MRCS: matrix recoverybased communicationefficient compressive sampling on temporalspatial data of dynamicscale sparsity in largescale environmental IoT networks
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Abstract
In the past few years, a large variety of IoT applications has been witnessed by fast proliferation of IoT devices (e.g., environment surveillance devices, wearable devices, citywide NBIoT devices). However, launching data collection from these mass IoT devices raises a challenge due to limited computation, storage, bandwidth, and energy support. Existing solutions either rely on traditional data gathering methods by relaying data from each node to the sink, which keep data unaltered but suffering from costly communication, or tackle the spacial data in a proper basis to compress effectively in order to reduce the magnitude of data to be collected, which implicitly assumes the sparsity of the data and inevitably may result in a poor data recovery on account of the risk of sparsity violation.
Note that these data collection approaches focus on either the fidelity or the magnitude of data, which can solve either problem well but never both simultaneously. This paper presents a new attempt to tackle both problems at the same time from theoretical design to practical experiments and validate in real environmental datasets. Specifically, we exploit data correlation at both temporal and spatial domains, then provide a crossdomain basis to collect data and a lowrank matrix recovery design to recover the data. To evaluate our method, we conduct extensive experimental study with real datasets. The results indicate that the recovered data generally achieve SNR 10 times (10 db) better than compressive sensing method, while the communication cost is kept the same.
Keywords
Device data gathering Data compression Matrix recoveryAbbreviations
 BS
Base station
 CS
Compressed sensing
 IoT
Internet of Things
 LRMR
Lowrank matrix recovery
 MR
Matrix recovery
 SNR
Signal to noise ratio
1 Introduction
As InternetofThings (IoT) applications are proliferating rapidly in recent years, the vastly distributed IoT devices and the resultant large volume of sensing data attract lots of research effort and have triggered a wide range of applications, such as smart city, transportation, and agriculture, owing to its capability of completing complex social and geographical sensing applications. Such social and geographical sensing usually requires large amounts of participants (usually IoT devices) to sense the surrounding environment and collect data from the sensing devices to a data sink due to limited computation, storage, and energy support.
Due to the scale of mass data generated in IoT networks, it is difficult to continuously gather the original data from the network, since such collection usually requires considerable effort of communication and storage at intermediate nodes. A traditional way of solving this problem includes waveletbased collaborative aggregation [1], clusterbased aggregation and compression [2, 3], and distributed source coding [4, 5]. All of them utilize the spatial correlation of device readings among device nodes. But they may meet robustness issues when dealing with crossdomain (temporal and spatial) event readings and behave limited capacity in compression. In recent years, it is suggested that compressed sensing (CS) may benefit the compression in data aggregation scenarios. It avoids introducing excessive computational, communication, and storage overheads at each device. Therefore, it meets the capacity limitation at each sensing device and is viewed as a promising technology for data gathering in IoT networks.
However, compressive sensing is based on constant sparsity, which means a stable/fixed transform basis (though unnecessary to known) is required according to prior information of sensed data. Such a situation hardly holds in real cases, and data with changing sparsity would impact the recovery quality significantly. In order to address this problem, Wang et al. [13] proposed an adaptive data gathering scheme based on CS. The “adaptive” here has twofold meaning: for one, the CS reconstruction becomes adaptive to the sensed data, which is accomplished by the adjustment of autoregressive (AR) parameters in the objective function, and for the other, the number of measurements required to the sensed data is turned adaptively according to the variation of data. To further deal with the varied sensing data, Wang et al. suggested that each time when the reconstruction is accomplished at sink node, the result is approximately evaluated and forms a feedback to the device nodes. The intuition here is that the temporal correlation between historically reconstructed data could help estimate current reconstruction result at sink node. It is notable that compression of original readings with CSbased method [6, 7, 8] or matrix completionbased method [9, 10] will reduce the quality of recovered data at sink node and routing the raw data to sink to preserve the fidelity brings considerable overhead. There is a conflict between high compression ratio and high fidelity.
Data gathering and recovery with event readings is another problem studied in compressive sensingbased data gathering. A wellknown method to tackle this problem is to decompose data d into d_{n}+d_{α}, assuming d_{α} is sparse in time domain since the abnormal readings are usually sporadic. However, when environment changes occur, it may result in a significant amount of readings beginning to change, which would further make d_{α} not sparse in spatial domain. Besides, though d_{n} is sparse in spatial domain under a proper basis and d_{α} is sparse in time domain, they are not necessarily sparse at the same time under the same basis. Therefore, it is doubtful that the sparsity of d could preserve cross time and space domain. Furthermore, the proper basis may vary in accordance with different events.

