Explicit matrices for sparse approximation

Khajehnejad, Amin; Tehrani, Arash Saber; Dimakis, Alexandros G.; Hassibi, Babak

doi:10.1109/isit.2011.6034170

Cited by 18 publications

(16 citation statements)

References 27 publications

(43 reference statements)

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“…Note that girth is an even number not smaller than 4. Deterministic matrices with RIP or NSP have no general ways to be definitely constructed or efficiently verified [9]. Instead, in [7], [9], girth is used to evaluate the performance of sparse binary measurement matrices under basis pursuit.…”

Section: Introductionmentioning

confidence: 99%

“…Deterministic matrices with RIP or NSP have no general ways to be definitely constructed or efficiently verified [9]. Instead, in [7], [9], girth is used to evaluate the performance of sparse binary measurement matrices under basis pursuit. It was shown that binary measurement matrices with girth Ω(log n), uniform row weights and uniform column weights have robust recovery guarantees under basis pursuit with high probability.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Reconstruction guarantee analysis of binary measurement matrices based on girth

Liu

Xia

2013

2013 IEEE International Symposium on Information Theory

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Binary 0-1 measurement matrices, especially those from coding theory, were introduced to compressed sensing (CS) recently. Good measurement matrices with preferred properties, e.g., the restricted isometry property (RIP) and nullspace property (NSP), have no known general ways to be efficiently checked. Khajehnejad et al. made use of girth to certify the good performances of sparse binary measurement matrices. In this paper, we examine the performance of binary measurement matrices with uniform column weight and arbitrary girth under basis pursuit. Explicit sufficient conditions of exact reconstruction are obtained, which improve the previous results derived from RIP for any girth g and results from NSP when g/2 is odd. Moreover, we derive explicit l1/l1, l2/l1 and l∞/l1 sparse approximation guarantees. These results further show that large girth has positive impacts on the performance of binary measurement matrices under basis pursuit, and the binary paritycheck matrices of good LDPC codes are important candidates of measurement matrices.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Reconstruction guarantee analysis of binary measurement matrices based on girth

Liu

Xia

2013

2013 IEEE International Symposium on Information Theory

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“…This runs counter to the intuition in LDPC decoding, where one wishes to design binary matrices with large girth. Indeed, in [47], the authors build on an earlier paper [48] and develop a message-passing type of decoder that achieves order-optimality using a binary matrix. The binary matrices that is used in [47] all have large girth Ω(log n) which is the theoretical upper bound.…”

Section: Discussionmentioning

confidence: 99%

Compressed Sensing Using Binary Matrices of Nearly Optimal Dimensions

Lotfi

Vidyasagar

2020

IEEE Trans. Signal Process.

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In this paper, we study the problem of compressed sensing using binary measurement matrices, and 1 -norm minimization (basis pursuit) as the recovery algorithm. We derive new upper and lower bounds on the number of measurements to achieve robust sparse recovery with binary matrices. We establish sufficient conditions for a column-regular binary matrix to satisfy the robust null space property (RNSP), and show that the sparsity bounds for robust sparse recovery obtained using the RNSP are better by a factor of (3 √ 3)/2 ≈ 2.6 compared to the restricted isometry property (RIP). Next we derive universal lower bounds on the number of measurements that any binary matrix needs to have in order to satisfy the weaker sufficient condition based on the RNSP, and show that bipartite graphs of girth six are optimal. Then we display two classes of binary matrices, namely parity check matrices of array codes, and Euler squares, that have girth six and are nearly optimal in the sense of almost satisfying the lower bound. In principle randomly generated Gaussian measurement matrices are "order-optimal." So we compare the phase transition behavior of the basis pursuit formulation using binary array code and Gaussian matrices, and show that (i) there is essentially no difference between the phase transition boundaries in the two cases, and (ii) the CPU time of basis pursuit with binary matrices is hundreds of times faster than with Gaussian matrices, and the storage requirements are less. Therefore it is suggested that binary matrices are a viable alternative to Gaussian matrices for compressed sensing using basis pursuit.

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“…Besides these progresses concerning RIP, there are also some other related contributions. For example, using the PEG algorithm for constructing the parity-check matrices of LDPC codes, Tehrani et al proposed a family of explicit NSP matrices with m = O(k log(n/k)) that can recover most k-sparse signals under 1 -minimization with probability 1 − 1/n [29,30]. In [49], we showed that for a binary matrix H, spark(H) ≥ d(C), where d(C) denotes the minimum distance of the binary code C defined by H. Thus, we can construct an explicit m × n matrix H with m = O(n) by using an LDPC code C with d(C) = O(n) (e.g.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Deterministic Constructions of Binary Measurement Matrices From Finite Geometry

Xia

Liu

Jiang

et al. 2015

IEEE Trans. Signal Process.

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Abstract-Deterministic constructions of measurement matrices in compressed sensing (CS) are considered in this paper. The constructions are inspired by the recent discovery of Dimakis, Smarandache and Vontobel which says that parity-check matrices of good low-density parity-check (LDPC) codes can be used as provably good measurement matrices for compressed sensing under 1-minimization. The performance of the proposed binary measurement matrices is mainly theoretically analyzed with the help of the analyzing methods and results from (finite geometry) LDPC codes. Particularly, several lower bounds of the spark (i.e., the smallest number of columns that are linearly dependent, which totally characterizes the recovery performance of 0-minimization) of general binary matrices and finite geometry matrices are obtained and they improve the previously known results in most cases. Simulation results show that the proposed matrices perform comparably to, sometimes even better than, the corresponding Gaussian random matrices. Moreover, the proposed matrices are sparse, binary, and most of them have cyclic or quasi-cyclic structure, which will make the hardware realization convenient and easy.

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Explicit matrices for sparse approximation

Cited by 18 publications

References 27 publications

Reconstruction guarantee analysis of binary measurement matrices based on girth

Reconstruction guarantee analysis of binary measurement matrices based on girth

Compressed Sensing Using Binary Matrices of Nearly Optimal Dimensions

Deterministic Constructions of Binary Measurement Matrices From Finite Geometry

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