1-bit compressed sensing (CS) is an important class of sparse optimization problems, which has been studied since 2008. In this chapter, we focus on the case where the amplitude information of a sparse signal is lost and only sign measurements of the signal are available. Some relaxation models were proposed for 1-bit CS in the literature. However, the solution of these models might not be consistent with the sign measurements of the target signal. In order to develop a model which satisfies the consistency requirement, we show that the 1-bit CS with sign measurements can be reformulated equivalently as an ℓ0-minimization problem with linear constraints. This reformulation promotes a natural LP-based decoding ...
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