A Generalized LDPC Framework for Robust and Sublinear Compressive Sensing

TL;DR

This paper introduces a generalized LDPC-based measurement matrix for compressive sensing, achieving near-optimal measurement efficiency and robustness for sparse signal recovery, including discrete and continuous cases.

Contribution

It proposes a novel LDPC-inspired measurement matrix design that improves measurement efficiency and robustness in compressive sensing, with theoretical guarantees.

Findings

Requires O(k log n) measurements for discrete signals

Achieves arbitrarily small error with O(k log^2 n) measurements for continuous signals

Provides a new error propagation graph for residual error analysis

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 $k$-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 $O(k \log^2 n)$ measurements and computational complexity, the reconstruction error can be made arbitrarily small with high probability.