In IoT sensing and data aggregation scenarios, note that either the fidelity problem or the magnitude problem can be solved well but never both simultaneously. This paper presents a new attempt to tackle both problems at the same time in largescale IoT networks with diverse time/spacescale events and reduce globalscale communication cost without introducing intensive computation or complicated transmissions at each IoT device.

The experiments of this paper on a real environmental IoT sensing network observe that constant sparsity hardly holds in real cases with diverse time/spacescale events, while lowrank property may be true. This observation may provide a fresh vision for research in both compressive sampling applications and IoT sensing and data aggregation scenarios. This paper further generalizes the lowrankbased optimization design to a nuclear normbased optimization design, to make the proposed approach more general and robust.

Theoretical analysis indicates that our matrix recoverybased method is robust over diverse time/spacescale event readings. The extensive experimental results show that event readings are almost kept unaltered under the proposed design and our method outperforms typical compressive sensing [11] in terms of SNR by 10 times (10 db) generally in the meanwhile.
This paper is organized as follows. Section 2 introduces the preliminaries and the network model. Section 3 proposes the data gathering and recovery design. Section 4 analyzes communication overhead of the proposed method with comparison to compressive sensing. Section 5 presents the experimental results with real environmental IoT sensing datasets from [12]. Then, we summarize related work in Section 6. Finally, we give out the discussion in Section 7 and conclude this paper in Section 8.
2 Preliminaries and assumptions
2.1 Matrix recovery
Let X denote the original data, where X is a M×N matrix and is no longer to be sparse even in a proper basis (different from compressive sensing). Let rank(X)=r, where r is assumed to be much smaller than min{M,N}.
According to [13], in order to recover X from a linear combinations of X_{ij}, the number of the combinations needed is no larger than cr(M+N), where c is a constant.
2.2 Network model
We consider a participatory IoT sensing network in which a base station (BS) continuously collects data from participatory IoT devices. Due to the scale of mass data and the limited ability in computation, storage, bandwidth, and energy at each device, these devices need to compress data with light computation overhead before data transmission. Suppose there are N resourceconstrained IoT devices in the network, whose positions can be determined after deployment via a selfpositioning mechanism such as those proposed in [14, 15, 16]. Then, the data collection path could be predetermined by the base station and be aware by each device. We further assume that the clocks of all nodes are loosely synchronized [17, 18, 19]. In particular, t_{1},t_{2},⋯,t_{i},⋯,t_{j},⋯ are used to represent the time instants in the network, where t_{i}<t_{j} given i<j, i,j∈Z^{+}. Every time instance, a device generates a reading. \(\mathcal {N}(u)\) is denoted as the n nearest neighbors of an open neighborhood of u. Note that \(\mathcal {N}(u)\) could be the onehop neighborhood or any neighboring area containing more nodes.
In this paper, we assume that participatory IoT devices follow a semihonest model [20]. Specifically speaking, they are honest and follow the protocol properly except that they may record intermediate results. We assume that the messages are securely transmitted within the network, which can be achieved via conventional symmetric encryption and key distribution schemes.
3 Temporalspatial compressive sampling design
3.1 Problem formulation
Given M time instances and N devices in the network, the original data in the network can be represented by a m×n matrix X, where each row represents the readings in the network at a time instant and each column represents the readings of an IoT device at a different time instant. X_{ij}(1≤i≤M,1≤j≤N) denotes the reading of each node j at time instant t_{i}.
where the first part is for noise and the second part is for low rank.
Remark 1
In IoT networks, devices may produce erroneous readings due to noisy environment or errorprone hardware. The erroneous readings usually occur at sporadic time and locations and thus may have few impacts on the data sparsity of the network. Thus, outlier/abnormal reading recovery/detection could still work in compressive sensingbased data gathering. However, device measurements on the same event usually have strong intercorrelations and geographically concentrated in a group of devices in close proximity. Such events may spread in diverse time and space scale and result in dynamic sparsity of the data, which would further violate the assumption of constant sparsity in compressive sensing and thus lead to poor recovery.
Remark 2
Given N M−dim signal vectors generated from N devices within M time instances, a good basis to make these vectors sparse may not be easy to find. Interestingly, [22] has analyzed different sets of data from two independent device network testbeds. The results indicate that the N×Mdim data matrix may be approximately low rank under various scenarios under investigation. Therefore, such N×M temporalspatial signal gathering problem with diverse scale event data that cannot be well addressed by CS method could be tackled under the lowrank framework^{1}.
3.2 Path along compressive collection
In this paper, we provide a generalization of current data gathering methods on temporalspatial signals with diverse scale events, during which device readings are compressively collected along the relay paths, e.g., chaintype or mesh topology, to the sink.
