Compressed sensing with sparse binary matrices: Instance optimal error guarantees in near-optimal time

This paper considers the problem of Quantitative Group Testing (QGT). Consider a set of N items among which K items are defective. The QGT problem is to identify (all or a sufficiently large fraction of) the defective items, where the result of a test reveals the number of defective items in the tested group. In this work, we propose a nonadaptive QGT algorithm using sparse graph codes over biregular bipartite graphs with left-degree ℓ and right degree r and binary t-error-correcting BCH codes. The proposed scheme provides exact recovery with probabilistic guarantee, i.e. recovers all the defective items with high probability. In particular, we show that for the sub-linear regime where K N vanishes as K, N → ∞, the proposed algorithm requires at most m = c(t)K t log 2 ℓN c(t)K + 1 + 1 + 1 tests to recover all the defective items with probability approaching one as K, N → ∞, where c(t) depends only on t. The results of our theoretical analysis reveal that the minimum number of required tests is achieved by t = 2. The encoding and decoding of the proposed algorithm for any t ≤ 4 have the computational complexity of O(K log 2 N K ) and O(K log N K ), respectively. Our simulation results also show that the proposed algorithm significantly outperforms a non-adaptive semi-quantitative group testing algorithm recently proposed by Abdalla et al. in terms of the required number of tests for identifying all the defective items with high probability.

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“…Next, we shall show that for any λ ≥ λ T (ℓ), we have p * > 0. From (13), it follows that λ = ln(1 − p * 1 ℓ−1 ) −p * .…”

Section: Proof Of Lemmamentioning

confidence: 99%

Sparse Graph Codes for Non-adaptive Quantitative Group Testing

Karimi

Kazemi

Heidarzadeh

et al. 2019

2019 IEEE Information Theory Workshop (ITW)

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“…Note that it's runtime is only O(m log 4 m), making it optimal up to log factors (recall Theorem 3). For the compressive sensing method we will utilize the following algorithmic result from [28]. Theorem 7 now follows easily from Theorem 3 with n = 0, and Theorem 8.…”

Section: Extension: Sublinear-time Phase Retrieval For Compressible Smentioning

confidence: 99%

Fast Phase Retrieval from Local Correlation Measurements

Iwen¹,

Viswanathan²,

Wang³

2016

SIAM J. Imaging Sci.

103

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We develop a fast phase retrieval method which can utilize a large class of local phaseless correlation-based measurements in order to recover a given signal x ∈ C d (up to an unknown global phase) in near-linear O d log 4 d -time. Accompanying theoretical analysis proves that the proposed algorithm is guaranteed to deterministically recover all signals x satisfying a natural flatness (i.e., non-sparsity) condition for a particular choice of deterministic correlation-based measurements. A randomized version of these same measurements is then shown to provide nonuniform probabilistic recovery guarantees for arbitrary signals x ∈ C d . Numerical experiments demonstrate the method's speed, accuracy, and robustness in practice -all code is made publicly available.In its simplest form, our proposed phase retrieval method employs a modified lifting scheme akin to the one utilized by the well-known PhaseLift algorithm. In particular, it interprets quadratic magnitude measurements of x as linear measurements of a restricted set of lifted variables, xixj, for |j − i| < δ d. This leads to a linear system involving a total of (2δ − 1)d unknown lifted variables, all of which can then be solved for using only O(δd) measurements. Once these lifted variables, xixj for |j − i| < δ d, have been recovered, a fast angular synchronization method can then be used to propagate the local phase difference information they provide across the entire vector in order to estimate the (relative) phases of every entry of x. In addition, the lifted variables corresponding to xjxj = |xj| 2 automatically provide magnitude estimates for each entry, xj, of x. The proposed phase retrieval method then approximates x by carefully combining these entry-wise phase and magnitude estimates.Finally, we conclude by developing an extension of the proposed method to the sparse phase retrieval problem; specifically, we demonstrate a sublinear-time compressive phase retrieval algorithm which is guaranteed to recover a given s-sparse vector x ∈ C d with high probability in just O(s log 5 s · log d)-time using only O(s log 4 s · log d) magnitude measurements. In doing so we demonstrate the existence of compressive phase retrieval algorithms with near-optimal linear-in-sparsity runtime complexities.

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“…Wang et al showed sparse random matrix was sufficient for data recovery, and the performance was comparable to the optimal k-term approximation [13]. The sparse binary matrices were studied in [14][15][16][17]. The class of matrices satisfied RIP-p property, which was a weaker form of RIP property [18].…”

Section: Related Workmentioning

confidence: 99%

Compressed sensing and mobile agent based sparse data collection in wireless sensor networks

Wang

Shen

et al. 2015

2015 IEEE International Instrumentation and Measurement Technology Conference (I2MTC) Proceedings

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In the paper, we study how the theory of compressed sensing is applied in wireless sensor networks to capture sparse signals of interest. Traditional data collection and compressive data gathering separately use client/server paradigm and dense random matrix, leading high computation load and energy consumption. We adopt sparse binary matrix for the measurement matrix. A mobile agent based compressed sensing paradigm is proposed. In the paradigm, the sensed data are stored locally, and it doesn't require network topology as prior knowledge. For the routing of mobile agents, we formulate a Mobile Agent Path Design Algorithm(MAPDA), which is designed by greedy algorithm. Finally Orthogonal Matching Pursuit(OMP) algorithm is used for signal recovery. The experimental results show the proposed algorithm can save energy than client/server paradigm based on different sparse binary matrix and plain-CS. In addition, we observe that sparse binary matrix has the same reconstruction performance as Gaussian random matrix(dense random matrix).

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Compressed sensing with sparse binary matrices: Instance optimal error guarantees in near-optimal time

Abstract: A compressed sensing method consists of a rectangular measurement matrix, M ∈

Cited by 30 publications

References 23 publications

Sparse Graph Codes for Non-adaptive Quantitative Group Testing

Sparse Graph Codes for Non-adaptive Quantitative Group Testing

Fast Phase Retrieval from Local Correlation Measurements

Compressed sensing and mobile agent based sparse data collection in wireless sensor networks

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