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Compressive sensing aims to recover a high-dimensional sparse signal from arelatively small number of measurements. In this paper, a novel design of themeasurement matrix is proposed. The design is inspired by the construction ofgeneralized low-density parity-check codes, where the capacity-achievingpoint-to-point codes serve as subcodes to robustly estimate the signal support.In the case that each entry of the $n$-dimensional $k$-sparse signal lies in aknown discrete alphabet, the proposed scheme requires only $O(k \log n)$measurements and arithmetic operations. In the case of arbitrary, possiblycontinuous alphabet, an error propagation graph is proposed to characterize theresidual estimation error. With $O(k \log^2 n)$ measurements and computationalcomplexity, the reconstruction error can be made arbitrarily small with highprobability.

A Generalized LDPC Framework for Robust and Sublinear Compressive Sensing