At each device s_{j}, given the reading produced from s_{j} at time instance t_{1}, s_{j} generates a random vector Φ_{1j} of length p, with time instance t_{1} and its ID s_{j} as the seed, and computes the vector X_{1j}Φ_{1j}. At the next time instance t_{2}, s_{j} generates a random vector Φ_{2j}, computes X_{2j}Φ_{2j}, and adds it to the previous vector X_{1j}Φ_{1j}. At time instance t_{M}, s_{j} computes X_{Mj}Φ_{Mj} and would have the summation \(S_{j}=\sum \limits ^{M}_{i=1} \boldsymbol {X}_{ij}\Phi _{ij}\).
In the network, each device s_{j} continuously updates its vector sum S_{j} till time instance t_{M}. After that, device s_{j} relays the vector S_{j} to the next device s_{i}. Then, s_{i} adds S_{j} with its vector sum S_{i} and forwards S_{i}+S_{j} to the next device. After the collection along the relay paths, the sink receives \(\sum \limits ^{M}_{i=1} \boldsymbol {X}_{ij}\Phi _{ij}\).
Remark 3
During data gathering, each node sends out only one vector of fixed length along the collection path, regardless of the distance to the sink (The property of the fixedlength vector will be discussed in Section 4).
Considering event data, recall that the row of data matrix X (the signal in the network) represents the data acquired at some time instance from all devices and each column of matrix X represents the data got from one device at different time instances.
Outlier readings could come from the internal errors at errorprone devices, for example, noise, systematic errors, or caused by external events due to environmental changes. Former internal errors are often sparse in spatial domain, while the latter readings are usually low rank in time domain. They both keep sparse at the corresponding domain but together may lead to dynamic changes of data sparsity.
Based on Eq. 1, [A,A] is a new linear map. The formulated problem could be solved in the framework of matrix recovery. That is, given the observation vector y∈R^{p}, the original data matrix X^{∗} could be recovered in R^{2M×N}.
3.3 A basic design of data recovery
This section provides the generalization of data recovery method from compressive sensing to the realm of matrix recovery. The advantages of such an extension are twofold: (1) it exploits the data correlation in both time and space domains and (2) the diverse scale of event data, which would mute the power of CS method due to sparsity changes, could be tackled with the proposed method.
where A(x)=ΦT(x) given T(·) as the transformation of a matrix to a vector by overlaying one column of x on another. Φ is a p×MN random matrix.
Note that Eq. 3 is the Lasso form of Eq. 2. In relaxed conditions, its solution is the solution of Eq. 2 [23]. Therefore, we consider Eq. 3 (Eqs. 3 and 5 are essentially same) instead of the original problem in Eq. 2.
where \(f(x)=\frac {1}{2} A(\boldsymbol {X})b^{2}_{F}\) and P(x)=μX_{∗}
Lemma 1
Proof
\(\nabla f(\boldsymbol {X})\nabla f(\boldsymbol {Y})_{F}^{2}=\Phi ^{T}(\Phi ^{*}T(\boldsymbol {X}\boldsymbol {Y}))^{2}_{2}\)
Set \(\Phi ^{T}\Phi = \left (\begin {array}{ccc} a_{11} & \cdots & a_{1,MN} \\ \vdots & & \vdots \\ a_{p1} &\cdots &a_{p,MN} \end {array}\right)\),
\(T(XY)=\left (\begin {array}{c} x_{11} \\ \vdots \\ x_{MN} \end {array}\right)\),
Remark 4
A much smaller L_{f} could be found in various real scenarios and may help converge quickly. The experimental results of this paper show that the L_{f} could be much smaller than the rough estimation above, given the matrix sampled from a Gaussian distribution.
where τ>0 is a given parameter, G=Y−τ^{−1}∇f(Y).
Since the above function of X is strong convex, it has a unique global minimizer.
where G∈R^{M×N}. Note that if G=Y−τ^{−1}A^{∗}(A(Y)−b), then the above minimization problem is a special case of Eq. 7 with \(f(X)=\frac {1}{2}A(X)b^{2}_{2}\) and P(X)=μX_{∗} when we ignore the constant term.

Step 1: Set \(Y_{k}=X_{k}+\frac {t^{k1}1}{t^{k}}\left (X_{k}X_{k1}\right)\)

Step 2: Set G_{k}=Y_{k}−(τ_{k})^{−1}A^{∗}(A(Y_{k})−b). Compute \(S_{\tau _{k}}(G_{k})\) from the SVD of G_{k}

Step 3: Set \(X^{k+1}=S_{\tau _{k}}(G_{k})\)

Step 4: Set \(t_{k+1}=\frac {1+\sqrt {1+4(t_{k})^{2}}}{2}\)
Lemma 2
with X_{LS}=A^{∗}(AA^{∗})^{−1}b
Based on this lemma, we could reach a deterministic estimation of the procedure and speed of convergence of data recovery.
