , Compressive detection, Compr essive classiﬁcation, Quan- tized CS I . I N T R O D U C T I O N Over the last tw o decades, the wireless sensor network (WSN) technology has gained increasing attention by both the research community and actual users [1]–[5]. Applications of WSNs span a wide range including en vironmental monitoring and surveillance [6], [7], detection and classiﬁcation [8]–[11], target/object tracking [12]–[16], industrial applications [17], [18], and health care [19] to name a few . In addition to domain speciﬁc and task-oriented applications, WSN technology has been identiﬁed as one of the key components in designing future Internet of Things (IoT) platforms [20], [21]. A typical sensor network consists of multiple sensors of the same or different modalities/types deployed ov er a geo- This material is based upon work supported by the National Science Foundation under Grant No. ENG 60064237. Thakshila W imalajeew a is with B AE Systems, Burlington MA. This work was done when she was at Syracuse Univ ersity , Syracuse, NY . Pramod K V arshney is with the Department of Electrical Engineering and Computer Science, Syracuse University , Syracuse, NY , USA. E-mail: thakshila.wimalajeew a@ieee.org, varshney@syr .edu. graphical area for monitoring a phenomenon of interest (PoI). Once deployed, the distributed sensors are required to form a connected network without a backbone infrastructure as in cellular networks. Most of the sensors are power constrained since they are equipped with small sized batteries which are difﬁcult or impossible to be replaced especially in hostile en vi- ronments. At the same time, the available (limited) communi- cation bandwidth needs to be efﬁciently used while e xchanging information for efﬁcient fusion. Thus, sensor networks are inherently resource constrained and they starve for energy and communication efﬁcient protocols [1], [22]. While distributed sensor fusion under resource constraints has been a research topic in vestigated for decades, the emergence of ne w sensors of different modalities that are capable of generating huge amounts of data in heterogeneous en vironments makes real- time fusion increasingly challenging [23]–[25]. Thus, desirable (or lossless) data compression is very important in designing WSNs for task-oriented as well as IoT based information systems. Advances in compressiv e sensing (CS) have led to novel ways of thinking about approaches to design energy ef ﬁcient WSNs with low cost data acquisition. CS has emerged as a promising paradigm for efﬁcient high dimensional sparse sig- nal acquisition. In the CS framework, a high dimensional sig- nal can be reliably recov ered with a small number of random projections under certain conditions if the signal of interest is sufﬁciently sparse [26]–[29]. In particular, compression is a simple linear operation implemented using random projection matrices which is independent of the signal parameters. In order to reconstruct the original high dimensional signal from its compressed version, several reconstruction techniques have been proposed where each one is different from the other in terms of their recovery performance and computational complexity [30]–[33]. CS is well motiv ated for a variety of WSN applications due to se veral reasons. Due to inherently scarce ener gy and communication resources in WSNs, data compression prior to transmission within WSNs is vital. On the other hand, sparsity is a common characteristic of many signals of interest that can be observed in various forms/dimensions. Thus, an immediate use of CS in WSNs is data gathering with reduced rate samples, as required by many en vironment and infrastructure monitoring applications. CS based data gathering may either exploit temporal, and/or spatial sparsity . Signal reconstruction with compressed data in CS was initially dev eloped for a single measurement vector (SMV) which was later extended to estimate multiple sparse signals sharing joint structures using multiple measurement v ectors (MMVs) [34], [35]. Direct use 2 of CS with SMV or MMVs may not be desirable due to communication constraints and speciﬁc application require- ments in large scale WSNs. In particular, recent extensions and modiﬁcations of CS to cope with communication/energy constraints and variations of sensor readings can be exploited to better utilize CS based techniques in WSNs. These exten- sions/modiﬁcations beyond the standard SMV/MMV include the design of adapti ve and sparse projection matrices to compress data at distributed nodes while meeting the desired communication constraints, and distributed and decentralized solutions for signal reconstruction considering different net- work models. The simple and univ ersal low rate data acquisition scheme provided by CS enables the design of new approaches to solve a varie ty of inference problems by suitable fusion of WSN data. In solving inference problems with compressed data, complete signal reconstruction, as employed in the standard CS framework, may not be necessary . Instead, constructing a decision statistic directly in the compressed domain is sufﬁcient to make a reliable inference decision, for example, in intruder detection, early detection of natural disasters in smart en vironments, estimation of parameters such as energy radiated by cell stations in smart cities, and object tracking. Moreov er , when applying CS techniques to perform dif ferent tasks in WSNs, their robustness in the presence of issues such as fading channels, physical layer secrecy concerns and quantization needs to be understood. This is because, the desirable conditions need to be satisﬁed by the standard CS framew ork can be violated under such practical aspects. Thus, to make CS ideas practically implementable for different tasks in a variety of WSN applications, above mentioned factors beyond the standard CS framework need to be understood. Over the last sev eral years, there has been extensiv e research efforts in this direction. A. Overview of the Curr ent P aper The goal of this re view paper is to discuss in some detail how the extensions and modiﬁcations done to the original CS framew ork can be utilized to solve a v ariety of problems in WSNs under practical considerations. Our revie w is based on the following classiﬁcation of existing work. W e believ e that this classiﬁcation allows us to gather most of the recent modiﬁcations/extensions to the CS frame work to meet WSN speciﬁc objectiv es and would provide the reader a compre- hensiv e understanding on the use of CS in WSN speciﬁc applications. In particular , our discussion covers: i) Extensions of the CS framework to operate under com- munication constraints for data gathering considering • form of sparsity exploited: temporal, spatial, spatio- temporal • data acquisition/collection techniques: sparse, adap- tiv e, and structured projection matrices, single-hop and multi-hop data collection • reconstruction techniques: centralized, and de- centralized implementation of different recon- struction algorithms including optimization based, greedy/iterativ e and Bayesian algorithms ii) Extensions of the CS frame work to solve a variety of in- ference problems without signal reconstruction including • detection, classiﬁcation, parameter estimation, source localization, and sensor management iii) Incorporation of practical communication issues into the CS frame work including • channel fading • physical layer secrecy constraints and • quantization iv) Lessons learned and future directions. In the following, we discuss the most related existing revie w/survey papers and highlight the contribution of the current paper compared to the existing papers. B. Comparison with Related Surve y/Review Articles CS ideas have gained signiﬁcant interest in a variety of applications such as imaging [36], [37], video processing [37], [38], [39], cognitiv e radio networks [40], [41], machine- type communications [42], radar signal processing [43], and physical layer operations in communication systems such as channel estimation in wireless networks [44]–[49], channel estimation in power line communication [50], to name a fe w . In early revie w papers/book chapters related to CS, theory , algorithms and general applications of CS hav e been discussed [27], [33], [51]. There are also few recent surve y papers that discuss recent adv ances in CS algorithms [52], [53]. Applications of CS in WSNs have been discussed to some extent in sev eral related papers. There are survey/re view papers av ailable in the literature on CS in communication systems in general where sensor networks are treated as one application and some results can be easily applied to sensor networks as special cases. In [57], the application of CS for compressed data gathering, distrib uted compression and source localization has been brieﬂy re vie wed under the general topic of CS for communications and networks. Similarly , the use of CS for communication networks has been revie wed in [63] focusing on getting different physical, network and application layer tasks done. Speciﬁc to WSNs, several topics such as com- pressed data gathering exploiting temporal and spatial sparsity , and compressed data routing in a centralized setting are re- viewed in [63]. In a recent surve y paper [58], the authors have emphasized on the factors to be considered when applying CS for channel estimation, interference cancellation, symbol detection, support identiﬁcation in wireless communication as applicable to different application scenarios. Some of the operations discussed in [58] such as simultaneous sparse signal recov ery using MMVs and source localization are applicable to WSNs as well. When considering existing surve y papers that speciﬁcally focus on WSNs, most of them discuss how CS can be utilized for compressed data gathering using the centralized architecture. In a centralized setting, the direct use (by direct use, we mean that there is no additional work done on designing projection matrices and/or reconstruction algorithms beyond the standard CS framew ork) of CS reduces to the MMV problem if temporal sparsity is e xploited and the SMV problem if spatial sparsity is exploited. In [59], the direct 3 T ABLE I: Related survey/re view papers Aspect References Y ear Contributions CS Recovery algorithms [53] 2015 A survey on sparse signal recovery algorithms with a single measurement vector; with SMV/MMV discusses different types of reconstruction algorithms along with a comparative study [54] 2011 A survey on sparse signal recovery algorithms with multiple measurement vectors; discusses optimization and greedy/iterative based simultaneous sparse approximation algorithms considering the JSM-2 model [55] [56] 2014 A review on CS reconstruction algorithms; discusses CS reconstruction algorithms for different distributed network models CS for communications and networks [57] 2013 A survey on theory and applications of CS; discusses compressed data gathering, distributed compression and source localization under CS in communications and networks CS for wireless communication [58] 2017 A survey on factors to be considered when applying CS for channel estimation, interference cancellation, symbol detection, support identiﬁcation in wireless communication Compression techniques in WSNs [59] 2013 A survey on compression techniques used in WSNs for data gathering; compares the use of CS based techniques and the conv entional compression schemes such as transformed based and distributed source coding CS for WSNs [60] 2011 A survey on CS for WSNs; discusses the improvements in factors such as lifetime, delay , cost and power CS for image/video data [39] 2013 A tutorial on a distributed compressive video sensing; discusses the advantages of compression in WSNs CS based video processing vs traditional techniques in a distributed manner CS for cognitive radio networks [61] 2013 A survey on wideband spectrum sensing techniques; discusses Nyquist and sub-Nyquist (CS based) techniques for spectrum sensing [40] 2016 A survey on application of CS in cognitive radio networks; discusses CS based wideband spectrum sensing, CS based CR-related parameter estimation and the use of CS for radio environment map construction [62] 2016 A survey on CS based techniques for cognitive radio networks; discusses the use of CS for a variety of CR applications including spectrum sensing, channel estimation, and multiple-input multiple-output based CR use of CS for data gathering exploiting spatial sparsity has been considered. A comparati ve study on the advantages and disadvantages of CS based compression with inter- and intra-signal correlation compared to con ventional compression schemes employed for WSNs has been presented. In [60], the direct use of CS exploiting spatial sparsity of data collected at multiple nodes for distributed compression has been con- sidered. The authors have discussed as to how certain sensor network parameters such as lifetime, delay , cost and power can be improved using CS. Ho wever , the work related to the exploitation of other forms of sparsity , incorporation of com- munication related issues when designing projection matrices and reconstruction techniques for data gathering is missing in [59], [60]. In [54], [56], work related to dev elopment of reconstruction algorithms taking different network models and communication architectures has been revie wed. In particular , the authors in [56] hav e discussed the distributed development of some reconstruction algorithms for sev eral signal models, and network topologies. Simultaneous sparse approximation algorithms revie wed in [54] are applicable for WSNs with the JSM-2 model (as deﬁned in [55] and discussed in detail in Section III-A1). While some of the algorithms discussed in this revie w paper for data gathering have an overlap to some extent with the ones that are discussed in [54], [56], our discussion is more comprehensi ve with respect to recent dev elopments focusing on centralized as well as decentralized settings and complexity analysis. Moreov er , [54], [56] did not consider data gathering e xploiting spatio-temporal sparsity , design of data acquisition and reconstruction schemes to meet communication constraints and the use of CS to solve any other inference tasks. The re view papers [40], [61], [62] hav e focused mainly on cognitiv e radio networks, while some of the algorithms discussed are applicable for data gathering in sensor network applications as well. T o the best of the authors’ knowledge, there is no any revie w paper that discusses CS based inference or impact of practical aspects on CS based processing in WSNs. A summary of survey/re vie w papers most related to this paper is gi ven in T able I. The or ganization of the paper is summarized belo w . C. P aper Or ganization The roadmap of the paper is sho wn in Fig. 1. In Section II, CS basics and motiv ation behind its use in se veral WSN applications are discussed. Application of CS in ef ﬁcient data gathering exploiting temporal and spatial sparsity is revie wed in Section III. Furthermore, data gathering tech- niques and algorithms developed in centralized as well as in distributed/decentralized settings are discussed. CS based inference including detection, classiﬁcation and localization is revie wed in Section IV. In Section V, a revie w on CS based signal processing under practical communication con- siderations such as channel fading, physical layer secrecy and quantization is giv en. Finally future research directions are discussed in Section VI and concluding remarks are giv en in Section VII. 4 I I I . C S f o r D i s t r i b u t e d D a t a G a t h e r i n g I V . C S f o r D i s t r i b u t e d I n f e r e n c e I V . A C o m p r e s s i v e D e t e c t i o n I V . A . 1 S p a r s e s i g n a l s w i t h s p a t i a l s p a r s i t y I V . A . 1 S p a r s e s i g n a l s w i t h s p a t i a l s p a r s i t y V . A I m p a c t o f F a d i n g C h a n n e l s I V . B C o m p r e s s i v e C l a s s i f i c a t i o n V . P r a c t i c a l A s p e c t s V I . L e s s o n s L e a r n e d a n d F u t u r e D i r e c t i o n s V . B P h y s i c a l L a y e r S e c r e c y C o n s t r a i n t s V . C Q u a n t i z e d C S A p p l i c a t i o n o f C S T e c h n i q u e s i n W S N s I I . B a c k g r o u n d I I . A C S B a s i c s I I . B A p p l i c a t i o n s o f C S i n W i r e l e s s S e n s o r N e t w o r k s : M o t i v a t i o n a n d C h a l l e n g e s I I I . A E x p l o i t T e m p o r a l S p a r s i t y I I I . B E x p l o i t S p a t i a l S p a r s i t y I I I . A . 1 C e n t r a l i z e d I I I . A . 1 C e n t r a l i z e d I I I . A . 2 D e c e n t r a l i z e d ( J S M - 2 ) I I I . A . 2 D e c e n t r a l i z e d ( J S M - 2 ) I I I . B . 1 C e n t r a l i z e d : U s e o f S p a r s e M a t r i c e s I I I . B . 1 C e n t r a l i z e d : U s e o f S p a r s e M a t r i c e s I I I . B . 3 D e c e n t r a l i z e d S o l u t i o n s I I I . B . 3 D e c e n t r a l i z e d S o l u t i o n s I I . A . 1 C o m p r e s s i o n I I . A . 1 C o m p r e s s i o n I I . A . 3 R e c o n s t r u c t i o n A l g o r i t h m s I I . A . 3 R e c o n s t r u c t i o n A l g o r i t h m s I I . A . 2 M e a s u r e m e n t m a t r i x I I . A . 2 M e a s u r e m e n t m a t r i x J S M - 2 M o d e l J S M - 2 M o d e l J S M - 1 M o d e l J S M - 1 M o d e l O p t i m i z a t i o n b a s e d O p t i m i z a t i o n b a s e d G r e e d y / i t e r a t i v e G r e e d y / i t e r a t i v e B a y e s i a n B a y e s i a n O t h e r O t h e r O p t i m i z a t i o n b a s e d O p t i m i z a t i o n b a s e d G r e e d y / i t e r a t i v e G r e e d y / i t e r a t i v e B a y e s i a n B a y e s i a n A n a l y s i n g d i f f e r e n t a l g o r i t h m s A n a l y s i n g d i f f e r e n t a l g o r i t h m s I I I . B . 2 C e n t r a l i z e d : U s e o f H y b r i d / a d a p t i v e T e c h n i q u e s I I I . B . 