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This work examines the compressed sensor caching problem in wireless sensor networks and devises efficient distributed sparse data recovery algorithms to enable collaboration among multiple caches. In this problem, each cache is only allowed to access measurements from a small subset of sensors within its vicinity to reduce both cache size and data acquisition overhead. To enable reliable data recovery with limited access to measurements, we propose a distributed sparse data recovery method, called the collaborative sparse recovery by anchor alignment (CoSR-AA) algorithm, where collaboration among caches is enabled by aligning their locally recovered data at a few anchor nodes. The proposed algorithm is based on the consensus alternating direction method of multipliers (ADMM) algorithm but with message exchange that is reduced by considering the proposed anchor alignment strategy. Then, by the deep unfolding of the ADMM iterations, we further propose the Deep CoSR-AA algorithm that can be used to significantly reduce the number of iterations. We obtain a graph neural network architecture where message exchange is done more efficiently by an embedded autoencoder. Simulations are provided to demonstrate the effectiveness of the proposed collaborative recovery algorithms in terms of the improved reconstruction quality and the reduced communication overhead due to anchor alignment., Comment: v1 was submitted to IEEE Transactions on Signal Processing on Sept. 18, 2023

Conventional sparse phase retrieval schemes can recover sparse signals from the magnitude of linear measurements only up to a global phase ambiguity. This work proposes a novel approach that instead utilizes the magnitude of affine measurements to achieve ambiguity-free signal reconstruction. The proposed method relies on two-stage approach that consists of support identification followed by the exact recovery of nonzero signal entries. In the noise-free case, perfect support identification using a simple counting rule is guaranteed subject to a mild condition on the signal sparsity, and subsequent exact recovery of the nonzero signal entries can be obtained in closed-form. The proposed approach is then extended to two noisy scenarios, namely, sparse noise (or outliers) and non-sparse bounded noise. For both cases, perfect support identification is still ensured under mild conditions on the noise model, namely, the support size for sparse outliers and the power of the bounded noise. Under perfect support identification, exact signal recovery can be achieved using a simple majority rule for the sparse noise scenario, and reconstruction up to a bounded error can be achieved using linear least-squares (LS) estimation for the non-sparse bounded noise scenario. The obtained analytic performance guarantee for the latter case also sheds light on the construction of the sensing matrix and bias vector. In fact, we show that a near optimal performance can be achieved with high probability by the random generation of the nonzero entries of the sparse sensing matrix and bias vector according to the uniform distribution over a circle. Computer simulations using both synthetic and real-world data sets are provided to demonstrate the effectiveness of the proposed scheme.

Sparse subspace clustering (SSC) relies on sparse regression for accurate neighbor identification. Inspired by recent progress in compressive sensing, this paper proposes a new sparse regression scheme for SSC via two-step reweighted $\ell_1$-minimization, which also generalizes a two-step $\ell_1$-minimization algorithm introduced by E. J. Cand\`es et al in [The Annals of Statistics, vol. 42, no. 2, pp. 669-699, 2014] without incurring extra algorithmic complexity. To fully exploit the prior information offered by the computed sparse representation vector in the first step, our approach places a weight on each component of the regression vector, and solves a weighted LASSO in the second step. We propose a data weighting rule suitable for enhancing neighbor identification accuracy. Then, under the formulation of the dual problem of weighted LASSO, we study in depth the theoretical neighbor recovery rates of the proposed scheme. Specifically, an interesting connection between the locations of nonzeros of the optimal sparse solution to the weighted LASSO and the indexes of the active constraints of the dual problem is established. Afterwards, under the semi-random model, analytic probability lower/upper bounds for various neighbor recovery events are derived. Our analytic results confirm that, with the aid of data weighting and if the prior neighbor information is enough accurate, the proposed scheme with a higher probability can produce many correct neighbors and few incorrect neighbors as compared to the solution without data weighting. Computer simulations are provided to validate our analytic study and evidence the effectiveness of the proposed approach.

This paper proposes a fusion-based cooperative support identification scheme for distributed compressive sparse signal recovery via resource-constrained wireless sensor networks. The proposed support identification protocol involves: (i) local sparse sensing for economizing data gathering and storage, (ii) local binary decision making for partial support knowledge inference, (iii) binary information exchange among active nodes, and (iv) binary data aggregation for support estimation. Then, with the aid of the estimated signal support, a refined local decision is made at each node. Only the measurements of those informative nodes will be sent to the fusion center, which employs a weighted $\ell_1$-minimization for global signal reconstruction. The design of a Bayesian local decision rule is discussed, and the average communication cost is analyzed. Computer simulations are used to illustrate the effectiveness of the proposed scheme.

To strike a balance between energy efficiency and data quality control, this paper proposes a sensor censoring scheme for distributed sparse signal recovery via compressive-sensing based wireless sensor networks. In the proposed approach, each sensor node employs a sparse sensing vector with known support for data compression, meanwhile enabling making local inference about the unknown support of the sparse signal vector of interest. This naturally leads to a ternary censoring protocol, whereby each sensor (i) directly transmits the real-valued compressed data if the sensing vector support is detected to be overlapped with the signal support, (ii) sends a one-bit hard decision if empty support overlap is inferred, (iii) keeps silent if the measurement is judged to be uninformative. Our design then aims at minimizing the error probability that empty support overlap is decided but otherwise is true, subject to the constraints on a tolerable false-alarm probability that non-empty support overlap is decided but otherwise is true, and a target censoring rate. We derive a closed-form formula of the optimal censoring rule; a low complexity implementation using bi-section search is also developed. In addition, the average communication cost is analyzed. To aid global signal reconstruction under the proposed censoring framework, we propose a modified l_1-minimization based algorithm, which exploits certain sparse nature of the hard decision vector received at the fusion center. Analytic performance guarantees, characterized in terms of the restricted isometry property, are also derived. Computer simulations are used to illustrate the performance of the proposed scheme., Comment: 30 pages, 9 figures