Let δ(x) denote dist(0,∂(f(x))+μX_{∗}), where δ(x) represents the convergence speed of data recovery. It is easy to see that the process naturally stops when δ(x) is small enough.
Since X_{∗} is not differential, it may not be easy to compute δ(x). However, there is a good upper bound for δ(x) provided by APG designs [24].
4 Advanced design of data recovery
This section provides a generalization of previous lowrankbased matrix recovery design to a nuclearformbased design. Suppose X_{0} denotes an M×N matrix with rank r given the singular value decomposition (SVD) UΣV^{∗}, where M≤N, Σ is r×r, U is M×r, and V is N×r.
Theorem 1
Given X_{0}, an arbitrary M×N rankrmatrix, and ·, the matrix nuclear norm, considering a Gaussian mapping Φ with m≤c·r(3M+3N−5r) for some c>1, the recovery is exact with probability at least 1−−2e^{(1−c)n/8}, where n=max(M,N) [13].
Here the Gaussian mapping Φ is an M×N random matrix with i.i.d., zeromean Gaussian entries with variance 1/p. It adopts a linear operator where \([\Phi (Z)]_{i} = \text {tr}(\Phi _{i}^{*}\cdot Z)\).
Remark 5
According to this theorem, each device sends out only one vector of fixed length of cr(3M+3N−5r) along the collection path at the end of time M, with an overwhelming recovery probability of the original data.
Comparing with compressive sensing (CS)based data gathering design, the length of vector sent by each device at each time instance with CS based design is O(logN). Based on recent results on the bounds for lowcomplexity recovery models [25], the total amount of vectors collected during all M time instances will be O(MN log(N)) in compressive sensing.
When M is larger than O(N/ log(N)), the proposed design will exhibit advantage in communication overhead. When M and N have the same order of magnitude, the proposed method has similar communication overhead compared with CSbased method.
Before estimating the recovery error and its upper bound, we first introduce the restricted isometry property (RIP):
Definition 1
If δ_{r} is bounded by a sufficiently small constant between 0 and 1, we say that A satisfies the RIP at rank r.
Theorem 2
Suppose X^{∗} is the solution of the recovery method. Given the noise z satisfying Φ^{∗}(z)≤ε and ΦT(z)_{∞}≤η, for some ε≤η, if \(\delta _{r}<\frac {1}{3}\) with r≤2, then X−X^{∗}_{F}≤(ε+η)_{+}
Based on this theorem, given a random matrix Φ properly chosen from i.i.d. zeromean Gaussian distribution with variance 1/p, the error of the proposed method could be bounded under the noise.
Based on the above analysis, we could see that (1) the vector kept by each device is bounded by cr(3M+3N−5r) and (2) the communication overhead of the network, i.e., the total number of message, is O(Ncr(3M + 3N − 5r)). Comparing with the overhead of CSbased data gathering method, which are M·(2cs logN + s) and MN·(2cs logN+s), respectively, for lowrank data [25], it is easy to see that the proposed method could outperform CSbased data gathering methods in terms of communication given a large collection period M. To compare with matrix completionbased method, we take STCDG proposed in [22] as an example under the same assumption that N nodes are deployed randomly. According to [13], the overhead of STCDG can be derived as O(n^{5/4}N^{1/2}r logn)(n=max(M,N)). STCDG may suffer a much larger overhead compared with our method under largescale IoTs.
Remark 6
According to the analysis, the larger the sampling period M at each device, the better the communication overhead efficiency the proposed method has.
5 Results
To evaluate data recovery quality and robustness of the proposed method, we conduct experiments on both artificial datasets and real sensor datasets. Artificial datasets are constructed by a 100×100 matrix representing a random deployed sensor network of 100 nodes within a 100h duration. The real sensor datasets are extracted from CitySee project [12], which has deployed a largescale wireless sensor network consisting of multiple subnetworks in a urban area in Wuxi, China. Specially, we compare the proposed method with a compressive sensing (CS)based method proposed in [11] on the temperature and humidity data generated from 55 sensors in 115 h.
The CSbased method proposed in [11] generates sampling matrix randomly and keeps original readings sparse in DCT domain. To detect abnormal readings, [11] decomposes the original reading d=d_{0}+d_{s} where d_{0} contains the normal readings and d_{s} contains the deviated values of abnormal readings and constructs a sparse basis for d=[d_{0},d_{s}]. The sink reconstructs sensor readings with linear programming (LP) techniques [26]. We generate sampling matrix with the same distribution in CSbased method and our proposed method with sampling rate about 47% on original readings (115×55 matrix).