2 C e n t r a l i z e d : U s e o f H y b r i d / a d a p t i v e T e c h n i q u e s I I I . C E x p l o i t S p a t i o - T e m p o r a l S p a r s i t y I I I . C . 1 U s e o f M a t r i x c o m p l e t i o n t e c h n i q u e s I I I . C . 1 U s e o f M a t r i x c o m p l e t i o n t e c h n i q u e s I I I . C . 2 U s e o f s t r u c t u r e d / K r o n e c k e r C S I I I . C . 2 U s e o f s t r u c t u r e d / K r o n e c k e r C S I V . A . 2 S p a r s e s i g n a l s w i t h t e m p o r a l s p a r s i t y I V . A . 2 S p a r s e s i g n a l s w i t h t e m p o r a l s p a r s i t y I V . A . 3 N o n - s p a r s e s i g n a l s I V . A . 3 N o n - s p a r s e s i g n a l s I V . A . 5 D e s i g n o f m e a s u r e m e n t m a t r i c e s I V . A . 5 D e s i g n o f m e a s u r e m e n t m a t r i c e s I V . C C o m p r e s s i v e E s t i m a t i o n I V . C . 1 P a r a m e t e r e s t i m a t i o n I V . C . 1 P a r a m e t e r e s t i m a t i o n I V . C . 2 L o c a l i z a t i o n I V . C . 2 L o c a l i z a t i o n I V . D S e n s o r M a n a g e m e n t V . C . 1 1 - b i t C S V . C . 1 1 - b i t C S V . C . 2 G e n e r a l q u a n t i z e d C S V . C . 2 G e n e r a l q u a n t i z e d C S V . C . 3 D i s t r i b u t e d / d e c e n t r a l i z e d s o l u t i o n s V . C . 3 D i s t r i b u t e d / d e c e n t r a l i z e d s o l u t i o n s V I . A S c a l a b i l i t y w i t h H i g h - D i m e n s i o n a l H e t e r o g e n e o u s D a t a V I . B D i s t r i b u t e d / D e c e n t r a l i z e d P r o c e s s i n g w i t h Q u a n t i z e d C S V I . C C S B a s e d F u s i o n w i t h M u l t i - M o d a l D e p e n d e n c i e s V I . D F u r t h e r D e v e l o p m e n t s U n d e r P r a c t i c a l C o n s t r a i n t s V I . E T e s t b e d E x p e r i m e n t s a n d P e r f o r m a n c e E v a l u a t i o n I V . A . 4 S u b s p a c e s i g n a l s I V . A . 4 S u b s p a c e s i g n a l s V . D O t h e r I s s u e s Fig. 1: Roadmap of the paper D. Notation and Abbr eviations Throughout the paper , we use the following notation. Scalars (in R ) are denoted by lower case letters and symbols, e.g., x and θ . V ectors (in R N ) are written in boldface lower case letters and symbols, e.g., x and β . Matrices are written in boldface upper case letters or symbols, e.g., A and Ψ . By 0 and 1 , we denote the vectors with appropriate dimension in which all elements are zeros and ones, respectively . The identity matrix with appropriate dimension is denoted by I . The transpose of matrix A is denoted by A T . The i -th row vector and the j -th column vector of matrix A are denoted by a i and a j , respectively . The ( i, j ) -the element of matrix A is denoted by A i,j or A [ i, j ] . The i -th element of vector x is denoted by x [ i ] or x i . The l p norm is denoted by || . || p while | . | is used for both the cardinality (of a set) and the absolute value (of a scalar). The Frobenius norm of A is denoted by || A || F . The Hadamard (element-wise) product is denoted by  while the Kronecker product is denoted by ⊗ . The notation x ∼ N ( µ , Σ) means that the random vector x is distributed as multiv ariate Gaussian with mean µ and co v ariance matrix Σ . The abbre viations used in the paper are summarized in T able II. 5 T ABLE II: Abbre viations used throughout the paper Abbreviation Description Abbreviation Description A CIE Alternating Common and Innovation Estimation IST Iterativ e Soft Thresholding ADM Alternating Direction Method JSM Joint Sparsity Model AFC Analog Fountain Codes LASSO Least Absolute Shrinkage and Selection Operator ALM Augmented Lagrangian Multiplier LMS Least-Mean Squares AMP Approximate Message Passing MA C Multiple Access Channel A WGN Additiv e White Gaussian Noise MAP Maximum a posteriori BCD Block-Coordinate Descent ML Maximum Likelihood BCS Bayesian CS MMV Multiple Measurement V ector BIHT Binary IHT MSP Matching Sign Pursuit BP Basis Pursuit MT -BCS Multitask BCS BPDN Basis Pursuit Denoising NHTP Normalized HTP BSC Binary Symmetric Channel NIHT Normalized IHT BSBL Block SBL NP Neyman Pearson CB-DIHT Consensus Based Distributed IHT OMP Orthogonal Matching Pursuit CB-DSBL Consensus Based Distributed SBL PoI Phenomenon of Interest CoSaMP Compressiv e Sampling Matching Pursuit RIP Restricted Isometry Property CS Compressiv e Sensing RLS Recursiv e Least Squares CRLB Cram ` e r-Rao Lower Bound RSSI Receiv ed Signal Strength Indicator CWS Compressiv e Wireless Sensing R VM Rele vance V ector Machines DBS Distributed BP SBL Sparse Bayesian Learning DC-OMP Distributed and Collaborativ e OMP SCoSaMP Simultaneous CoSaMP DCSP Distrib uted and Collaborative SP SCS Sequential CS DCT Discrete Cosine T ransform SDP SemiDeﬁnite Programming DIHT Distributed IHT SHTP Simultaneous HTP DiOMP Distributed OMP SIHT Simultaneous IHT DiSP Distributed SP SiOMP Side information based OMP DO A Direction of Arriv al SMV Single Measurement V ector DOI Difference-of-Inno vations SNHTP Simultaneous NHTP DR-LASSO Decentralized Row-based LASSO SNIHT Simultaneous NIHT DWT Discrete W avelet Transform SNR Signal-to-Noise Ratio FOCUSS FOCal Underdetermined System Solver SOCP Second-Order Cone Programming FPC Fixed Point Continuation S-OMP Simultaneous OMP GAMP Generalized AMP SP Subspace Pursuit GLR T Generalized Likelihood Ratio T est SR Sparse Representation GMM Gaussian Mixture Model SSP Simultaneous SP GMP Greedy Matching Pursuit TDO A T ime-difference-of-arri val GSM Gaussian Scale Mixture TECC T ranspose Estimation of Common Component GSP Generalized SP VBEM V ariational Bayesian expectation-maximization HTP Hard Thresholding Pursuit VQ V ector Quantizers IHT Iterativ e Hard Thresholding WSN W ireless Sensor Network IoT Internet of Things I I . B AC K G RO U N D W e start our discussion by presenting some background material on CS along with motiv ating factors behind its application in WSNs. A. CS Basics Let x ∈ R N be a discrete time signal v ector . When represented in an appropriate basis Φ ∈ R N × N so that x = Φs , x is said to be sparse (with respect to the basis Φ ) if s contains only a fe w nonzero elements; i.e., || s || 0  N . The support of s (also known as the sparsity pattern/sparse support set) is deﬁned as the set U ∈ { 1 , · · · , N } such that U := { i ∈ { 1 , · · · , N } | s [ i ] 6 = 0 } where s [ i ] denotes the i -th element of s for i = 1 , · · · , N . 1) Compression: In the CS framework, compression of x is performed using the following linear operation: y = Ax (1) where A is a M × N linear projection (measurement) matrix with M < N . In the presence of noise, (1) can be represented in a more general form, y = Ax + v (2) where v denotes the M × 1 additi ve noise vector . With a SMV , one aims to solve for sparse s (equiv alently x since x = Φs with kno wn Φ ) from (1) or (2). 6 Recov ering s from its compressed form y in (1) (or (2)) is ill-conditioned when M < N , howe ver , it has been shown that it is possible to reconstruct s under certain conditions on the measurement matrix A if s is sufﬁciently sparse [26], [28], [29]. Reconstruction of s is exact when there is no noise and approximate when there is noise. 2) Requirements for the measur ement matrix: Several ma- trix properties ha ve been discussed to establish necessary and sufﬁcient conditions satisﬁed by the matrix A so that s can be recovered from y [26], [28], [29], [33]. One such property is the r estricted isometry pr operty (RIP) property . The matrix A is said to satisfy RIP of order k if there exists a δ k ∈ (0 , 1) such that (1 − δ k ) || x || 2 2 ≤ || Ax || 2 2 ≤ (1 + δ k ) || x || 2 2 (3) for all x ∈ Σ k where Σ k = { x : || x || 0 ≤ k } . It has been sho wn that when the entries of A are chosen according to a Gaussian (mean 0 and variance 1 M ), Bernoulli ( + 1 √ M or − 1 √ M with equal probability) or in general from a sub- Gaussian distribution, A satisﬁes RIP with high probability when M = O ( k log( N /k )) [33]. 3) Reconstruction algorithms: In order to recover s from y , the natural choice is to solve the follo wing optimization problem (with no noise) [26], [28], [29], [33]: min s || s || 0 suc h that y = AΦs . (4) Unfortunately , this l 0 norm minimization problem is generally computationally intractable. In order to approximately solve (4), se veral approaches hav e been proposed. One of the com- monly used approaches is to replace the l 0 norm in (4) with a con ve x l 1 norm. Greedy pursuit and iterati ve algorithms are also promising in approximately solving (4). In the following, we brieﬂy discuss these approaches. • Conv ex relaxation: A fundamental approach for signal reconstruction proposed in CS theory is the so-called basis pursuit (BP) [64] in which the l 0 term in (4) is replaced by the l 1 norm to get min s || s || 1 suc h that y = AΦs . (5) Under some fav orable conditions, the solution to (4) coincides with that in (5). In the presence of noise, basis pursuit denoising (BPDN) [64] aims at solving min s || s || 1 suc h that || y − AΦs || 2 ≤  1 (6) and least absolute shrinkage and selection operator (LASSO) [30], [65], [66] solves min s || y − AΦs || 2 suc h that || s || 1 ≤  2 (7) or equi v alently min s λ || s || 1 + 1 2 || y − AΦs || 2 (8) where λ is the penalty parameter and  1 ,  2 > 0 . In order to further enhance the performance of the l 1 norm minimization based approach, the authors in [67] hav e proposed to optimize the reweighted l 1 norm. The reweighted l 1 norm form of LASSO in (8) reduces to min s N X i =1 w i | s [ i ] | + 1 2 || y − AΦs || 2 (9) where w i > 0 denotes the weight at index i . While con vex optimization based techniques are promising in providing optimal and/or near optimal solutions to the sparse signal recovery problem, their computational com- plexity is relativ ely high. For example, the computational complexity of BP when interior point method is used scales as O ( M 2 N 3 / 2 ) [68]. T o reduce computational complexity of sparse signal recovery , greedy and iterativ e algorithms as discussed below ha ve been proposed. • Greedy and iterativ e algorithms: Greedy/iterativ e algo- rithms aim to solve (4) (or its noise resistant extension) in a greedy/iterativ e manner which are in general less computationally complex than the optimization based approaches. Examples of such algorithms include or- thogonal matching pursuit (OMP) [31], subspace pur- suit (SP) [68], Compressive sampling matching pursuit (CoSaMP) [69], iterativ e hard thresholding (IHT) [32], [70] and their variants such as regularized OMP [71], and stagewise OMP [72], Normalized IHT (NIHT) [73] and Hard threshoding pursuit (HTP) [74]. The OMP and SP algorithms can be implemented with a computational complexity in the order of O ( kM N ) [57], [68] while the complexities of CoSaMP and IHT scale as O ( M N ) [57] and O ( M N T r ) [32], [70], respecti vely where T r denotes the number of iterations required by the IHT algorithm for con ver gence. The recovery performance of OMP and IHT is comparable to the l 1 norm minimization based approach when the signal is sufﬁciently sparse ( k is very small) and the SNR is relativ ely large [31], [68], [70]. On the other hand, SP can perform comparable to the l 1 norm minimization based approach ev en with relativ ely large k depending on the distribution of non-zero coef ﬁcients of the sparse signal. • Bayesian algorithms: Another class of sparse recovery algorithms falls under the Bayesian formulation. In the Bayesian frame work, the sparse signal reconstruction problem is formulated as a random signal estimation problem after imposing a sparsity promoting probability density function (pdf) on x in (2). A widely used sparsity promoting pdf is the Laplace pdf [75]. With Laplace prior , x is imposed with the pdf, p ( x | ρ ) =  ρ 2  ρ/ 2 e − P N i =1 | x [ i ] | (10) where ρ > 0 . When the noise v in (2) is modeled as Gaussian with mean 0 and covariance matrix σ 2 v I , the solution in (8) corresponds to a maximum a posteriori (MAP) estimate of x with the prior (10). Computation of the MAP estimator in closed-form with the Laplace prior is computationally intractable, and sev eral compu- tationally tractable algorithms have been proposed in the literature. Sparse signal recov ery using sparse Bayesian learning (SBL) algorithms has been proposed in [76]. In 7 [75], a Bayesian CS framework has been proposed where relev ance vector machines (R VM) [77] are used for signal estimation after introducing a hierarchical prior which shares similar properties as the Laplace prior , yet, pro- viding tractable computation. Babacan et.al. in [78] ha ve also considered a hierarchical form of the Laplace prior of which R VM is a special case. CS via belief propagation has been considered in [79]. Bayesian CS by means of expectation propagation has been considered in [80]. An interesting characteristic of the Bayesian formulation is that it lets one exploit the statistical dependencies of the signal or dictionary atoms while dev eloping sparse signal recovery algorithms. The authors in [81] have considered the problem of sparse signal reco very in the presence of correlated dictionary atoms in a Bayesian framew ork. While Bayesian approaches provide more ﬂexibility in designing recovery algorithms than deter- ministic approaches their computational cost is relati vely high compared to greedy and iterativ e techniques. For example, the computational complexity of SBL scales as O ( N 3 ) [58]. Comparisons of different sparse signal reco very algorithms in terms of computational comple xities and the minimum number of measurements needed can be found in [53], [57]. In the following section, we discuss the motiv ation behind applying CS techniques in WSN applications. B. Applications of CS in W ireless Sensor Networks: Motiva- tion and Challeng es In a WSN deployed to collect ﬁeld information in dif- ferent application scenarios such as environment monitoring and surveillance, gathering sensed information at distributed sensors in an energy ef ﬁcient manner is critical to the operation of the sensor network for a long period of time. Since the energy consumption of distrib uted sensors is mostly dominated by the radio communication [82] and sensing [83], data compression prior to transmission is vital. Further , the data collected at multiple nodes can be redundant in temporal or spatial domains, thus transmitting raw data may be inefﬁcient. Data compression in WSNs has been studied for a long time focusing on redundancy in temporal and/or spatial domains. A nice revie w on different compression schemes proposed for WSNs can be found in [82]. Most of the existing schemes can be categorized as transform coding [82], [84] and distributed source coding [55], [85] which basically suf fer from the requirement of in-network computation and control overheads. T o that end, the CS frame work, as a successful way of data compression not only for data gathering but also for solving other inference tasks, has been found to be attractiv e in the recent years. CS based compression does not require intensiv e computation at sensor nodes and complicated transmission control compared to con ventional compression techniques. In order to further reduce the computation and communication costs at the local nodes, quantized CS [86]–[90] can be utilized. As such, an immediate application of CS in WSNs is com- pressiv e data gathering. When merging CS and data gathering, one of the main issues to be considered is ho w to design the two parts; compression side and the reconstruction side, under communication constraints. In WSNs, data collected at mul- tiple nodes may hav e low dimensional properties in different dimensions; e.g., spatial, temporal, or spatio-temporal. On the compression side, practical design of compression schemes via random projections depends on ho w the sparsity is exploited. CS theory was initially developed for estimating a single sparse signal using SMV [26]–[29], and then it was e xtended to estimate multiple sparse signals using MMVs [34], [35], [91] under certain assumptions. While the CS reconstruction techniques dev eloped for the SMV and MMV cases can be applied for CS based data gathering in ideal situations, e xten- sions to the standard CS framework was required to account for communication related aspects. In early works on applying CS based techniques for data gathering, it was mainly assumed that the sensor nodes communicate with a fusion center , and signal reconstruction is performed at the fusion center [92]. In this approach, multi-hop communication between the sensors and the fusion center incurs a signiﬁcant communication cost. In order to reduce the communication overhead, one of the approaches explored widely is to design sparse, structured or adaptive measurement matrices [93] which can be dif- ferent from ’good’ CS measurement matrices. When using such matrices for compression, it is required to establish the conditions under which the successful recovery is guaranteed. Another approach is to minimize the communication overhead is to perform reconstruction in a decentralized manner which requires sensor nodes to communicate only with their one- hop neighbors. Decentralized processing is attractiv e in WSN applications since it improves the scalability and robustness compared to centralized processing. This approach requires the extension of the CS recov ery algorithms into the decentralized setting. In certain WSN applications where compression via CS seems to be promising, complete signal reconstruction as required for data gathering is not required. For example, in detection, classiﬁcation and parameter estimation problems, it is more important to understand the amount of informa- tion retained in the compressed domain so that a reliable inference decision can be obtained without complete signal reconstruction. In these inference problems with compressed data, the performance and the speciﬁc design principles depend on how the signals or