LDPC Codes for Compressed Sensing

We present a mathematical connection between channel coding and compressed sensing. In particular, we link, on the one hand, channel coding linear programming decoding (CC-LPD), which is a well-known relaxation of maximum-likelihood channel decoding for binary linear codes, and, on the other hand, c...

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Vydáno v:	IEEE transactions on information theory Ročník 58; číslo 5; s. 3093 - 3114
Hlavní autoři:	Dimakis, A. G., Smarandache, R., Vontobel, P. O.
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York, NY IEEE 01.05.2012 Institute of Electrical and Electronics Engineers The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Témata:	Applied sciences Approximation guarantee Approximation methods basis pursuit Channel coding Channels Coding theory Coding, codes Compressed Compressed sensing Data compression Decoding Detection Exact sciences and technology graph cover Information theory Information, signal and communications theory Linear programming linear programming decoding Low density parity check codes Mathematical analysis Matrices Matrix Matrix methods Maximum likelihood decoding Maximum likelihood method pseudocodeword pseudoweight Sampling, quantization Signal and communications theory sparse approximation Sparse matrices Telecommunications and information theory Vectors zero-infinity operator Performance evaluation Approximation guarantee High density pseudocodeword Linear channel Binary code Linear code Relaxation graph cover Linear equation zero-infinity operator Sparse representation Parity check codes Deterministic approach Binary channel Linear programming linear programming decoding basis pursuit Channel coding Maximum likelihood decoding sparse approximation pseudoweight Error correcting code Compressed sensing Parity check
ISSN:	0018-9448, 1557-9654
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Popis
Shrnutí:	We present a mathematical connection between channel coding and compressed sensing. In particular, we link, on the one hand, channel coding linear programming decoding (CC-LPD), which is a well-known relaxation of maximum-likelihood channel decoding for binary linear codes, and, on the other hand, compressed sensing linear programming decoding (CS-LPD), also known as basis pursuit, which is a widely used linear programming relaxation for the problem of finding the sparsest solution of an underdetermined system of linear equations. More specifically, we establish a tight connection between CS-LPD based on a zero-one measurement matrix over the reals and CC-LPD of the binary linear channel code that is obtained by viewing this measurement matrix as a binary parity-check matrix. This connection allows the translation of performance guarantees from one setup to the other. The main message of this paper is that parity-check matrices of "good" channel codes can be used as provably "good" measurement matrices under basis pursuit. In particular, we provide the first deterministic construction of compressed sensing measurement matrices with an order-optimal number of rows using high-girth low-density parity-check codes constructed by Gallager.
Bibliografie:	SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 ObjectType-Article-2 content type line 23
ISSN:	0018-9448 1557-9654
DOI:	10.1109/TIT.2011.2181819