In data gathering and recovery problem, event readings usually result in dynamic and diverse sparsity changes in both time and space domains, which may seriously undermine the foundation of CSbased method during environment changes. Prior works [6, 27, 28, 29, 30] have made an attempt to tackle data recovery with smallscale event readings with CSbased methods, e.g., events reported from several devices brought by device accidents or smallrange environment change. However, when events spread in large range and various time scales, it is doubtful whether CSbased data gathering method could deal with it or not. In this paper, we conduct the experiments to study the data recovery quality on the data with both smallrange events and largerange events on both CSbased method and the proposed method.
5.1 Recovery quality and robustness study on data with largescale events
In the meanwhile, CSbased method could not recover the data as shown in Figs. 1b and 2b. The recovered data in the event area are almost overwhelmed in the noise due to the changes of the sparsity foundation of CSbased method. And the recovery quality of CS method in other areas (except event area) is affected by event readings due to the violation of static sparsity. Therefore, CSbased method has limited recovery capability and less robustness against largescale event compared with the proposed method.
5.2 Recovery quality and robustness study on data with smallscale events
5.3 Recovery quality and robustness study on data without events
As shown in Fig. 10b, compressive sensingbased method can recover the temperature data in some degree. It is interesting to observe that CSbased method could hardly keep recovery quality stable, while the proposed method can achieve much better recovery quality as well as robustness. It can be further confirmed with the comparison of the SNR result of both methods at each sensor. The proposed method outperforms CSbased method in SNR with about 10 times (10 db) as shown in Fig. 3a.
6 Related work
In device networks, data gathering usually results in considerable communication overhead. Traditional approaches dealing with such problem include distributed source coding [31, 32], innetwork collaborative wavelet transform [33, 34, 35], holistic aggregation [36], and clustered data aggregation and compression [37, 38]. Though these approaches to some extent utilize the spatial correlation of device readings, they lack the ability to support the recovery of diversescale events.
In the past decade, compressive sensing (CS) has gained increasing attention due to its capacity of sparse signal sampling and reconstruction [39, 40] and triggered a large variety of applications, ranging from image processing to gathering geophysics data [41].
In terms of data gathering, various CSbased approaches have been proposed to the decentralized data compression and gathering of networked devices, aiming to efficiently collect data among a vast number of distributed nodes [6, 27, 28]. Liu et al. [7] present a novel compressive data collection scheme for IoT sensing networks adopting a powerlaw decaying data model verified by real data sets. Zheng et al. [8] propose another method handling with data gathering in IoT sensing networks by random walk algorithm. Xie and Jia [42] develop a clustering method that uses hybrid CS for device networks reducing the number of transmissions significantly. Li et al. [43] apply compressive sensing technique into data sampling and acquisition in IoT sensing networks and Internet of Things (IoT). Mamaghanian et al. [44] propose the potential of the compressed sensing for signal acquisition and compression in low complexity ECG data in wireless body device networks (WBSN). Zhang et al. [29] propose a compressive sensingbased approach for sparse target counting and positioning in IoT sensing networks. Tian and Giannakis [45] utilize compressed sensing technique for the coarse sensing task of spectrum hole identification. In addition, there are several papers researching in CS for device network focusing on throughput, routing, video streaming processing, and sparse event detection in [30, 46, 47, 48].
Cheng et al. [49] focus on dealing with continuous sensed data. Extracting kernel or dominant dataset from big sensory data in WSN provides another compressing method in [50, 51].
In recent years, lowrank matrix recovery (LRMR) extends the vectors’ sparsity to the low rank of matrices, becoming another important method to obtain and represent data after CS given only incomplete and indirect observations [10]. Keshavan et al. compared the performance of three matrix completion algorithms based on lowrank matrix completion with noisy observations [52]. Zhang et al. [53] present a spatiotemporal compressive sensing framework on Internet traffic matrices. Yi et al. [9] take advantage of both the lowrankness and the DCT compactness features improving the recovery accuracy. Compared with prior work based on LRMR, our method achieves better compression ratio and lower communication overhead.
7 Discussion
According to the analysis and experimental study, it is interesting to observe that the proposed method enables IoT networks the ability of dealing with both fidelity problem and magnitude problem simultaneously with diverse time/spacescale events and reduce globalscale communication cost without intensive edge computation.
The experiments of this paper on a real environmental IoT sensing network also reveal that constant sparsity hardly holds in real cases with diverse time/spacescale events, while lowrank property may be true. While events may violate constant sparsity in compressive sensing and reduce the recovery quality severely, the recovery quality of the proposed method still keeps the fidelity of event readings, which is about 10 times (10 db) better than typical compressive sensing [11] in terms of SNR. This observation may provide a fresh vision for research in both compressive sampling applications and IoT sensing and data aggregation scenarios.
However, it is worth noting that there is still limitation in the cases that low rank property does not hold in the network. To deal with this problem, this paper further generalizes the lowrankbased optimization design to a nuclear normbased optimization design, to make the proposed approach more general and robust. In future work, we would like to focus on enhancing the performance of our method in IoT networks with events.