the parameters are modeled. Further , the robustness and the accurac y of the CS framew ork under other practical communication considerations such as fading channels, secrec y issues, measurement quantization, link fail- ures and missing data need to be understood to make CS based techniques feasible in WSN applications. In the recent literature, the CS framework has been extended to cope with a variety of practical aspects. The rest of the sections are dev oted to a discussion of recent ef forts of such extensions in some detail. I I I . C S F O R D I S T R I B U T E D D A TA G AT H E R I N G In this section, we discuss the CS based data gathering problem formulated as a sparse signal reconstruction problem 8 Proj A 1 Proj A 2 Proj A L  [  [ / [ Sensor 1 Sensor 2 Sensor L    [ $ \    [ $ \ / / / [ $ \ Fig. 2: Acquisition of compressed measurements of observa- tions with temporal sparsity exploiting temporal, spatial and spatio-temporal sparsity in WSNs. W e describe the modiﬁcations and extensions to the CS framew ork to take communication constraints into account fo- cusing on compression side as well as the recovery algorithms considering centralized and distributed/decentralized settings. A. CS Based Data Gathering Exploiting T emporal Sparsity Consider a WSN with L sensor nodes observing a PoI as illustrated in Fig. 2. The time samples collected at the j -th node are represented by the v ector x j ∈ R N . In data gathering, sensor readings need be collected efﬁciently . Transmitting raw data vectors x j ’ s is inefﬁcient in many applications since sensor data is compressible with many types of sensors. Data collected at acoustic, seismic, IR, pressure, temperature, etc., can hav e sparse representation in a certain basis. For example, audio signals collected by acoustic sensors (microphones) can be sparsely represented in DCT and D WT [94]. Formally , with an orthonormal basis Ψ j ∈ R N × N , when x j is represented as x j = Ψ j s j for j = 1 , · · · , L , x j is said to be sparse when s j contains only a few nonzero elements compared to N . Thus, CS is readily applicable to compress temporal sparse data. In particular, only a small number of random projections obtained via y j = A j x j at the j -th node for j = 1 , · · · , L (Fig. 2) are sufﬁcient to be transmitted where A j ∈ R M × N , M < N is the measurement matrix used at the j -th node. When applying CS to compress temporal sparse data at a giv en node, the data streams collected at the node are compressed via random projections independently . The goal is to reconstruct [ x 1 , · · · , x L ] based on their compressed v ersions communicated through the network where the reconstruction techniques depend on the speciﬁc communication architecture used to combine y j ’ s. 1) Centralized solutions for simultaneous sparse signal r e- covery exploiting temporal sparsity: First, we focus on recov- ering X ≡ [ x 1 , · · · , x L ] jointly based on Y = [ y 1 , · · · , y L ] in a centralized setting. In this setting, the nodes transmit their compressed measurements to a central fusion center with long- range single-hop communication. The MMVs collected at the fusion center in matrix form with the same projection matrix so that A j = A for j = 1 , · · · , L can be represented by Y = AX . (11) For the more general case with dif ferent projection matrices, the observation matrix at the fusion center can be expressed as Y = [ A 1 x 1 , · · · , A L x L ] . (12) In the simultaneous sparse approximation framework, X needs to be reconstructed when Y (or its noisy version) and A (with the same projection matrix) or A 1 , · · · , A L (with different matrices) are giv en. In order to jointly estimate X , joint sparse structures of X can be exploited. There are sev eral such structures discussed with applications to sensor networks [55], [95], [96]. It is worth noting that estimation of X sometimes refers to estimating the support of X by some authors. While the techniques and performance guarantees are not the same for estimating X and its support, sometimes it is sufﬁcient to estimate only the support jointly . This is because, once the support set is known, estimating indi vidual coef ﬁcients reduces to a linear estimation problem. Common support set model (JSM-2): The widely consid- ered joint sparse model for sensor network data is the common support set model which is termed as the JSM-2 model in [55]. In this model, the sparse signals observed at multiple nodes, x j ’ s, have the same but unknown sparsity pattern with respect to the same basis. Howe ver , the corresponding amplitudes can be different in general. The JSM-2 model with the same measurement matrix as in (11) is commonly termed as the MMV model [91], [97]. Without loss of generality , in the rest of the section, we assume that x j ’ s are sparse in the standard canonical basis unless otherwise speciﬁed. While dev eloping algorithms and ev aluating performance with the JSM-2 model, sev eral measures have been deﬁned. T o measure the number of nonzero elements of X , || · || row − 0 norm is widely used where || X || row − 0 = | ro wsupp( X ) | (13) with ro wsupp( X ) = { i ∈ [1 , · · · , N ] : X i,j 6 = 0 for some j } . (14) The natural approach to solv e for sparse X from Y in (11) is to solve the following optimization problem: min X || X || row − 0 suc h that Y = AX (15) with no noise or min X || X || row − 0 suc h that 1 2 || Y − AX || F ≤  (16) in the presence of noise. Since the problem (15) (and (16)) is NP hard, one often solves (15) (and (16)) by using the mix ed norm approach which is an extension of the conv ex relaxation method for the SMV case to the MMV case. Con vex relaxation: A large class of relaxation versions of || X || row − 0 aims at solving an optimization problem of the following form [54]: min X J p,q ( X ) suc h that Y = AX (17) in the noiseless case and min X 1 2 || Y − AX || 2 F + λJ p,q ( X ) (18) 9 in the noisy case (also known as R-LASSO [98]) where λ is a penalty parameter and J p,q ( X ) = X i ( || x i || p ) q (19) where typically p ≥ 1 and q ≤ 1 . It is noted that the value q promotes the common sparsity proﬁle of X , p is a measure to weight the contribution of multiple signals to the common sparsity proﬁle and (19) is con ve x whenev er q = 1 and p ≥ 1 . Different approaches hav e been discussed to solve (17) and (18) for different values of p and q . The most widely consid- ered scenario is p = 2 and q = 1 which is commonly known as M-BP [53], [54], [97], [99]–[104]. This case is also dubbed as the mixed l 2 /l 1 norm minimization approach. When L = 1 , this case reduces to the BP (or LASSO) formulation in (5) (or (8)). The works in [53], [54], [97], [99]–[104] have focused on dev eloping different algorithms, and establishing recov ery guarantees. In [97], M-FOCUSS (FOCal Underdetermined System Solver) has been dev eloped for the noiseless case, and the regularized M-FOCUSS for the noisy case. M-FOCUSS is an iterative algorithm that uses Lagrange multipliers and is also applicable when q ≤ 1 . The average case analysis on recov ery guarantees using multichannel BP has been discussed in [101]. In [102], an alternating direction methods (ADM) based approach has been proposed to solve (18) which is called MMV -ADM. The M-BP problem as a special case of group LASSO, with the block coordinate descent algorithm called M-BCD, has been discussed in [54], [103], [104]. In [100], the mixed l 2 /l 1 type norm minimization problem has been solved via a semi-deﬁnite program which is sho wn to reduce to solving a second-order cone program (SOCP)-a special type of semi-deﬁnite program. In [34], the case where p = ∞ and q = 1 has been considered and the authors hav e discussed the conditions under which the con vex relaxation is capable of ensuring recovery guarantees. Some known theoretical results on recov ery guarantees of the SMV case have been generalized to the MMV case in [91] considering q = 1 and arbitrary p . Recovery guarantees using fast thresholded Landweber algorithms considering p = 1 , 2 , ∞ , and q = 1 have been discussed in [105]. A comparison of different simultaneous sparse approximation methods with different values for p and q can be found in [34], [54]. Gr eedy algorithms: In order to solve (15), se veral greedy and iterati ve algorithms have been proposed which typically hav e less computational complexity than conv ex relaxation based approaches. In particular , most of these approaches are extensions of their SMV counterparts. The extension of the OMP algorithm to the MMV case with the common support set model, S-OMP , has been considered in [35]. Performance analysis of the S-OMP algorithm including noise rob ustness has been presented in a recent paper [124]. An MMV e xtension of the IHT algorithm, SIHT has been considered in [107], [108]. Some variants of SIHT such as SNIHT , and extensions of other greedy algorithms such as HTP , NHTP , and CoSaMP hav e also been discussed in [107], [109]. Generalized SP as an extension of the SP algorithm with MMVs has been considered in [110] which is shown to outperform the natural extension of SP , i.e., simultaneous SP (S-SP). Bayesian algorithms: Solutions to the MMV problem with the JSM-2 model in the Bayesian setting ha ve been discussed in [111]–[117], [120]. The MSBL algorithm, which is an ex- tension of the SBL algorithm with SMV , has been de veloped in [111]. Multitask BCS (MT -BCS) has been developed in [115] in which applications of multitask learning to solve the MMV problem in the Bayesian framework have been discussed. In [116], the authors have extended the MT -BCS frame work to take the intra-task dependency into account. In [117], the MMV problem in a Bayesian setting has been considered focusing on DOA estimation where the prior probability distribution is modeled with a Gaussian scale mixture (GSM) model. The proposed approach is dubbed M-BCS-GSM. Belief propagation based methods ha ve been proposed in [112], [113] in which the approximate message passing (AMP) framework is used to jointly estimate the sparse signals. In [114], the MMV problem has been solv ed when the nonzero elements of a giv en column of X are correlated and the two Bayesian learning algorithms, called T -SBL and T -MSBL hav e been dev eloped. Other appr oaches: The MMV problem with the JSM-2 model has been treated as a block sparse signal recov ery prob- lem after v ectorizing X to a block sparse signal in [119]. Then, the algorithms developed for block sparse signal recovery with SMV can be used to solve the MMV problem. In [120], block SBL (BSBL) algorithms, which take intra-block correlations into account, hav e been developed where the MMV problem is treated as a block sparse signal recov ery problem. In [118], the authors ha ve sho wn that most simultaneous sparse recovery algorithms discussed above are rank blind. They have proposed rank aware algorithms with mixed norm minimization as well as greedy algorithms which show better performance than rank blind algorithms under certain conditions. In [125], real temperature, humidity and light data hav e been used to validate some algorithms de veloped under the JSM-2 model exploiting spatial and time correlations among network data. An adaptiv e sparse sensing framework has been proposed in [126] which precisely recovers spatio-temporal physical ﬁelds by optimizing for sparse sampling patterns. Common support set + innovation model (JSM-1): Another widely considered joint sparse support set model with many WSN applications is the common support set + innovation model, which is termed as the JSM-1 model in [55]. It is assumed that each sparse signal x j can be expressed as x j = x C + ˜ x j where x C is a common (to all x j ’ s) component which is sparse and ˜ x j is called the innov ation component which is also sparse but different from x C ’ s sparsity pattern. Recov ery of X with the JSM-1 model has been discussed in [55]. In [55], weighted l 1 norm minimization for jointly estimating X is discussed. Considering the signal vector at one node as side information, the authors in [123] ha ve de- veloped the Difference-Of-Inno vations (DOI) and T exas DOI algorithms for simultaneous sparse signal approximation for the JSM-1 model. In [121], centralized (as well as distributed) implementation of the ADM algorithm with MMV has been presented. In [122], side information based OMP (SiOMP) has been proposed. In SiOMP , the distributed CS is performed for the JSM-1 model where the estimate at one node via OMP is 10 T ABLE III: Centralized solutions for joint sparse signal recovery with MMVs exploiting temporal sparsity Approach Algorithms and References JSM-2 Con ve x relaxation, p = 2 , q = 1 in (19) Group LASSO (with BCD) [54], [103], [104], SDP-SOCP [100], MMV -ADM [102], MMV prox [106] Con ve x relaxation, p = 2 , q ≤ 1 in (19) M-FOCUSS [97] Con ve x relaxation, p = 1 , 2 , ∞ , q = 1 in (19) Landweber algorithms [105] Con ve x relaxation, p = ∞ , q = 1 in (19) [34] (Algorithm is implemented via standard mathematical software) Greedy and iterative S-OMP [35], SIHT , SNIHT [107], [108], SHTP [109], SNHTP [107], SCoSaMP [107], Generalized SP [110] Bayesian MSBL [111], MMV -AMP [112], [113], T -MSBL [114], MT -BCS [115], [116], M-BCS-GSM [117] Other approaches Rank-aware algorithms [118], block sparse signal recovery based algorithms [119], BSBL [120] JSM-1 Con ve x relaxation weighted l 1 norm minimization [55], l 1 norm minimization using ADM [121] Greedy SiOMP [122] Greedy/optimization based DOI and T exas DOI [123] (to estimate sparse innov ation components, greedy/optimization based recovery algorithms can be used, valid for JSM-3 model as well) JSM-3 Greedy/optimization based TECC and ACIE [55] (to estimate sparse innovation components, greedy/optimization based recovery algorithms can be used) considered to be the initial value for the estimate at the next node. Although not as interesting as the JSM-1 and JSM-2 models, some works can be found for the JSM-3 model discussed in [55]. In the JSM-3 model, the signal observed at each node is assumed to be composed of a nonsparse common component + sparse innov ation component. Applications of this model in sensor network settings hav e been discussed in [55]. Further , two algorithms called Transpose Estimation of Common Component (TECC) and Alternating Common and Innov ation Estimation (ACIE) have been dev eloped in [55]. In these algorithms, the nonsparse common support is computed jointly and using that the sparse innovation components are computed by running sparsity aware algorithms. The algo- rithms developed in [123] for the JSM-1 model are applicable to the JSM-3 model as well. In T able III, we summarize the simultaneous sparse signal recov ery algorithms classifying under dif ferent categories as discussed abov e with MMVs for dif ferent JSM models. 2) Decentralized solutions for simultaneous sparse signal r ecovery exploiting temporal sparsity: While centralized so- lutions are attractive in terms of the performance, their imple- mentation can be prohibitiv e in resource constrained WSNs since the power costs for long-range transmission can be still quite high. Further, centralized solutions are not robust to node and link failures. Decentralized solutions are attractiv e and are sometimes necessary in resource constrained sensor networks in which distributed nodes e xchange (locally processed) mes- sages only with their neighbors. They can signiﬁcantly reduce the overall communication power and bandwidth compared to centralized processing. In the decentralized setting, each node is required to obtain the centralized solution (or an approximation to it) by only communicating within one-hop neighbors. Since the initial CS frame work was de veloped for the SMV or the MMV case with centralized architecture, there was a need to extend it to the distributed and decentralized framew ork to address the necessities arising out of WSN applications. In de veloping decentralized algorithms exploiting temporal sparsity , where the goal is to estimate a set of sparse signals using their compressed versions, the joint sparse structures play an important role. In fact, most of the decentralized solutions dev eloped so far consider the JSM-2 model for multiple sparse signals. First, we revie w the decentralized extensions of the optimization based algorithms where the goal is to solve equations of the form (17) or (18) in decentralized manner . Optimization based approac hes: W ith the JSM-2 model, the matrix X has only a small number of nonzero rows. In other words, when the common support set is estimated jointly , the indi vidual estimates of x j ’ s can be estimated by the j - th node. For the JSM-2 model with different measurement matrices at multiple nodes, the ro w-based LASSO (R-LASSO) formulation has been extended to the decentralized setting in [98]. Recall that R-LASSO in (18) with different measurement matrices reduces to min X 1 2 L X l =1 || y l − A l x l || 2 2 + LλJ 1 /p p,q ( X ) (20) where J p,q ( X ) is as deﬁned in (19). In [98], a decentralized implementation of (20) has been proposed with p = 2 and q = 1 . In particular , the authors have reformulated (20) as a consensus optimization problem and dev eloped an itera- tiv e algorithm. It is noted that consensus optimization is a powerful tool that can be utilized in designing decentralized networked systems with distributed nodes. In [98], the l -th node estimates the common support set and the individual coefﬁcient vectors by solving a local optimization problem via augmented Lagrangian multiplier (ALM) method based on the information recei ved from its local neighborhood. The algorithm requires each node to communicate a length N message at each iteration to its one-hop neighbors, thus the communication burden is proportional to the total number of iterations required and N . In [127], with the same problem 11 model as in [98], a non-conv ex optimization approach has been discussed