Array LDPC Code-based Compressive Sensing

Abstract

In this paper, we focus on the problem of compressive sensing using binary measurement matrices, and basis pursuit as the recovery algorithm. We obtain new lower bounds on the number of samples to achieve robust sparse recovery using binary matrices and derive sufficient conditions for a binary matrix with fixed column-weight to satisfy the robust null space property. Next we prove that any column-regular binary matrix with girth 6 has nearly optimal number of measurements. Then we show that the parity check matrices of array LDPC codes are nearly optimal in the sense of having girth six and almost satisfying the lower bound on the number of samples. Array code parity check matrices demonstrate an example of binary matrices that achieve guaranteed recovery via robust null-space property and in practice for $n \leq 10^{6}$ provide faster recovery compared to the Gaussian counterpart. This is an extended abstract without proofs. The full paper with additional details can be found in [1].