A generalized LDPC framework for robust and sublinear compressive sensing

Abstract

Compressive sensing aims to recover a high-dimensional sparse signal from a relatively small number of measurements. In this paper, a novel design of the measurement matrix is proposed. The design is inspired by the construction of generalized low-density parity-check codes, where the capacity-achieving point-to-point codes serve as subcodes to robustly estimate the signal support. In the case that each entry of the n-dimensional ft-sparse signal lies in a known discrete alphabet, the proposed scheme requires only O(k log n) measurements and arithmetic operations. In the case of arbitrary, possibly continuous alphabet, an error propagation graph is proposed to characterize the residual estimation error. With O(k log2 n) measurements and computational complexity, the reconstruction error can be made arbitrarily small with high probability.