8 Conclusion
In this paper, we have shown the effectiveness and validity of crossdomain matrix recovery in data compression, gathering, and recovery through the study on environmental IoT sensing datasets. It is obvious that the proposed method could be further extended to a large variety of other IoT application scenarios. In particular, we have demonstrated the capacity of the proposed MRCS method dealing with both data fidelity and magnitude problems simultaneously in data gathering of IoT networks, via both theoretical analysis and experimental study. The results show that the proposed MRCS method outperforms the original CS method in terms of recovery quality. Our work provides a new approach in both compressive sampling applications and IoT networks with diverse time/spacescale events and suggests a general design given the relaxation from lowrankbased optimization to nuclear normbased optimization.
Footnotes
Notes
Acknowledgements
Not applicable.
Funding
This work was financially supported by NSFC 61332004.
Availability of data and materials
The experiment is based on CitySee project [12]. We conduct the experiment on a subnetwork of 55 sensors within 115 h duration. The data can be found at: https://github.com/oleotiger/experimentaldata.
Authors’ contributions
The authors have contributed jointly to the manuscript. All authors have read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
 1.A. L. S. Orozco, J. R. Corripio, J. C. HernandezCastro, Source identification for mobile devices, based on wavelet transforms combined with sensor imperfections. Computing. 96(9), 829–841 (2014).CrossRefGoogle Scholar
 2.P. Kasirajan, C. Larsen, S. Jagannathan, A new data aggregation scheme via adaptive compression for wireless sensor networks. ACM Trans. Sens. Netw.9(1), 1–26 (2012).CrossRefGoogle Scholar
 3.G. Yang, M. Xiao, S. Zhang, Data aggregation scheme based on compressed sensing in wireless sensor network. Netw. J.8(1), 556–561 (2013).Google Scholar
 4.C. Tapparello, O. Simeone, M. Rossi, Dynamic compressiontransmission for energyharvesting multihop networks with correlated sources. IEEE ACM Trans. Netw.22(6), 1729–1741 (2014).CrossRefGoogle Scholar
 5.A. Zahedi, J. Ostergaard, S. H. Jensen, P. Naylor, S. Bech, in Data Compression Conference (DCC), 2014. Distributed remote vector gaussian source coding for wireless acoustic sensor networks (IEEE, 2014), pp. 263–272.Google Scholar
 6.J. Haupt, W. U. Bajwa, M. Rabbat, R. Nowak, Compressed sensing for networked data. IEEE Signal Process. Mag.25(2), 92–101 (2008).CrossRefGoogle Scholar
 7.X. Y. Liu, Y. Zhu, L. Kong, C. Liu, Cdc : Compressive data collection for wireless sensor networks. IEEE Trans. Parallel Distrib. Syst.26(8), 2188–2197 (2015).CrossRefGoogle Scholar
 8.H. Zheng, F. Yang, X. Tian, X. Gan, X. Wang, S. Xiao, Data gathering with compressive sensing in wireless sensor networks: a random walk based approach. IEEE Trans. Parallel Distrib. Syst.26(1), 35–44 (2014).CrossRefGoogle Scholar
 9.K. Yi, J. Wan, T. Bao, L. Yao, A DCT regularized matrix completion algorithm for energy efficient data gathering in wireless sensor networks. Int. Distrib. J. Sensor Netw.2015(1), 96 (2015).Google Scholar
 10.S. Maok, Y. Xie, Maximum entropy lowrank matrix recovery. IEEE J. Sel. Top. Sign. Process. (2017). IEEE.Google Scholar
 11.C. Luo, F. Wu, J. Sun, C. W. Chen, in Proceedings of the 15th annual international conference on Mobile computing and networking. Compressive data gathering for largescale wireless sensor networks (ACM, 2009), pp. 145–156.Google Scholar
 12.X. Mao, X. Miao, Y. He, X. Y. Li, Y. Liu, in INFOCOM, 2012 Proceedings IEEE. Citysee: urban CO2 monitoring with sensors (IEEE, 2012), pp. 1611–1619.Google Scholar
 13.E. J. Candes, B. Recht, Exact matrix completion via convex optimization. Found. Comput. Math.9(6), 717 (2009).MathSciNetCrossRefGoogle Scholar
 14.X. Cheng, A. Thaeler, G. Xue, D. Chen, in INFOCOM 2004. Twentythird AnnualJoint Conference of the IEEE Computer and Communications Societies. vol. 4. TPS: a timebased positioning scheme for outdoor wireless sensor networks (IEEE, 2004), pp. 2685–2696. https://doi.org/10.1109/INFCOM.2004.1354687.