extending the reweighed l q norm minimization approach proposed in [67] for simultaneous sparse signal recov ery in a decentralized manner . Similar to [98], the authors hav e reformulated the decentralized sparse signal recovery problem as a consensus optimization problem and only a length N weight v ector needs to be e xchanged among neigh- boring nodes during each iteration. In [127], the ADM method has been used to solve the corresponding optimization problem at a giv en node. A decentralized extension to the reweighted l 1 norm minimization approach has also been discussed in a recent work in [128], where the authors propose to solve the problem using the iterativ e soft thresholding algorithm (IST). This algorithm requires each node to transmit a length N messages in its local neighborhood at each iteration and the number of iterations depends on the IST con ver gence rate. A special case of the JSM-2 model is the case where all the nodes observe the same sparse signal (same support and same coefﬁcients). W e refer to this case as the common signal model where there is only a single sparse signal to be estimated with MMVs. Decentralized algorithms for this model hav e been dev eloped in [129]–[133]. It is worth noting that the work reported in [130], [132] is application speciﬁc to spectrum sensing in cognitive radio networks, while the algorithms are applicable for WSNs with MMVs. In [132], [133], the distributed LASSO (D-LASSO) algorithm has been employed to compute the common LASSO estimator by collaboration among nodes. In [130], a similar formulation and approach has been adopted as in [132], ho wev er, their algorithm has faster con ver gence rate via proper fusion. In [129], [131], it is assumed that each node has access to only partial information on random projections of the common signal. In [129], a decentralized gossip based algorithm has been dev eloped to implement the l 1 norm minimization approach in a decen- tralized manner . Each node solves the optimization problem via the projected-gradient approach by communicating only with one-hop neighbors. Formulating the l 1 norm minimiza- tion problem as a bound-constrained quadratic program, the gradient computation at each step of the coordinate descent algorithm is expressed as a sum of quantities computed at each node applying distributed consensus algorithm. In [131], distributed basis pursuit (DBS) has been dev eloped based on the ADM method. There are a few works that extend the optimization tech- niques considering the JSM-1 model to the decentralized set- ting. In [121], the authors have proposed a distributed version of the ADM algorithm (dubbed as D ADMM) to implement an equiv alent version of (20) in a decentralized manner where communication is only with one hop neighbours. Gr eedy/iterative appr oaches: Distributed and decentralized versions of greedy/iterati ve algorithms can be found in [56], [122], [134]–[137], [141], [144], [145]. In [134], [137], dis- tributed SP (DiSP) and Distributed OMP (DiOMP) have been dev eloped which are applicable to both JSM-1 and JSM-2 models. In DiSP , at each iteration an estimate to the support set is updated based on the local support set estimates of the neighboring nodes. In each iteration, it is required to transmit the whole support set among the neighborhood. It is noted that, the number of iterations required for conv ergence depends on the network topology and is fairly close to the sparsity index. In [140], the distributed parallel pursuit (DIPP) algorithm has been proposed for the JSM-1 model. In DIPP , fusion is performed to improve the estimate of the common support set via local communication, and the estimated support set is used as side information for complete signal reconstruc- tion. Embedding fusion within OMP iterations, in [136], the authors have proposed two decentralized versions of the OMP algorithm, called DC-OMP 1 and DC-OMP 2 for the JSM- 2 model. In DC-OMP 1, an estimate of the support index is computed similar to the standard OMP and those computed indices at all the nodes are fused at each iterations to get an more accurate set of indices for the support set. In DC-OMP 2, fusion is performed at two stages; to improve the initial estimate of the support set at each node, measurement fusion is done within the one-hop neighborhood. Similar to DC-OMP 1, index fusion is performed to get a more accurate index set. Both DC-OMP 1 and DC-OMP -2 can terminate with less than k number of iterations and require multi-hop communication for the index fusion stage since global knowledge is required. In [138], [139], the distributed and collaborativ e subspace pursuit (DCSP) algorithm has been developed for the JSM- 2 model. The ideas de veloped in [138], [139] are similar to that in [136] with OMP , ho wev er, communication overhead requirements are slightly higher in [138], [139] than that for DC-OMP 1 and DC-OMP 2. The algorithm requires each node to communicate with its neighbors twice in each iteration to update the common support set. Distributed IHT (D-IHT) and a consensus based distributed IHT named (CB-DIHT) hav e been proposed in [141], [145] for the common signal model. Bayesian appr oaches: Distributed and decentralized Bayesian algorithms for sparse signal recovery have been proposed in [142], [143], [146]. In [142], an approximate message passing (AMP) based decentralized algorithm, AMP- DCS, is dev eloped to reconstruct a set of jointly sparse signals with JSM-2. The sparse signals are assumed to hav e sparsity inducing Bernoulli-Gaussian signal prior . The algorithm requires each node communicate M N messages at each iteration. In [146] a decentralized Bayesian CS framework has been proposed for the JSM-1 model where applying variational Bayesian approximation, the common support component and the inno v ation component are decoupled. Formulating the decoupled reconstruction problem as a set of decentralized problems with consensus constraints, each node estimates its innovation component independently and the common component jointly via local communication. In [143], a Consensus Based Distributed Sparse Bayesian Learning (CB-DSBL) algorithm has been proposed. The authors also exploit ADM to solve the consensus optimization problems in the sparse Bayesian learning framework at each node. Analysing differ ent types of decentralized solutions: The abov e discussed decentralized algorithms are summarized in T able IV. The three cases in T able IV correspond to the JSM-2 model, common signal model and the JSM-1 model. Selection of one algorithm ov er the other mainly depends on the desired performance requirements and tolerable communication and 12 T ABLE IV: Decentralized estimation of sparse signals with multiple measurements exploiting joint sparse structures; Different cases are for different joint structures of multiple sparse signals, Case 1: JSM-2 with different coefﬁcients, Case 2: JSM-2 with the same coef ﬁcients (common signal model), Case 3: JSM-1 Algorithm/References Applicability Commun. complexity Features/average computational complexity per node ( # of transmitted messages per node ) Optimization based DR-LASSO [98] Case I O ( N T 1 I t ) ALM to solve the optimization problem / O ( I t ( N 2 M T 1 + n 0 N T 2 )) reweighted l 1 norm min. [128] Case I O ( N I t ) IST to solve the optimization problem/ O ( I t ( N 2 M + N 2 + N M )) reweighted l q norm min. [127] Case I O ( N I t ) Use non-conv ex sum-log-sum penalty , ADM to solve the optimization problem / O ( I t (( N 2 + M 3 + N M 2 ) T 3 + n 0 N )) Distributed ADMM [121] Case III O (2 N I t ) ADM complexity D-LASSO [130], [132] Case III O ( N I t ) ADM complexity Distributed BP [131] Case II O ( N I t ) or O ( N T 3 I t ) ADM complexity l 1 norm min. [129] Case II O ( N I t ) Projected-gradient approach/dominated by O (( N 2 n 0 + N ) I t ) Greedy and iterative DiOMP [134] Case I/Case III O ( k 2 ) O ( k c ( T 4 M N )) DC-OMP 1 [135], [136] Case I O ( I † t ) Fusion of estimated indices at each iteration / O ( I t ( M N + L )) DC-OMP 2 [136] Case I O ( I † t + N I t ) Fusion of estimated indices as well as correlation coefﬁcients in the neighborhood at each iteration / O ( I t ( M N + n 0 N + L )) DiSP [137] Case I/Case III O ( kI t ) SP operations, fusion via majority rule/ O ( I t kM N ) DCSP [138], [139] Case I O ( kI t ) SP operations, fusion via majority rule/ O ( I t ( M N + n 0 )) GDCSP [138], [139] Case I O (( k + N ) I t ) SP operations, fusion of indices, measurements/ O ( I t ( M N + n 0 N )) DIPP [140] Case III O ( kI t ) Modiﬁcation of SP , fusion via consensus, expansion/ O ( I t ( N M T 5 + kn 0 )) D-IHT [141] Case II O ( N † I t ) IHT operations/ O ( M p N I t ) Bayesian DCS-AMP [142] Case I O ( M N I t ) Bernoulli Gaussian signal prior is used/ O ( I t ( M N + n 0 N )) CB-DSBL [143] Case I O ( N T 3 I t ) O ( I t (( N 2 + M 3 + N M 2 + N T 3 n 0 )) 1) N : dimension of the unknown sparse signals , M : number of compressed measurements per node, k : number of nonzero elements of the sparse signals in JSM-2 (sparsity index), k c ≤ k : number of nonzero elements with commons support set in JSM-1, L : number of total nodes in the network, n 0 : av erage number of on-hop neighbors per each node 2) I t denotes the number of iterations required for conv ergence which is a algorithm dependent variable 3) T 1 and T 2 denote the number of inner loop iterations required by DR-LASSO [98] 4) T 3 : a variable to denote inner loop iterations in ADM 5) T 4 : number of inner loop iterations in DiOMP 6) T 5 : number of inner loop iterations in DiPP 7) In DC-OMP 1, DC-OMP 2, DCSP , and GDCSP: I t ≤ k 8) † denotes that messages need to be communicated globally via multi-hop communication 9) M p ≤ M : number of partial measurements at each node computational complexities. In terms of computational and communication complexities, it is worth mentioning that al- most all the optimization based techniques require a relatively large number of iterations to conv erge. Thus, the commu- nication cost per each node is relativ ely high. Further, the computational cost at each node scales at least quadratic with respect to N resulting in a relati vely large computational cost when dealing with high dimensional signals. Similarly , the de- centralized versions of Bayesian algorithms also require a rela- tiv ely high computational cost in processing high dimensional signals. On the other hand, most of the greedy approaches require only few iterations for algorithm termination [134], [136]–[139] that are comparable to the sparsity inde x k . This results in relati vely small communication o verhead compared to implementing optimization based algorithms in a decentral- ized manner . In particular , DC-OMP 1 and DC-OMP 2 have a faster conv ergence rate which require less than k number of iterations. Further, the in-node computational complexity of greed algorithms is much smaller (mostly linear with N ) than that with optimization based and Bayesian approaches. In terms of performance, optimization based techniques with a suitable penalty parameter performs better than the other types of algorithms especially when k is large and SNR is relativ ely small. While both Bayesian and optimization based methods show similar order of computational complexity per node, Bayesian methods are more ﬂexible in terms dependency on parameters than the optimization based techniques. Bayesian CS techniques can be implemented fully automated since all the unknown parameters are estimated during the execution of the algorithm while the optimization based techniques require the tuning of parameters such as penalty parameter and noise statistics for optimal performance. Thus, the optimization based techniques may require a larger communication ov er- head than the Bayesian techniques since the communication ov erhead depends on the number of iterations of the algorithm. Moreov er , the Bayesian approach provides a framew ork to 13 SN 1 SN 2 SN L 1 x L x 2 x 1 1 , x i A 2 2 , x i A L L i x , A ¦ i v i y Fusion Center i k k i L k i v x y  ¦ , 1 A Fig. 3: MA C model for CS based data gathering exploiting spatial sparsity [84], [147] account for the spatial and temporal statistical dependencies of the joint sparse signals. Comparing all types of algorithms, the greedy and iterative algorithms appear to be quite promising in decentralized processing of temporal sparse signals under sev ere network resource constraints although a certain per- formance loss can be expected compared to the optimization based and Bayesian approaches. B. CS Based Data Gathering/Reconstruction Exploiting Spa- tial Sparsity T o exploit CS techniques for data gathering e xploiting spatial sparsity , compression of spatial data (over the nodes) via random projections at a giv en time step is needed. In contrast to temporal data compression as considered in Section III-A where indi vidual nodes use random projection matrices independently , in this case, random projections hav e to be implemented in a distributed manner . Several architectures hav e been explored to implement distrib uted random projec- tions so that a compressed version of sparse spatial data is made av ailable at the fusion center . One of the architectures, with one-hop communication between the sensors and the fusion center is kno wn as compressi ve wireless sensing (CWS) as proposed in [84], [147]. The CWS framework enables distributed compression with random projections employing synchronous multiple access channels (MA Cs). W ith MA C, individual nodes transmit their scaled observ ations to the fusion center coherently . This architecture is depicted in Fig. 3 for one MA C transmission. W ith this model, the observation at the j -th sensor, denoted by x j , is multiplied by a scalar factor , and transmitted to the fusion center via a MAC channel. The receiv ed signal at the fusion center after the i -th MA C transmission is gi ven by , y i = L X j =1 A i,j x k + v i (21) where A i,k is the scalar factor , and v i is the recei ver noise. After M < L such MA C transmissions, the observation vector at the fusion center has the form of (2) where the ( i, j ) -th element of A is giv en by A i,j for i = 1 , · · · , M and k = 1 , · · · , L , x = [ x 1 , · · · , x L ] T and v = [ v 1 , · · · , v M ] T . W ith this model, the fusion center receiv es a compressed version of the sparse (or compressible) signal x . In worth noting that, to av oid ambiguity in the paper , in this section, we denote the signal under compression as a length L vector in contrast to a length N vector as considered in Section III-A. Note that the CWS architecture requires synchronization among multiple sensors during each MA C transmission. T o alleviate this requirement, another architecture widely con- sidered is employing random projections through multi-hop transmission [92], [148]–[151]. In this framework, the data aggregation process is implemented with multi-hop routing as illustrated in Fig. 4. In [92], [148], [151], a tree based architecture is used to implement the multi-hop routing pro- tocol so that the fusion center recei ves the observation vector y = Ax . In the baseline multihop routing approach (without any compression), the j -th sensor transmits its reading x j along with the messages recei ved from the previous node to the next node, thus, the nodes located closer to the sink consume a large amount of energy . In contrast, in the CS based multihop routing approach in Fig. 4, each node has to transmit only M = O ( k log L ) messages where k is the sparsity inde x with respect to an appropriate basis as deﬁned before. In both single-hop or multi-hop architectures, the problem of compressive data gathering exploiting spatial sparsity can be formulated as a CS recovery problem at the fusion center with a SMV . The works reported in [84], [147] and [92] are the ﬁrst fe w papers that demonstrated the savings in commu- nication and computations costs using direct application of CS techniques in large scale networks in contrast to the traditional sample-then-process approach. Rob ustness of CS based data gathering with this set-up in the presence of link failures and outlying sensor readings is further studied in [151]. While directly applying random projections to compress spatial data sav es communication po wer costs compared to gathering raw data, still the transmission power costs can be high since all the nodes participate in the data aggregation process. In particular, when a dense Gaussian matrix is used for compression in the multi-hop routing architecture, the communication cost in terms of the total number of message transmissions and/or the maximum number of message transmissions of any single node can be high. In order to take limited communication and energy resources into account in CS based data gathering ex- ploiting spatial sparsity in large scale sensor networks, further research has been done mainly focusing on controlling the amount of information transmitted by each node (equiv alently designing projection matrices) and de veloping reconstruction algorithms. 