LDPC Codes for Compressed Sensing

• Published in: IEEE transactions on information theory Vol. 58; no. 5; pp. 3093 - 3114
• Main Authors: Dimakis, A. G., Smarandache, R., Vontobel, P. O.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.05.2012; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: 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
• DOI: 10.1109/TIT.2011.2181819

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.