2012/01/31 by Alexandros G. Dimakis, Roxana Smarandache, Pascal O. Vontobel · 155 citations
Computer Science · Engineering · Mathematics · #Algorithm #Basis pursuit #Block code #Compressed sensing #Computer science #Concatenated error correction code #Decoding methods #Energy Harvesting in Wireless Networks #Error Correcting Code Techniques #Linear code #Linear programming #List decoding #Low-density parity-check code #Matching pursuit #Mathematics #Matrix (chemical analysis) #Parity-check matrix #Sparse and Compressive Sensing Techniques #Theoretical computer science
paper · open access · doi:10.1109/tit.2011.2181819
published in IEEE Transactions on Information Theory 58(5), 3093-3114 (Institute of Electrical and Electronics Engineers)
openalex publication_date 2012/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.