1) Centralized pr ocessing: Use of sparse matrices for spa- tial data gathering: When dense random projections such as Gaussian random variables are used, each node has to forward all of its scaled observations which can incur a signiﬁcant transmission power cost. When the projection matrix is made sparse, not all the sensors need to transmit all the local data under both single-hop and multi-hop architectures. The use of sparse random projections has the potential of reducing both communication and computational costs at sensor nodes. In CS theory , the signal recovery guarantees with sparse random matrices are widely discussed [51], [93], [152]–[157]. It is noted that many desirable properties that need to be satisﬁed to enable reliable reconstruction are violated when the projection matrix is made very sparse. Nev ertheless, the authors in [157] 14 SN 1 SN 2 SN 3 SN L Fusion Center/ Sink ¦ ¦ ¦ L j j j L j j j L j j j M x x x 1 , 1 1 , 2 1 , . . A A A ¦ ¦ ¦ 2 1 , 1 2 1 , 2 2 1 , . . j j j j j j j j j M x x x A A A 1 1 , 1 1 1 , 2 1 1 , . . x x x M A A A Fig. 4: Compressi ve data gathering with multi-hop routing e xploiting spatial sparsity [92], [148] hav e shown that very sparse random matrices can be used for data compression with a small reco very performance loss. Some widely considered sparse random projections are sparse Bernoulli and sparse Gaussian matrices as deﬁned belo w . In sparse Bernoulli matrices, the ( i, k ) -th element of A is drawn from [93] A i,k = r 1 γ    1 with prob γ 2 0 with prob 1 − γ − 1 with prob γ 2 (22) while in sparse Gaussian matrices A i,k is chosen as [155] A i,k =  N (0 , 1 γ ) with prob γ 0 with prob 1 − γ (23) with 0 < γ < 1 . The CWS scheme with sparse random projections in a centralized architecture is depicted in Fig. 5 for one MA C transmission. Using the sparse matrix (22), it has been shown that it is sufﬁcient to hav e O ( poly ( k , log L )) random projections (equiv alently MA C transmissions) to reconstruct the original signal with high probability when only O (log L ) nodes on an av erage transmit their scaled observations during a gi ven MA C transmission [93]. Instead of randomly selecting participating nodes as in [93], design of which nodes to transmit or equiv alently constructing routing trees minimizing the ov erall energy consumption has been considered in [158]. While the optimal solution is NP-hard, sev eral suboptimal approaches hav e been proposed with signiﬁcantly less amount of data to be transmitted compared to the case where raw data forwarding. The minimum number of MA C transmissions required to en- able signal reconstruction with high probability in the presence of fading has been analyzed by [153] using the matrix (22) while the authors in [154] hav e provided a similar analysis with the matrix (23) assuming phase coherent transmission. Impact of fading channels on signal reconstruction is further discussed in Section V -A. In [148], design of the matrix A so that the total commu- nication cost kept under a desired le vel has been in vestigated. The matrix should be designed so that reliable recovery is guaranteed. In particular, the authors have proposed to split the measurement matrix and incorporated sparsity to one side so that the RIP property is not signiﬁcantly violated. Thus, the designed matrix with smaller communication overhead is able to provide closer performance to the case using a dense matrix which consumes a large communication b urden. The authors in [149] have also considered multi-hop routing based CS data gathering, considering more general network models. In particular , a single sink as well as multi-sink models were considered. The capacity; the maximum rate that the sink can receiv e the data generated at nodes, and the delay; the time between the data sampling at nodes and the recei ving at the sink were analyzed with both single sink and multi- sink models exploiting sparse random projections. In [159], the measurement matrix has been made as sparsest as possible where each ro w contains only one element. Since this matrix is not capable of providing reliable reco very guarantees when representing sensor data in a common transform basis, the au- thors hav e designed the transform basis so that the correspond- ing projection matrix is capable of providing reliable recov ery guarantees. While this method reduces the communication cost signiﬁcantly it adds computational cost in designing the transform basis as required for signal reconstruction. In [150], a non-uniform version of the sparse Bernoulli matrix (22) has been used where each column is generated using sparsity parameter 0 < γ j < 1 for j = 1 , · · · , L . Recovery guarantees with such non-uniform sparse matrices hav e been established in [150]. In terms of recovery performance, it is comparable to the case with direct application of CS as in [92] with large M , howe ver , it leads to a performance de gradation compared to [92] when M is small. Incorporating practical constraints such as interference in adjacent transmissions into sparse projections based data gathering frame work has been addressed in [160], [161]. Hybrid and adaptiv e approaches to design the measurement matrices and/or the reconstruction algorithms to minimize communication cost in data gathering exploiting spatial spar - sity are also attracti ve. Recent de velopments of adapti ve CS are exploited in these approaches which will be discussed next. 2) Centralized pr ocessing: Hybrid and adaptive appr oaches for CS based spatial data gathering: In [162], a hybrid data aggregation approach has been proposed combining the energy efﬁcienc y aspect of applying CS to data collection in WSNs focusing on multihop networking scenarios. In particular, by joint routing and compressed aggregation, the energy con- sumption is reduced compared to applying CS directly as in [92]. Ideas of adapti ve compressive sensing [163], where the goal is to design projections in an iterative manner to maximize the amount of information and minimize the num- ber of projections, have been exploited in [164], to design 15 Fusion center ¦ i v i y 1 1 x i A 5 5 x i A m im x A Distributed sensors Fig. 5: Compressive W ireless Sensing with sparse random projections; only few sensors transmit during a given MAC transmission. projection matrices. In particular, the subsequent projections hav e been designed taking into both energy constraints and the information gain into account so that the desired reco very performance is achiev ed at the sink. Simulated as well as testbed (to measure temperature) setups have been used to adaptiv ely collect sensor data in [164]. Adaptiv e techniques for CS based data gathering ha ve also been explored in [165]–[167]. In [165], both measurement scheme and the reconstruction approach are adapti vely updated to cope with the variation of sensed data as well as the application require- ments. T estbed data has been used to illustrate the v ariation of senor readings due to external event and internal errors as well as to show the performance of the proposed adapti ve techniques. In [166], a testbed for wildﬁre monitoring has been used to illustrate the performance of the proposed adaptiv e data compression scheme exploiting spatial sparsity . In [167], a feed-back controlled adapti ve technique has been proposed to cope with the v ariation of sensor readings. The performance has been ev aluated using a testbed designed to monitor the luminosity of the environment. In T able V, we summarize the centralized data gathering approaches discussed abo ve exploiting spatial sparsity . 3) Decentralized solutions for sparse signal reco very ex- ploiting spatial sparsity: In contrast to forwarding the com- pressed data to a sink node for reconstruction, decentralized implementation is to estimate the sparse vector of interest in a distributed/decentralized manner when a gi ven node has access to only local or some partial information about the information that would go to a sink node in a centralized setting. This requires the communication to be only within one-hop neigh- borhood of each node thus reducing the communication cost signiﬁcantly compared to multi-hop transmission and long- range single-hop transmission as considered in Section III-B. Most of the decentralized algorithms de veloped exploiting spatial sparsity are application speciﬁc since the constraints, av ailable local information and the communication o verhead depend on the application scenario. The work in [168] models the sparse ev ent monitoring task as a sparse signal reconstruc- tion problem where compression is achieved letting only few activ e sensors collect data. W ith L distributed sensors each having a position denoted by r i for i = 1 , · · · , L , the sources of ev ents are conﬁned to sensor points; i.e., an ev ent occurs only at a sensor point as shown in Fig. 6. The magnitude First Dimension Second Dimension Fig. 6: Sparse event monitoring; ( [168], [169]), locations of sources (stars) coincide with the locations of sensors (circles), solid stars - active sources, void stars-inacti ve sources of the ev ent at the r i -th location is denoted by a positiv e scalar s i . If no e vent occurred at the location r i , s i = 0 , then those sensors are called inactive sensors. Forming a vector s = [ s 1 , · · · , s L ] T , the measurement at the sensor located at r i is gi ven by , y i = X j Ψ i,j s j + v i (24) where Ψ i,j denotes the inﬂuence of a unit-magnitude ev ent at the position r j on the sensor position at r i for i, j = 1 , · · · , L . The observation vector at M < L activ e sensors in vector - matrix notation has the form of (2) where y = [ y 1 , · · · , y M ] T , A in (2) is replaced by AΨ with Ψ being an L × L matrix where Ψ i,j is the ( i, j ) -th element, and A being an M × L matrix which selects the M rows of Ψ corresponding to activ e sensors and v = [ v 1 , · · · , v M ] T . In this formulation, the problem of estimating the active sources is formulated as estimating the sparse vector s so that [168] arg min s λ 2 || AΨs − y || 2 2 + || s || 1 suc h that s ≥ 0 (25) where λ is a penalty parameter which is a generalization of the LASSO formulation. The main idea of the decentralized implementation in (25) is to estimate the s i at the i -th node and the elements corresponding to the inactive sensors in the i -th node’ s one hop neighborhood by collaborating with the 16 T ABLE V: Centralized solutions for sparse signal reco very exploiting spatial sparsity Architecture Features References Single-hop Use of MAC, consider A WGN channels [84], [147] Use of MAC, consider fading channels [153], [154] Multi-hop Direct application of CS [92], [151] Design of sparse random matrices to meet communication constraints [148]–[150], [157], [159]–[161] Use adaptive, and hybrid approaches [162], [164]–[167] neighboring activ e nodes exploiting consensus optimization and ADM ideas. In this framework, in contrast to estimating the whole length N sparse vector at each node, the coef ﬁcients corresponding to itself and neighboring nodes are estimated. This reduces the amount of information to be exchanged among nodes in the one hop neighborhood. In [169], a similar problem formulation as in [168] has been considered where the authors have proposed to use partial consensus based and the Jacobi approaches to further reduce the communication cost in the decentralized implementation. C. Data Gathering Exploiting Spatio-T emporal Sparsity In Section III-A, we considered the case were random pro- jections are used to compress a single high dimensional sparse vector of time samples collected at a given node. The goal was to simultaneously estimate multiple such high dimensional sparse signals exploiting joint structures. In particular , the sparsity of the signal is considered with respect to the trans- form basis for temporal data. On the other hand, in Section III-B, the spatial samples collected at multiple nodes at a given time instant were compressed using random projections in a distributed manner . In this case, sparsifying basis was taken with respect to the spatial data. In both Sections III-A and III-B, sparsifying basis was considered with respect to a single vector . In many applications, network data generally exhibits spatio-temporal correlations [170], thus, data compression exploiting sparsity in both dimensions (spatial and temporal) leads to better performance than the case where compression is done considering only one dimension [171]. In the following, we discuss CS techniques that can be utilized to process high dimensional data e xploiting spatio-temporal sparsity . Recall that X deﬁned in Section III-A is a N × L matrix in which each column contains the length- N data vector (uncom- pressed) collected at a giv en node. When exploiting spatio- temporal sparsity , it is desired that X is compressed both row- and column-wise. When implementing such compression schemes in WSN settings, there have been research efforts which e xploit some concepts de veloped in CS theory for sparse matrix compression [83], [171]–[173] 1) Use of matrix completion techniques: In [83], [171], the authors hav e exploited the ideas of matrix completion [174] inspired by CS theory to dev elop efﬁcient data gathering schemes exploiting spatio-temporal sparsity . In particular , the matrix X can be assumed to be low rank [175] in the presence of spatio-temporal correlations which can be recov ered reliably from a compressed version of it (or a subset of its entries) using matrix completion ideas [174]. Lo w rank feature of data collected by WSNs has been demonstrated using testbed experiments in [171]. Mathematically the X can be found by solving the optimization problem min X rank( X ) suc h that f A ( X ) = Y (26) where f A ( · ) denotes a linear operation and Y is the received data matrix at the sink. Solving (26) in its current form is NP hard, thus there have been several attempts to ﬁnd ap- proximate solutions for (26) where nuclear norm minimization as discussed in [176] is shown to be a promising approach. In the data gathering scheme presented in [171], each sensor forwards its observ ed data to the sink with a certain probability leading to sparsity in the matrix X . By deﬁning f A ( · ) = X  Q where Q is a N × L matrix consisting of 1 ’ s and 0 ’ s, the data gathering problem is solved via the nuclear norm minimization approach which has been shown to outperform the case where only spatial sparsity is exploited as in the plain CS setting [92]. The performance e v aluation is done using testbed data reported in [177] as well as with a testbed setup implemented by the authors in [171] to collect temperature, light, humidity , and v oltage data. 2) Use of structur ed/Kronec ker CS: Another approach con- sidered to exploit spatio-temporal sparsity in data gathering is to exploit ideas in Kronecker CS [178]. The Kronecker CS framew ork can be used to combine the individual sparsifying bases in both spatial and temporal domains to a single transfor - mation matrix. In particular, if each column in X has a sparse representation in the basis Φ c ∈ R N × N and each row has a sparse representation in the basis Φ r ∈ R L × L , the vectorized 1 version of X denoted by vec( X ) has a sparse representation in Φ r ⊗ Φ c ∈ R N T × N T [178]. Kronecker CS ideas have been used for spatio-temporal data gathering in [172], [173], [179], [180] with enhanced performance compared to the use of CS only considering a single domain. It is noted that with the Kronecker based formulation, the communication and computational complexities of the data transmission and recov ery algorithms increase due to the need of handling a N T × N T matrix. In order to reduce communication complexity , a Kronecker based approach has been proposed in [172] where nodes communicate with only limited number of nodes in data forwarding. In [173], a sequential approach has been discussed which e xploits Kronecker sparsifying bases (in spatial and temporal domains) with improved performance compared to using Kronecker CS ideas [178] directly . Advances of CS techniques discussed abov e to solve the data gathering problem exploiting spatio-temporal sparsity are summarized in T able VI. 1 vectorized v ersion is obtained by stacking columns of X one after the other . 17 T ABLE VI: Data gathering exploiting spatio-temporal sparsity T echnique Features References Matrix completion techniques Centralized processing, exploit low-rank property [83], [171] of spatio-temporal data, nuclear norm minimization structured/Kronecker CS Centralized processing, combine spatial and temporal [172], [173], [179], [180] sparsifying bases to a single transformation matrix Irrespectiv e of the form of sparsity exploited, the data gathering problem as discussed in this section deals with complete signal reconstruction be it centralized or distributed. In the next section, we discuss other applications of WSNs that do not necessarily require signal reconstruction, howe ver , CS can still be utilized I V . C S F O R D I S T R I B U T E D I N F E R E N C E In WSNs, CS techniques can be used as a means of data compression while the end result is not signal reconstruction but solving an inference problem. For example, in order to solve a variety of inference problems such as detection, classiﬁcation, estimation of parameters and tracking, it is sufﬁcient to construct a reliable decision statistic based on compressed data without completely recovering the original signal. Beyond the standard CS framework, this requires the in vestigation of different metrics for performance analysis and quantiﬁcation of the amount of information preserved under compression to obtain a reliable inference decision. Further , in contrast to complete signal reconstruction, sparsity prior is not necessary and the performance and the speciﬁc design principles depend on how the signals or the noise parameters are modeled. Thus, research on the applications of CS to solve such inference problems under resource constraints is attractiv e in WSNs. In the following, we discuss detection, classiﬁcation, and and parameter estimation problems with CS. A. Compressive Detection Detection is one of the fundamental tasks performed by WSNs [181]. In order to solve a detection problem efﬁciently , a decision statistic needs to be computed by intelligently combining multisensor data under resource constraints. In order to minimize the amount of information to be transmitted by sensor nodes, processing sensor data locally is of great importance. The overall decision statistic further depends on how the signals of interest and noise are modeled. In the following, we discuss how CS can be beneﬁcial in solving detection problems. Based on the speciﬁc application and the approach employed, we classify the existing CS based detec- tion techniques into ﬁv e categories which will be presented in Subsections IV -A1 to IV -A5. 