LDPC Codes for Compressed Sensing

TL;DR

This paper establishes a mathematical link between channel coding and compressed sensing, showing how LDPC codes can be used as effective measurement matrices for basis pursuit, enabling deterministic constructions with optimal size.

Contribution

It introduces a tight connection between LP decoding in channel coding and compressed sensing, enabling the transfer of performance guarantees and proposing LDPC codes as measurement matrices.

Findings

Parity-check matrices of good channel codes are effective measurement matrices.

First deterministic construction of compressed sensing matrices with optimal size.

High-girth LDPC codes can be used for basis pursuit.

Abstract

We present a mathematical connection between channel coding and compressed sensing. In particular, we link, on the one hand, \emph{channel coding linear programming decoding (CC-LPD)}, which is a well-known relaxation o maximum-likelihood channel decoding for binary linear codes, and, on the other hand, \emph{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 under-determined system of linear equations. More specifically, we establis 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 (LDPC) codes constructed by Gallager.