Reconstruction Guarantee Analysis of Basis Pursuit for Binary Measurement Matrices in Compressed Sensing

Recently, binary 0-1 measurement matrices, especially those from coding theory, were introduced to compressed sensing. Dimakis et al. found that the linear programming (LP) decoding of LDPC codes is very similar to the LP reconstruction of compressed sensing, and they further showed that the sparse...

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Veröffentlicht in:IEEE transactions on information theory Jg. 63; H. 5; S. 2922 - 2932
Hauptverfasser: Liu, Xin-Ji, Xia, Shu-Tao, Fu, Fang-Wei
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York IEEE 01.05.2017
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:
basis pursuit
binary matrix
Codes
Compressed sensing
Decoding
Detection
deterministic construction
girth
LDPC codes
Linear programming
Low density parity check codes
Lower bounds
l₁-minimization
Matrices (mathematics)
measurement matrix
minimum BP weight
Parity check codes
Reconstruction
reconstruction guarantee
Sparks
Sparse matrices
Weight measurement
ISSN:0018-9448, 1557-9654
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Zusammenfassung:Recently, binary 0-1 measurement matrices, especially those from coding theory, were introduced to compressed sensing. Dimakis et al. found that the linear programming (LP) decoding of LDPC codes is very similar to the LP reconstruction of compressed sensing, and they further showed that the sparse binary parity-check matrices of good LDPC codes can be used as provably good measurement matrices for compressed sensing under basis pursuit (BP). Moreover, 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 BP. For a fixed measurement matrix, we first introduce a performance indicator w min BP called minimum BP weight, and show that any k-sparse signals could be exactly recovered by BP if and only if k ≤ (w min BP - 1)/2. Then, lower bounds of w min BP are studied. Borrowing ideas from the tree bound for the LDPC codes, we obtain several explicit lower bounds of w BP min , which improve on the previous results in some cases. These lower bounds also imply explicit ℓ 1 /ℓ 1 , ℓ 2 /ℓ 1 and ℓ ∞ /ℓ 1 sparse approximation guarantees, and further confirm that large girth has positive impacts on the performance of binary measurement matrices under BP.
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2017.2677965