1) Sparse events detection exploiting spatial sparsity: An immediate application of CS in detection is to e xploit spatial sparsity in sparse ev ent detection problems. Detection of the presence of rare ev ents by a sensor networks has applications in div erse areas such as environment monitoring and alarm systems [182]. When the number of active events is lees than the number of all possible events, the ev ent detection problem can be reformulated as a sparse recovery problem. CS ideas hav e been e xploited in sparse e vent detection in [168], [169], [183], [184]. Spatial sparsity can be exploited in sparse event detection in se veral ways. In [183], the sparse ev ent detection problem is formulated in the following sense. Let there be altogether K sources in which k (out of K ) are active. In particular , the k ev ents are assumed to occur simultaneously . The measurement vector at SN 1 SN 2 SN M First Dimension Second Dimension Fig. 7: Sparse event monitoring; ( [183]), dark stars-activ e sources, v oid stars-inacti ve sources, circles-sensor nodes M activ e sensors (out of L ) has the form of (2) where x is a sparse vector with x [ i ] ∈ { 0 , 1 } . The ( i, j ) -th element of A is gi ven by A i,j = r − α/ 2 i,j | h i,j | (27) where r i,j denotes the distance from the i -th sensor to the j -th source, α is the propagation loss factor and h i,j is the fading coef ﬁcient of the channel between the i -th sensor and the j -th source. In this frame work, the sparse event monitoring problem reduces to estimating the sparse vector x from (2) when the elements of A as giv en in (27). Note that here A is not a user deﬁned random matrix satisfying RIP properties as desired by the standard CS framework. In particular, the randomness arises due to the random locations and fading coefﬁcients. The authors in [183], have dev eloped Bayesian algorithms exploiting marginal likelihood maximization for sparse event detection. In [184], the problem of sparse event detection has been addressed in an adaptive manner using sequential compressive sensing (SCS). SCS based event detec- tion is shown to outperform the Bayesian approach considered in [183] in the low SNR region. The sparse e vent detection problem has been formulated from the perspectiv e of coding theory by modeling the detection problem as a decoding of Analog Fountain Codes (AFCs) in [185]. The decentralized approach proposed in [168] for signal recovery exploiting spatial sparsity as discussed in Section III-B3 can also be used 18 to ev ent detection when the ev ents are assumed to be conﬁned to sensor locations. 2) Detection exploiting temporal sparsity: Consider the following detection problem with the time samples collected at the j -th node given by H 1 : x j = θ j + v j H 0 : x j = v j (28) where θ j ∈ R N is the (unkno wn) signal observ ed by the j - th node and v j is the additi ve noise for j = 1 , · · · , L . The hypotheses H 1 denotes that the signal is present while H 0 denotes the signal is absent. Let the signal to be detected be sparse in the basis Ψ so that θ j = Ψs j with s j having only a few nonzero elements. With the common support set model, the coefﬁcients s j for j = 1 , · · · , L share the same support. Instead of transmitting x j ’ s, let the nodes transmit only a compressed version by applying random projections to the observations. When the detection problem is solved using compressed data, it is important to understand ho w to compute decision statistics along with their performance. After compression, the detection problem needs to be solv ed based on y j = A j x j (29) for j = 1 , · · · , L . In particular , the goal is to decide between hypotheses H 1 and H 0 based on (29). The detection problem in (29) can be expressed as: H 1 : y j = B j s j + ˜ v j H 0 : y j = ˜ v j (30) for j = 1 , · · · , L where ˜ v j = A j v j ∼ ( 0 , σ 2 v I M ) when AA T = I M and v j ∼ σ 2 v I N . When the support of s j ’ s denoted by U is exactly known, (30) reduces to H 1 : y j = B j ( U ) s j ( U ) + ˜ v j H 0 : y j = ˜ v j (31) for j = 1 , · · · , L where B j ( U ) denotes the M × k submatrix of B j = A j Ψ in which columns are index ed by the ones in U , and s j ( U ) is a k × 1 vector containing nonzero elements in s j index ed by U for j = 1 , · · · , L . When B j ( U ) is kno wn, (31) is the subspace detection problem which has been addressed previously [186]–[188]. Depending on how the unknown coef- ﬁcient v ector s j ( U ) is modeled, different detectors ha ve been proposed. In [186], a generalized likelihood ratio test (GLR T) based detector has been proposed when s j ( U ) is assumed to be deterministic. In [187], the analysis has been e xtended to the case when s j ( U ) is modeled as random. The problem with multiple observation vectors has been addressed in [188] where the authors hav e proposed adaptive subspace detectors when the coef ﬁcients { s j ( U ) } L j =1 follow ﬁrst and second order Gaussian models. In the case of sparse signal detection, it is unlikely that the exact knowledge of U is available a priori . In other words, sparse signal detection needs to be performed when U is unkno wn. When the sparsity prior is ignored, one of the common approaches would be to consider the GLR T where ML estimator of s j is found as [189] ˆ s j = ( B T j B j ) − 1 B T j y j . (32) When s j is sparse, the ML estimate is inaccurate since it is not capable of providing a sparse solution. Motiv ated by CS theory , some sparsity aw are algorithms hav e been de veloped to detect sparse signals based on (30) by ﬁnding a better estimate for ˆ s j for j = 1 , · · · , L [135], [190]–[193] with enhanced per- formance compared to the ML based approach. In particular , the standard OMP algorithm has been modiﬁed in [191] to detect the presence of a sparse signal based on a SMV . In [193], the sparse signal detection problem has been addressed with MMV . The authors hav e derived the minimum fraction of the support set to be estimated to achie ve a desired detection performance. Further, distrib uted algorithms for detection with only partial support set estimation via OMP are developed. In [135], a heuristic algorithm has been proposed for sparse signal detection in a decentralized manner based on partial support set estimation via OMP at individual nodes. Sparse detection problem in a Bayesian framework has been treated in [194] without complete signal reconstruction. 3) Detection of non-sparse signals: Let us revisit the detection problem presented in (28). When the signals to be detected, θ j ’ s are known, the optimal detector which minimizes the average probability of error is given by the matched ﬁlter . Consider the same problem in the compressed domain based on (29). This problem has been treated with a single sensor in [195]. When θ j ’ s are known and assuming v j ∼ N (0 , σ 2 v I ) , the decision statistic of the matched ﬁlter , Λ c , in the compressed domain is gi ven by Λ c = 1 σ 2 v L X j =1 y T j ( A j A T j ) − 1 A j θ j . (33) This results in the following probability of detection of the NP detector with probability of false alarm less than α 0 [196]: P c d = Q   Q − 1 ( α 0 ) − 1 σ v v u u t L X j =1 || P A T j θ j || 2 2   (34) where P A T j = A T j ( A j A T j ) − 1 A j and Q ( · ) denotes the Gaus- sian Q-function. When A j is selected to be random and an orthoprojector so that A j A T j = I , (34) can be approximated by P c d ≈ Q Q − 1 ( α 0 ) − r M N SNR ! (35) where SNR = P L j =1 || θ j || 2 2 σ 2 v . W ith uncompressed observ ations, the matched ﬁlter results in the following probability of detection, P u d : P u d = Q  Q − 1 ( α 0 ) − √ SNR  . (36) Thus, the impact of performing known signal detection in the compressed domain appears on the probability of detection via the argument of the Q function. As discussed in [195], [196], when SNR is large, compressi ve detection is capable 19 T ABLE VII: CS based solutions for detection in WSNs Problem Features References Sparse event detection Formulate the sparse event detection problem as a sparse signal [168], [169], [183]–[185] reconstruction problem, exploit spatial sparsity Detection of temporal sparse signals Compute decision statistics based on complete/partial [135], [190], [191] signal reconstruction, compression at each node, exploit temporal sparsity [192], [193] Detection of non-sparse signals Compression at each node, no signal reconstruction [195]–[198] Detection of subspace signals Compression at each node, known subspaces as well as [199]–[201] unknown subspaces are considered Design of measurement matrices Optimize for measurement matrices to achieve a gi ven detection performance [202]–[204] of providing similar performance as that of the uncompressed detector . In particular , transmitting only a compressed version of observations by each node does not, under certain con- ditions, result in performance loss compared to transmitting raw data. Thus, when the problem is detection (b ut not exact signal recovery), the CS measurement scheme can still be beneﬁcial even without having the sparsity prior . In [196], the performance of the compressed detector when θ j ’ s are random has also been discussed. In [205], the performance analysis of detection in a Bayesian framew ork with unequal probabilities for hypotheses has been presented. In recent papers [197], [206], the authors ha ve considered the problem of detection with multimodal dependent data analyzing the potential of CS in capturing the second order statistics of uncompressed data to compute decision statistics for detection. Performance analysis of sequential detection in the compressed domain has been considered in [198] with a single as well as multiple sensors. 4) Detection of subspace signals: Compressing time sam- ples via random projections at each node to detect a v ariety of signals lying in a kno wn low dimensional subspace (with- out complete signal reconstruction) has been inv estigated by sev eral other authors. In [199], the detection performance of random sparse signals has been deriv ed assuming that the subspace in which the signal is sparse is known. Subspace signal detection with compress data under realistic scenarios has been treated in [200]. Decision statistics have been deriv ed when in teh presence of unkno wn noise v ariance and imprecise measurements. The results have been extended to the case when the signal of interest lies in a union of subspaces. The authors in [201] have considered the sparse signal detection problem assuming the signal to be detected lies in a known subspace. When the corresponding subspace is unknown, detection is performed after estimating the subspace using some training data. 5) Design of measurement matrices for detection: When exploiting CS in signal compression to solve detection prob- lems, random matrices that satisfy RIP properties are widely employed. Ho wev er, when the problem is to perform de- tection, it is interesting to see if the same conditions as required for complete signal reconstruction are necessary for the measurement matrices. Design of measurement matrices for compressiv e detection so that a giv en objectiv e function is optimized has been considered in [202]–[204]. In [202], measurement matrices are designed so that the worst case SNR and the average minimum SNR are maximized. In [204], the authors have considered the probability of detection of the NP detector as the objectiv e function and shown that the optimal measurement matrices depend on the signal being detected. Detection with the designed measurement matrices in [202]–[204] is shown to outperform that with random mea- surement matrices satisfying RIP properties as used for signal reconstruction, howe ver , at the expense of some additional computational cost. CS based solutions for detection problems in WSNs are summarized in T able VII. B. Compressive Classiﬁcation CS based classiﬁcation work treated thus far in the literature has mainly dealt with the temporal data. A classiﬁcation prob- lem with C classes can be formulated as a multiple hypothesis testing problem with C hypotheses. Let the observation vector at the j -th node under the i -th hypothesis be H i : x j = s ( i ) j + v j (37) for j = 1 , · · · , L and i = 1 , · · · , C where s ( i ) j ∈ R N is the sig- nal of interest under H i . If all the nodes transmit their length N observation vectors to the fusion center, the fusion center makes the classiﬁcation decision based on x = [ x T 1 , · · · , x T L ] T (or its noisy version). W ith CS, the receiv ed compressed signal vector at the fusion center from the j -th node is gi ven by y j = A j x j + w j (38) where w j is the noise vector at the fusion center . Classiﬁcation is then performed using y = [ y T 1 , · · · , y T L ] T instead of x = [ x T 1 , · · · , x T L ] T . The speciﬁc classiﬁcation method and the impact of the compression on the performance depend on ho w the signals s ( i ) j ’ s and the noise vectors v j ’ s are modeled. For example, when s ( i ) j ’ s are deterministic and known, and elements of each v j are iid Gaussian with mean zero and variance σ 2 v , the maximum likelihood classiﬁer decides the true hypothesis (class) to be ˆ i = arg max i p ( x |H i ) = arg min i L X j =1 || x j − s ( i ) j || 2 2 . (39) based on x . W ith compressed data in (38) and assuming that each projection matrix is an orthoprojector , the ML classiﬁer 20 reduces to ˆ i = arg min i L X j =1 || y j − A j s ( i ) j || 2 2 . (40) Thus, as discussed in [207] (for the case of L = 1 ), classiﬁ- cation performance with compressed and uncompressed data depends only on how the distance measures are distorted by random projections. In particular, when A j ’ s are orthoprojec- tors, the distance between any two points in the compressed domain is reduced by approximately a factor of M / N com- pared to that with compared to uncompressed data. In [208], the authors hav e established the relationships with several probabilistic distance measures with compressed as well as uncompressed data considering Gaussian as well as non- Gaussian distributions for uncompressed data. The distance measures can be used to ev aluate how good the compressive classiﬁcation is. In [209], design of projection matrices to improv e the compressive classiﬁcation performance is dis- cussed. Nonparametric approaches for classiﬁcation exploiting CS have been discussed in [210]–[212] although not directly focusing on sensor data. C. Compressive Estimation Estimation is another important task performed by sensor networks in addition to detection and classiﬁcation. In this section, we describe recent work on CS based estimation. CS in fact deals with an estimation problem using an underdeter- mined linear system to recov er the projected high dimensional sparse signal. Ho we ver , there are se veral applications where we are interested in estimating some parameters or functions without complete signal reconstruction. W e ﬁrst focus on the use of CS for parameter estimation in general and then discuss the CS based location estimation (or source localization) problem. 1) Compressive parameter estimation: Instead of estimat- ing the complete projected signal, it is of interest to ex- plore how CS can be used to estimate a parameter (or a set of parameters) that govern the high dimensional signal. These types of problems arise in several applications such as time delay estimation [213] and frequency estimation for a mixture of sinusoids [214]–[216]. Sparsity of the signal is not a necessary requirement in such problems. Similar to the detection and classiﬁcation problems discussed earlier , we need to understand how much information is retained in the compressed domain so that a reliable estimator can be obtained. In general form, consider that the signal of interest x j ∈ R N at the j -th node is parameterized by some K -dimensional parameter vector ω = [ ω 1 , · · · , ω K ] T with K < N . The compressed observation vector at the j -th node (2) can be rewritten as y j = A j x j ( ω ) + v j . (41) In order to quantify the information retained in the compressed domain, the authors in [216] ev aluated the impact of the random projections on the estimation error . In particular , they have shown that the Cram ` e r-Rao Lower Bound (CRLB), which sets a lower bound on the variance of any unbiased estimator , is increased approximately by a factor of M / N with compressed data compared to its uncompressed counterpart when the noise is iid Gaussian. Howe ver , since the CRLB is a function of SNR, the performance of the CS based estimation can still be v ery close to that with uncompressed data when SNR is large. Estimation of time-difference-of-arri val (TDO A) using compressed data via the ML estimation method has been discussed in [213] without signal reconstruction. The TDO A estimates computed in the compressed domain can be used for source localization. These types of approaches are important when the sensors generate huge amounts of data and transmitting such large datasets is prohibitiv e due to communication constraints. Another interesting formulation of compressi ve estimation is to estimate a function of the data based on compressed measurements as discussed in [195] where complete signal reconstruction is not necessary . In this frame work with a SMV , the goal is to estimate a function of x , f ( x ) , based on y in (2). In [195], the authors have considered the case where f ( x ) = h g , x i where g ∈ R N where h·i denotes the inner product. The existing compressiv e parameter estimation techniques mostly focus on the SMV case. Howe ver , extensions to the MMV case are worth inv estigating so that they are applicable for WSN applications. 