 15.F. Liu, X. Cheng, D. Hua, D. Chen, in Wireless Sensor Networks and Applications. TPSS: a timebased positioning scheme for sensor networks with short range beacons (Springer, 2005), pp. 175–193.Google Scholar
 16.A. Thaeler, M. Ding, X. Cheng, iTPS: an improved location discovery scheme for sensor networks with longrange beacons. Parallel. J. Distrib. Comput.65(2), 98–106 (2005).CrossRefGoogle Scholar
 17.K. Sun, P. Ning, C. Wang, Secure and resilient clock synchronization in wireless sensor networks. IEEE Sel. J. Areas Commun.24(2), 395–408 (2006).CrossRefGoogle Scholar
 18.L. Chen, J. Leneutre, in Parallel Processing Workshops, 2006. ICPP 2006 Workshops. 2006 International Conference on. A secure and scalable time synchronization protocol in ieee 802.11 ad hoc networks (IEEE, 2006), pp. 8–pp.Google Scholar
 19.K. Römer, in Proceedings of the 2nd ACM international symposium on Mobile ad hoc networking & computing. Time synchronization in ad hoc networks (ACM, 2001), pp. 173–182. https://doi.org/10.1145/501416.501440.
 20.O. Goldreich, Foundations of cryptography volume 2. 2(1), 1–14 (2004). Cambridge University Press.Google Scholar
 21.E. J. Candes, T. Tao, Nearoptimal signal recovery from random projections: universal encoding strategies?IEEE Trans. Inf Theory. 52(12), 5406–5425 (2004).MathSciNetCrossRefGoogle Scholar
 22.J. Cheng, Q. Ye, H. Jiang, D. Wang, C. Wang, STCDG: an efficient data gathering algorithm based on matrix completion for wireless sensor networks. IEEE Trans. Wirel. Commun.12(2), 850–861 (2013).CrossRefGoogle Scholar
 23.E. Richard, P. A. Savalle, N. Vayatis, Estimation of simultaneously sparse and low rank matrices. Int. Conf. Mach. Learn., 51–58 (2012).Google Scholar
 24.KC Toh, S Yun, An accelerated proximal gradient algorithm for nuclear norm regularized least squares problems. Pac. Optim. J.6(3), 615–640 (2009).MathSciNetzbMATHGoogle Scholar
 25.E. Candes, B. Recht, Simple bounds for lowcomplexity model reconstruction. Acta Bot. Gallica Bull. Soc. Bot Fr.156(3), 477–486 (2011).Google Scholar
 26.D. L. Donoho, Compressed sensing. IEEE Trans. Inf. Theory. 52:, 1289–1306 (2006).MathSciNetCrossRefGoogle Scholar
 27.C. Luo, F. Wu, J. Sun, C. W. Chen, in Proceedings of the 15th annual international conference on Mobile computing and networking. Compressive data gathering for largescale wireless sensor networks (ACM, 2009), pp. 145–156.Google Scholar
 28.J. Wang, S. Tang, B. Yin, X. Y. Li, in INFOCOM, 2012 Proceedings IEEE. Data gathering in wireless sensor networks through intelligent compressive sensing (IEEE, 2012), pp. 603–611.Google Scholar
 29.B. Zhang, X. Cheng, N. Zhang, Y. Cui, Y. Li, Q. Liang, in INFOCOM, 2011 Proceedings IEEE. Sparse target counting and localization in sensor networks based on compressive sensing (IEEE, 2011), pp. 2255–2263.Google Scholar
 30.J. Meng, H. Li, Z. Han, in Information Sciences and Systems, 2009. CISS 2009. 43rd Annual Conference on. Sparse event detection in wireless sensor networks using compressive sensing (IEEE, 2009), pp. 181–185.Google Scholar
 31.A. Zahedi, J. Ostergaard, S. H. Jensen, P. Naylor, S. Bech, in Data Compression Conference (DCC), 2014. Distributed remote vector Gaussian source coding for wireless acoustic sensor networks (IEEE, 2014), pp. 263–272.Google Scholar
 32.A. J. Aljohani, S. X. Ng, L. Hanzo, Distributed source coding and its applications in relayingbased transmission. IEEE Access. 4:, 1940–1970 (2016).CrossRefGoogle Scholar
 33.A. Ciancio, S. Pattem, A. Ortega, B. Krishnamachari, in Proceedings of the 5th international conference on Information processing in sensor networks. Energyefficient data representation and routing for wireless sensor networks based on a distributed wavelet compression algorithm (ACM, 2006), pp. 309–316.Google Scholar