2) Localization: Source localization has been an activ e research area for a long time which has applications in div erse ﬁelds. In the literature, the source localization problem has been treated using both parametric as well as nonparametric approaches. In a parametric framework, one of the commonly considered methods is the ML method which shows excel- lent statistical properties. Howe ver , the ML approach is in general computationally complex since obtaining a closed- form solution is difﬁcult. Nev ertheless, there hav e been some approaches proposed to exploit CS in estimating parameters such as the TDO A in the ML framework using compressed data as discussed in the previous subsection which are used to perform source localization [213]. Among sev eral other techniques for source localization, sparse representation (SR) based source localization has at- tracted much attention over the years due to its capability of achieving super-resolution compared to other suboptimal localization methods [99]. The problem of SR based source localization has a similar form as considered for sparse e vent monitoring in Section IV -A1. In particular, with redeﬁned notation, let there be K sources and M sensors. The emitted signals by the K sources are giv en in s = [ s 1 , · · · , s K ] T . Further , let r = [ r 1 , · · · , r K ] T denote the source locations. The recei ved signals at the M sensors can be represented as y = Ψ ( r ) s + v (42) where the ( i, j ) -th element of the M × K array manifold matrix Ψ ( r ) contains the delay and gain information from the j -th source located at r j to the i -th sensor . In the SR framework, (42) is represented as an ov ercomplete representation by constructing a ﬁne grid, to account for all possible source locations, o ver the region of interest. Let ˜ r = [ ˜ r 1 , · · · , ˜ r L ] T denote the grid locations where L >> K . Then, the matrix 21 T ABLE VIII: CS based solutions for parameter estimation and source localization in WSNs Problem Features References Parameter estimation Framework for estimating deterministic (scalar/vector) parameters [216] based on compressed data, deriv e CRLB, SMV Estimation of TDOA based on compressed data, use ML approach [213] Source localization SR with SMV , MMV , use l 1 norm minimization [99], [217] SR with additional pre-/post- processing, use l 1 norm minimization [218] SR with MMV , use OMP [219] SR with SMV , use GMP [220] SR MMV , use GMP [221] SR with SMV , use VBEM [222], [223] SR with SMV , use iterative BCS [224] Signal sparsity and spatial SR with MMV , distributed BCS [225] Ψ ( r ) is redeﬁned as a M × L matrix whose ( i, j ) -the element corresponds to the gain and delay information between the source location ˜ r j and the i -th sensor . The signal ﬁeld is giv en in ˜ s where the n -th element of ˜ s is nonzero if the n -th location has an activ e source while it is zero otherwise. W ith this representation, (42) can be rewritten as y = Ψ ( ˜ r ) ˜ s + v (43) where ˜ s is a sparse vector . In this formulation, the sparse localization problem reduces to estimating a sparse signal ˜ s based on y where the elements of the Ψ ( ˜ r ) are determined by the particular signal model under consideration. The works in [99], [217] hav e used l 1 norm minimization to impose sparsity to solv e the source localization problem with SMV as well as MMVs. It has been shown that the SR based approach outperforms the existing localization techniques such as MUSIC in terms of the resolution, robustness to noise, number of time samples needed, and existence of dependence among sources. Howe ver , the computational complexity of the l 1 norm based approach is quite signiﬁcant compared to other localization approaches. In [218], RSSI-based source local- ization problem has been considered using a similar spatial grid model used in (43) employing l 1 norm minimization. Compared to the work in [99], [217], pre-processing is used to induce incoherence needed in CS theory , and post-processing is included to compensate for the spatial discritization caused by the grid assumption in [218]. Greedy techniques for SR based source localization hav e been utilized in [219]–[221] which hav e reduced computational complexities compared to l 1 norm based approaches. In [220], a greedy matching pursuit algorithm (GMP), has been proposed for target counting and localization jointly using the SR frame work. It has been shown that the GMP algorithm is capable of providing a good trade-off between the localization performance and the computational complexity compared to other SR-based and non SR-based approaches for localization. The GMP algorithm has been extended to take MMVs into account in [221] with RSSI measurements. In [219], OMP has been employed exploiting the SR framework using RSSI measurements. While promising results hav e been illustrated compared to non-SR based localization approaches, performance comparison with other SR-based approaches is missing in [219]. Bayesian algorithms for SR based source localization have been ex- plored in [222]–[225]. A multiple target counting and local- ization framew ork using the variational Bayesian expectation- maximization (VBEM) algorithm has been developed in [222], [223]. It has been shown that, in contrast to l 1 norm minimiza- tion based and greedy algorithms, VBEM algorithm is more robust in localizing off-grid targets. In [224], the authors hav e proposed an adaptive BCS algorithm which selects the number of sensors to be participated in the localization process based on the feedback of estimated variances of noise at different nodes. In [225], a distributed source localization scheme have been proposed combining SR in spatial domain and signal sparsity at each node to reduce sampling cost. The algorithm has further been extended to take only 1-bit measurements of the compressed data in [225]. A summary of CS based general parameter estimation and SR based solutions for source localization is gi ven in T able VIII. When analyzing existing SR based source localization techniques, simulation results have been provided in most of the work to analyze adv antages and disadvantages ov er the performance and computational cost of different algorithms. Howe ver , theoretical analysis under practical conditions such as the presence of MMV , mismatches of spatial grid, and stability of estimates is lacking in most of the work which is worth further in vestigating. D. Sparsity A war e Sensor Mana gement Sensor management, also dubbed as sensor censoring , is identiﬁed as one of the solutions for energy limitations in WSNs [226]. This inv olves in selecting the best subset of sensors that guarantees a certain inference performance. Find- ing the optimal solution for this problem is intractable in general and a wide class of sub optimal approaches, that fall into conv ex optimization, greedy and heuristic approaches, hav e been proposed in the literature [12], [226]–[228]. The sensor selection problem has gained much attention lately in the context of sparsity aware processing since efﬁcient algorithms can be dev eloped beneﬁtting from the inherent sparsity in many WSN applications. In this subsection, we revie w the sensor management work that exploits sparsity aware techniques. Sensor management for jointly sparse sig- nal recovery in WSNs is addressed in [229] when sparse 22 measurement vectors are used to compress observations at distributed nodes. The goal is to design a ternary protocol at the sensors to decide whether to transmit the compressed measurement vector , transmit a 1-bit hard decision, or not transmit based on a error criterion deﬁned in terms of the ov erlap between the signal support and the support of the measurement v ector . Centralized and distributed algorithms for sparsity aware sensor selection for the problem of estimation with a linear model have been proposed in [230]. In [231], the sensor selection problem for linear estimation has been for- mulated as a sparsity aware optimization problem considering two types of sensor collaboration; information constrained and energy constrained collaborations. The authors have employed reweighted l 1 norm based ADM and the bisection algorithm to solve the optimization problem. In [232], the sensor selection problem has been formulated as the design of a sparse vector considering a nonlinear estimation frame work. In [233], sensor selection for tar get tracking has been considered in which the selection is performed by designing a sparse gain matrix. A probabilistic sensor management scheme for target tracking has been proposed in [234] where the MAC model with distributed sparse random projections as discussed in III-B is used to compress the spatially sparse data. In [235], the authors hav e exploited the CS framework to de velop a node selection mechanism for compressiv e sleeping WSNs to improv e the performance of data reconstruction accuracy , network lifetime, and spectrum resource usage. In Sections III and IV, exploitation of CS with and/or without reconstruction has been discussed categorizing the work by speciﬁc task of interest in WSNs. The follo wing section is devoted to discuss approaches to cope with practical aspects when applying CS in WSNs. It is noted that, different aspects discussed below are applied to any giv en task in general. V . P R AC T I C A L A S P E C T S As discussed in the Introduction section, for CS based solutions to be practical in different WSN applications, it is important to in vestig ate ho w well the CS based techniques perform in practical communication networks; i.e., in the presence of practical issues such as channel impairments, security related issues and measurement quantization. A. Impact of F ading Channel Impairments In practical communication networks, the communication channels between the sensor nodes and the fusion center undergo fading. The presence of fading affects the recov- ery/inference capabilities of CS based techniques. For exam- ple, consider the CWS frame work discussed in Section III-B. In the presence of fading, the observ ation at the fusion center after the i -th MAC transmission, (21), can be re written as y i = L X k =1 h i,k A i,k x k + v i (44) for i = 1 , · · · , M where h i,k is the fading coefﬁcient for the channel between the k -th sensor and the fusion center during the i -th transmission. In vector notation, (44) can be expressed as y = Bx + v (45) where B = H  A where H is the M × L channel matrix with H i,k = h i,k and  denotes the Hadamard (element-wise) product. With the model in (45), the effecti ve measurement matrix, B , has different properties compared to that of A . Due to this, the recov ery guarantees de veloped assuming A WGN channels may not be valid in the presence of fading since it leads to inhomogeneity and non Gaussian statistics in measurement matrices. In [153], [236], the problem of sparse signal recovery based on (45) has been addressed where the authors pro vide uniform reco very guarantees [237] based on RIP when A is chosen as a sparse Bernoulli matrix (22). The authors in [154] have extended the work in [153] to deri ve nonuniform recovery guarantees [237] in the presence of Rayleigh f ading in which the sparse Gaussian matrix (23) is used for distributed compression. In particular, the heterogeneousness and the heavy tailed behavior of the projection matrix due to fading requires the sensors to take slightly more measurements than that required with A WGN channels to guarantee reliable reco very [154]. The robustness of the CS framework in the presence of nonideal channels has also been discussed in [238], [239]. The impact of fading and noisy communication channels in random access CS, an efﬁcient method for data gathering [240], has been in vestigated in [241]. Another practical aspect to be dealt with is to understand the impact of security issues in WSNs on the CS frame work. B. Physical Layer Secr ecy Constr aints In WSNs, transmissions by distributed nodes may be ob- served by eavesdroppers. Further , the network may operate under malicious and Byzantine attacks. Thus, the secrecy of a networking system against eavesdropping attacks is of utmost importance [242]. In a fundamental sense, an eav esdropper can be selﬁsh and malicious, to compromise the secrecy of a given inference network. In the recent past, there has been a signiﬁcant interest in the research community in addressing eav esdropping attacks on distributed inference networks. Ho w- ev er, while there are a few recent works [196], [204], [243]– [246], a detailed study with respect to CS based techniques has not yet been done thus far . In [243], [244], the performance limits of secrecy of CS hav e been analyzed. The amplify-and-forward CS scheme is introduced in [245] as a physical layer secrecy solution for WSNs. The authors have studied the secrecy performance against a passiv e eavesdropper agent composed of several malicious and coordinated nodes. A CS encryption framework for resource-constrained WSNs has been proposed in [246]. The authors establish a secure sensing matrix, which is used as a key , by utilizing the intrinsic randomness of the wireless channel. In [196], the authors have considered the distrib uted compressiv e detection problem in the presence of eavesdrop- pers. The authors have deriv ed the optimal system parameters to maximize detection performance at the fusion center while 23 ensuring perfect secrecy at the eav esdropper . In [204], the problem of designing measurement matrices for compressive detection so that the detection performance of the netw ork is maximized while guaranteeing a certain le vel of secrec y has been discussed. Next, we discuss the impact of measurement quantization on CS based processing. C. Sparse Signal Reconstruction with Quantized CS Compression achieved via random projections at the sam- pling stage is desirable in resource constrained WSNs. How- ev er, further compression/quantization of the compressed mea- surements may be necessary in many WSN applications op- erating under severe resource constraints. Coarse quantization reduces the bandwidth requirements and computational costs at local nodes, and thus, is well motiv ated for practical WSNs. The traditional CS framework with real valued measurements has been extended to take quantization effect into account. In particular , one of the dri ving forces behind the development of a quantized CS frame work is the motiv ation to further reduce the amount of information to be communicated in resource constrained WSNs. Consider the compressed obser- vation model as giv en in (1) or (2). With quantized CS, one has access to z = Q ( y ) instead of y where Q ( · ) is an entry- wise (scalar) quantizer which maps real valued measurements to a discrete quantized alphabet of size Q . In particular, each element of y is quantized into Q le vels so that z i =            0 , if τ 0 ≤ y i < τ 1 1 , if τ 1 ≤ y i < τ 2 . . Q − 1 , if τ Q − 1 ≤ y i < τ Q (46) for i = 1 , 2 , · · · , M , where τ 0 , τ 1 , · · · , τ Q represent the quantizer thresholds with τ 0 = −∞ and τ Q = ∞ . With this approach, d log 2 Q e bits per measurement are required to transmit each element of y . In the special case with a 1-bit quantizer , the measurements are quantized into only two lev els such that Q = 2 . One example under this special case is to use only the sign information of the compressiv e measure- ments [86]–[90]. More speciﬁcally , the 1-bit CS scheme ﬁrst proposed in [86] with sign measurements is gi ven by z i =  1 , if y i ≥ 0 − 1 , otherwise (47) for i = 1 , 2 , · · · , M . Equiv alently , we may e xpress (47) by z = sign( y ) (48) where z = [ z 1 , · · · , z M ] T , and sign( · ) denotes the element- wise sign operation. Sparse signal processing with 1-bit quan- tized CS is attractive since 1-bit CS techniques are robust un- der different kinds of non-linearties applied to measurements and hav e less sampling complexities at the hardware level because the quantizer takes the form of a comparator [86], [88]. Now , the goal is to perform sparse signal recovery or solve inference tasks based on z instead of y . There are several factors that need to be taken into account when ev aluating the performance and de veloping algorithms with quantized CS especially as applicable to resource constrained WSNs; • Quantization introduces nonlinearity . Thus, the algo- rithms and the reco very guarantees developed for sparse signal processing with real valued CS may not be directly applicable for quantized CS. This provides the moti v ation to dev elop quantization schemes and reconstruction algo- rithms so that the performance with quantized CS is very close to (or ev en better than) that with the real valued CS. • Coarse quantization can reduce the ability of signal reconstruction or performing inference tasks compared to real valued CS. This provides the impetus to take more measurements to compensate for the loss due to quantization. Thus, inv estigation of the trade-off between the cost for quantization and the cost for sampling is important. • In distributed networks, MMVs appear naturally . This motiv ates us to extend the quantized CS based process- ing set-up to the distributed/decentralized settings with different communication architectures. Over the past few years, there hav e been sev eral research efforts that aim to consider quantized CS in different contexts. In the follo wing, we ﬁrst re view the w ork on 1-bit CS. 1) Algorithm development and performance guarantees with 1-bit CS: Most of the early work related to 1-bit CS considers the SMV case. In [86], the authors hav e introduced the 1-bit CS problem with sign measurements (48) for the noiseless case (i.e., y is as in (1)). The authors have de veloped an optimization based algorithm for sparse signal recovery using a variation of the ﬁxed point continuation (FPC) method [247]. In particular , they have considered solving min x || x || 1 suc h that ZAx ≥ 0 and || x || 2 = 1 (49) where Z = diag ( z ) . It is noted that, for an N × 1 vector x , diag ( x ) denotes a N × N diagonal matrix in which the main diagonal is composed of elements of x . The authors