 34.M. Crovella, E. Kolaczyk, in INFOCOM 2003. TwentySecond Annual Joint Conference of the IEEE Computer and Communications. IEEE Societies. Graph wavelets for spatial traffic analysis, vol. 3 (IEEE, 2003), pp. 1848–1857.Google Scholar
 35.X. H. Xu, X. Y. Li, P. J. Wan, S. J. Tang, Efficient scheduling for periodic aggregation queries in multihop sensor networks. IEEE/ACM Trans. Netw.20(3), 690–698 (2012).CrossRefGoogle Scholar
 36.J. Li, S. Cheng, Y. Li, Z. Cai, Approximate holistic aggregation in wireless sensor networks. ACM Trans. Sensor Netw.13(2), 11 (2017).CrossRefGoogle Scholar
 37.A. Sinha, D. K. Lobiyal, Performance evaluation of data aggregation for clusterbased wireless sensor network. Humancentric Comput. Inf. Sci.3(1), 13 (2013).CrossRefGoogle Scholar
 38.X. Xu, R. Ansari, A. Khokhar, A. V. Vasilakos, Hierarchical data aggregation using compressive sensing (HDACS) in WSNs. ACM Trans. Sensor Netw.11(3), 1–25 (2015).CrossRefGoogle Scholar
 39.E. J. Candès, J. Romberg, T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory. 52(2), 489–509 (2006).MathSciNetCrossRefGoogle Scholar
 40.D. L. Donoho, Compressed sensing. IEEE Trans. Inf. Theory. 52(4), 1289–1306 (2006).MathSciNetCrossRefGoogle Scholar
 41.S. Qaisar, R. M. Bilal, W. Iqbal, M. Naureen, S. Lee, Compressive sensing: from theory to applications, a survey. Commun. J. Netw.15(5), 443–456 (2013).CrossRefGoogle Scholar
 42.R. Xie, X. Jia, Transmissionefficient clustering method for wireless sensor networks using compressive sensing. IEEE Trans. Parallel Distrib. Syst.25(3), 806–815 (2014).CrossRefGoogle Scholar
 43.S. Li, L. D. Xu, X. Wang, Compressed sensing signal and data acquisition in wireless sensor networks and Internet of Things. IEEE Trans. Ind. Inform.9(4), 2177–2186 (2013).CrossRefGoogle Scholar
 44.H. Mamaghanian, N. Khaled, D. Atienza, P. Vandergheynst, Compressed sensing for realtime energyefficient ECG compression on wireless body sensor nodes. IEEE Trans. Biomed. Eng.58(9), 2456–2466 (2011).CrossRefGoogle Scholar
 45.Z. Tian, G. B. Giannakis, in IEEE International Conference on Acoustics, Speech and Signal Processing. Compressed sensing for wideband cognitive radios (Michigan Technological Univ Houghton, 2007), pp. 1357–1360.Google Scholar
 46.J. Luo, L. Xiang, C. Rosenberg, in Communications (ICC), 2010 IEEE international conference on. Does compressed sensing improve the throughput of wireless sensor networks? (IEEE, 2010), pp. 1–6.Google Scholar
 47.S. Lee, S. Pattem, M. Sathiamoorthy, B. Krishnamachari, A. Ortega, in International Conference on GeoSensor Networks. Spatiallylocalized compressed sensing and routing in multihop sensor networks (Springer, 2009), pp. 11–20.Google Scholar
 48.S. Pudlewski, A. Prasanna, T. Melodia, Compressedsensingenabled video streaming for wireless multimedia sensor networks. IEEE Trans. Mob. Comput.11(6), 1060–1072 (2012).CrossRefGoogle Scholar
 49.S. Cheng, Z. Cai, J. Li, Curve query processing in wireless sensor networks. IEEE Trans. Veh. Technol.64(11), 5198–5209 (2015).CrossRefGoogle Scholar
 50.S. Cheng, Z. Cai, J. Li, H. Gao, Extracting kernel dataset from big sensory data in wireless sensor networks. IEEE Trans. Knowl. Data Eng.29(4), 813–827 (2017).CrossRefGoogle Scholar
 51.S. Cheng, Z. Cai, J. Li, X. Fang, in Computer Communications (INFOCOM), 2015 IEEE Conference on. Drawing dominant dataset from big sensory data in wireless sensor networks (IEEE, 2015), pp. 531–539.Google Scholar
 52.R. H. Keshavan, A. Montanari, S. Oh, in Communication, Control, and Computing, 2009. Allerton 2009. 47th Annual Allerton Conference on. Lowrank matrix completion with noisy observations: a quantitative comparison (IEEE, 2009), pp. 1216–1222.Google Scholar
 53.Y. Zhang, M. Roughan, W. Willinger, L. Qiu, in ACM SIGCOMM Computer Communication Review. Spatiotemporal compressive sensing and Internet traffic matrices, vol. 39(4) (ACM, 2009), pp. 267–278.Google Scholar
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