hav e shown that the recovery performance can be signiﬁcantly improv ed with the proposed algorithm compared to employing classical CS algorithms when the measurements are quantized to 1-bit. One problem in (49) is that it requires the solution of a non-conv ex optimization problem. In [248], the authors hav e presented a prov able optimization algorithm to solve (49). A con vex formulation of the signal recovery problem from 1-bit CS has been presented in [249] which solv es for min ˜ x || ˜ x || 1 suc h that sign( A ˜ x ) = z and || A ˜ x || 1 = M . (50) It has been shown in [249] that, (50) can provide an arbitrarily accurate estimation of e very k -sparse signal x with high probability when M = O ( k log 2 ( N /k )) 1-bit measurements. According to their results, the required number of CS measure- ments matches the kno wn results with real v alued CS up to the exponent 2 of the logarithm and up to an absolute constant factor . In [250], [251], the log-sum penalty function, which has the potential to be much more sparsity-encouraging than the l 1 norm, has been used for sparse signal recovery with 1-bit 24 CS. They have developed an iterative reweighted algorithm which consists of solving a sequence of con ve x differentiable minimization problems for sparse signal recov ery . Greedy and iterati ve algorithms de veloped for classical CS hav e been extended to the 1-bit CS case in [89], [90], [252]. Modiﬁcation of the CoSAMP algorithm for the 1-bit case to produce Matching Sign Pursuit (MSP) has been presented in [90]. In [89], an extension of the IHT algorithm with 1-bit CS, called binary IHT (BIHT) has been dev eloped. The BIHT algorithm to incorporate additional information on the partial support has been considered in [252]. The authors in [253] hav e presented a Bayesian approach for signal reconstruction with 1-bit CS, and analyzed its typical performance using statistical mechanics. In [254], a Bayesian algorithm based on generalized approximate message passing (GAMP) has been dev eloped. Recov ery guarantees and consistency of the estimator for both Gaussian and sub-Gaussian random measurements have been established in [255] for 1-bit CS using the recently pro- posed k-support norm [256]. In [257], the sample comple xity of vector recovery using 1-bit CS has been discussed. In a recent work presented in [258], the authors have shown that 1-bit measurements allow for exponentially decreasing error with adapti ve thresholds. This framework is slightly dif ferent from the 1-bit CS model discussed with sign measurements in [86]. More speciﬁcally , the i -th element of z is gi ven by z i = sign( y i − ν i ) =  1 , if y i ≥ ν i − 1 , if y i < ν i (51) for i = 1 , · · · , M . This scheme allows the quantizer to be adaptiv e, so that ν i in (51) of the i -th entry may depend on the 1st , 2nd , · · · , ( i − 1)st quantized measurements. Adaptive one-bit quantization has also been considered in [259] where the authors have shown that when the number of one-bit measurements is sufﬁciently large, the sparse signal can be recov ered with a high probability with a bounded error . The error bound is linearly proportional to the l 2 norm of the difference between the thresholds and the original unquantized measurements. In another recent work by Knudson et.al. in [260], it has been shown that norm recov ery is possible with sign measurements of the form sign( Ax + b ) for random A and ﬁxed b which is impossible with sign( Ax ) . 2) Algorithm development and performance guarantees with quantized CS: The authors in [261]–[264] have consid- ered the sparse signal/support recovery problem with a giv en scalar qunatizer where 1-bit CS appears as a special case. In scalar quantization, each element of y is quantized indepen- dently as shown in (46). In [261], [262], sev eral algorithms and performance bounds have been deriv ed considering a SMV . More speciﬁcally , Zymnis et.al. in [261] have de veloped two algorithms for sparse signal recov ery with quantized CS by minimizing a differentiable con ve x function plus an l 1 regularization term. In [262], the authors have inv estigated the minimum number of CS measurements required for support recov ery with a gi ven quantizer (including 1-bit CS) with a SMV speciﬁcally focusing on support recovery . The effect of quantization has further been analyzed in [263]–[267]. The effects of precision in the measurements hav e been analyzed in [265] by considering the syndrome decoding algorithm for Reed-Solomon codes when applied in the context of compressed sensing as a reconstruction algorithm for V ander- monde measurement matrices. Universal scalar quantization with exponential decay of the quantization error as a function of the ov ersampling rate has been considered in [267]. In particular , the author has shown that non-monotonic quantizers achiev e exponential error decay in the ov ersampling rate using consistent reconstruction. Howe ver , reconstruction from such a quantization method is not straightforward, and the same author has proposed a practical algorithm for reconstruction using a hierarchical quantization approach in [266]. In [263], the authors ha ve presented a variant of basis pursuit denois- ing, based on l p norm rather than using the l 2 norm. They hav e prov ed that the performance of the proposed algorithm improv es with larger p . In [264], an adaptation of basis pursuit denoising and subspace sampling has been proposed for dealing with quantized measurements. Design and analysis of vector quantizers (VQ) with CS measurements hav e been considered in [268], [269]. In [268], the authors have addressed the design of VQ for block sparse signals using block sparse recovery algorithms. Inspired by the Gaussian mixture model (GMM) for block sparse sources, op- timal rate allocation has been designed for a GMM-VQ which aims to minimize quantization distortion. In [269], optimum joint source-channel VQ schemes have been developed for CS measurements. Necessary conditions for the optimality of vector quantizer encoder -decoder pair with respect to end-to- end MSE ha ve been deri ved. 3) Distributed and decentralized solutions with quantized CS: While most of the existing works on quantized CS are restricted to the SMV case, there are a few recent works that hav e extended the quantized CS framew ork to the multiple sensor setting. In [270], sparse signal reconstruction using 1- bit CS has been considered modeling communication between sensors and the fusion center via a binary symmetric channel (BSC). Instead of using the sign measurements as in (47), local binary quantizers are designed to cope with the bit ﬂipping caused by BSC. Se veral existing CS algorithms developed for real v alued CS ha ve been been extended to the quantized CS setting in [271]–[273] considering centralized as well as distributed/decentralized implementations. Solving the support recov ery problem in a decentralized setting with 1-bit CS (sign measurements) has been considered in [272], [273] where the authors have dev eloped sev eral decentralized versions of the BIHT algorithm. In [274], the authors employed a distrib uted variable-rate quantized CS methodology for acquiring cor- related sparse sources in WSNs. Optimality conditions that minimize a weighted sum of the average MSE distortion for signal recovery with complexity-constrained encoders have been deri ved. D. Other Issues In addition to the practical issues discussed above, ro- bustness of the CS scheme against missing or erroneous information, node failures, and other types of errors has been in vestigated by sev eral researchers. It has been shown that the 25 multi-hop data g athering scheme proposed in [84] employing randomized gossip algorithms is robust to node failures and changes in network conﬁgurations. In [275], compression of spatio-temporal data with missing information and in the presence of anomalies has been addressed. As discussed in Section III-C1, the data matrix has a low rank structure in the presence of spatio-temporal sparsity . When there are missing information and/or anomalies, assumptions made in general CS theory may be violated leading to inaccurate/degraded performance if the y are not properly taken care of. By proper decomposition of the data matrix to take the low-rank property , missing value interpolation and noise terms into account, the authors in [275] have developed a nuclear norm minimization based approach which is shown to be robust against missing information and anomalies. CS recovery techniques have also been used in [276] as a tool for estimating spatio-temporal data in the presence of missing information in WSNs without employing an y data compression as discussed in Section III-C1. While application of CS in WSNs has a quite rich literature as of now along most of the categories discussed above, there are still av enues for future research which will be discussed next. V I . L E S S O N S L E A R N E D A N D F U T U R E R E S E A R C H D I R E C T I O N S W ith the emergence of new technologies such as IoT where heterogeneous networking architectures need to be integrated to perform a variety of tasks, handling and integration of a large amount of data generated by the interaction of multiple factors/sensors keeps continuously challenging. In particular, future WSNs are expected to inte grate with a variety of other networks such as wireless mesh netw orks, W i-Fi, and vehicu- lar networks to make smart platforms for IoT applications. Understanding the role of CS, as a tool to cope with the problem of data deluge, is of great importance in making such applications realizable. As revie wed in this paper speciﬁcally focusing on WSNs, there are sev eral ways that CS techniques can be beneﬁ- cial. While there has been quite signiﬁcant amount of work done during last several years to make CS based techniques practically implementable in WSNs under network resource constraints focusing on different problems, there are still challenges and open issues worth further inv estigation in the areas discussed in Sections III, IV and V. In the follo wing, we discuss such challenges and future research directions. A. Scalable Network Pr ocessing Exploiting Sparsity in Multi- ple Dimensions with Heter ogeneous Data As discussed in Section III, in CS based data gathering, the attention is mainly giv en to the case when sparsity is deﬁned with respect to a single vector; temporal as in III-A and spatial as in III-B. T o exploit spatio-temporal correlation in data gathering, there are a fe w approaches as discussed in Section III-C which deﬁne sparsity in 2-dimensions (2-D). Mainly , the ideas of Kronecker CS and matrix completion [174], [178] have been exploited under restricted assumptions such as centralized processing. Further , when exploiting Kronecker CS ideas, reconstruction is performed after transforming 2- D (spatio-temporal) data to a single vector which requires large memory and computational costs. Such requirements are not desirable especially with on-line WSN applications. Thus, exploring approaches to reduce computational requirements when exploiting spatio-temporal sparsity is useful in many potential applications. One direction of research interest is to consider decentralized/cluster based settings where only partial information is processed at multiple sinks or clusters so that the o verall computational burden is distributed. Further , ideas dev eloped in [277], [278] can be utilized to solve these kind of problems in the matrix form with less computational cost without using the vector form. Beyond WSNs, when integrating different network archi- tectures for future IoT applications, large amount of data can be generated by the interaction of multiple factors/sensors and thus can be intrinsically represented by higher order tensors. Thus, exploitation of low dimensional properties of high order tensors in an efﬁcient manner is needed to better employ CS based compression techniques to get a variety of tasks done. In CS theory , there has been quite signiﬁcant attention to generalize the CS frame work for high dimensional multi-linear models [279], [280]. Due to the same reasons discussed as in Section II-B, the direct use of such techniques may not be desirable and extensions to cope with the av ailable network resources need to be explored. B. Distributed/Decentralized Processing with Quantized CS While quantized CS (especially 1-bit CS), as discussed in Section V -C is appealing for WSN applications to further reduce the communication cost, most of the existing work on quantized CS is restricted to the SMV case. Recently , the works reported in [271]–[273] have extended the 1-bit CS framew ork to the multiple-sensor setting. Howe ver , its de vel- opment is still in its infancy . Understanding the relationships among the network parameters, compression ratio (achie ved via sampling with CS) and the quantization lev els (achiev ed via quantizing CS samples) focusing on decentralized process- ing in both data gathering (signal reconstruction) and inference perspectiv es is necessary to better utilize the quantized CS framew ork in potential WSN applications. Further, quantifying the performance difference in terms of reconstruction and inference when using coarse quantized CS and real valued CS would help understand the trade-off between quantization and the performance. Thus, further efforts are required to ev aluate the beneﬁts of quantized CS in WSNs. C. CS Based Infer ence with Multi-Modal Data When performing inference tasks from compressed data without complete signal reconstruction, it is of importance to understand how much information is retained in the com- pressed domain so that a reliable inference decision can be made. This depends on how the signal and noise are modeled. When the (uncompressed) multisensor data is independent and corrupted by additi ve Gaussian noise, performance metrics for CS based detection, classiﬁcation, and parameter estimation 26 hav e been discussed in sev eral recent works as revie wed in Section IV. Dependence is one of the common characteristics exhibited in multiple sensor data which is hard to model analytically with multimodal data unless the data is Gaussian. Further studying CS based inference in terms of deriving performance metrics, de veloping communication efﬁcient al- gorithms, and designing projection matrices in the presence of inter-modal and intra-modal dependence will enable the efﬁcient use of CS in many potential applications. While there exist sev eral recent works that e xploit spatial (or inter-modal) dependence in detection problems in the Bayesian CS frame- work under restricted assumptions [197], [206], CS based multi-sensor dependent data fusion especially in the presence of non-Gaussian pdfs and spatio-temporal dependence is not well understood. Thus, exploitation of higher order depen- dence and structured properties of high dimensional data in CS based multi-sensor fusion is worth in vestigating. In addition to domain speciﬁc applications, when WSNs are used as a smart architecture for IoT applications, it is expected to process multi-modal data efﬁciently in dynamic, and heterogeneous en vironments to perform a variety of dif- ferent tasks. Understanding how the CS framework can be beneﬁcial in an adaptiv e manner to perform multiple tasks under such en vironments is essential in the realization of future IoT techniques. D. Further Developments Under Pr actical Considerations Understanding the robustness of the CS framew ork to prac- tical systems in the presence of practical issues such as fad- ing channels, interference, non-Gaussian noise impairments, synchronization errors, quantization errors, link failures, and missing data is important to make CS based techniques useful in WSNs. As discussed in Section V, many desirable con- ditions required to establish performance guarantees in the standard CS framework may not be satisﬁed in the presence of such practical issues. While some recent work exists in this area, further inv estigation of approaches to rectify the adverse impact of such practical aspects on the overall performance is needed. E. T estbed Experiments and P erformance Evaluation So far , most research on CS in WSNs is restricted to theoretical and algorithmic dev elopment. There are some testbed experiments and demonstrations done to incorporate CS based techniques speciﬁcally focusing on data gathering. As discussed in Section III-B2, several testbed experiments analysis with real data hav e been reported for CS assisted data gathering exploiting spatial sparsity for different applications [164]–[167]. Further, in [171], testbed experiments have been performed for CS based spatio-temporal data collection and recov ery . Ho wever , in order to bridge the theory and practice, further development is needed to validate the proposed tech- niques with real applications. In particular , testbed experiments with CS based techniques for a variety of applications with and/or without complete signal reconstruction, performance ev aluation, and robustness analysis against practical consid- erations with real data will be useful. V I I . C O N C L U S I O N In this surve y paper , our goal was to describe the research trends and recent work on the use of CS in a variety of WSN applications. By discussing the motiv ational f actors, we hav e identiﬁed several challenges that need to be addressed to enable practical implementation of CS based techniques in WSNs. T o that end, we ha ve revie wed the recent works that focus on de veloping centralized, distrib uted and decentralized solutions for data gathering and reconstruction exploiting temporal, spatial as well as spatio-temporal sparsity under communication resource constraints. 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Application of Compressive Sensing Techniques in Distributed Sensor Networks: A Survey

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