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One-bit compressive sensing gains its popularity in signal processing and communications due to its low storage costs and low hardware complexity. However, it has been a challenging task to recover the signal only by exploiting the one-bit (the sign) information. In this paper, we appropriately formulate the one-bit compressive sensing into a double-sparsity constrained optimization problem. The first-order optimality conditions for this nonconvex and discontinuous problem are established via the newly introduced <inline-formula><tex-math notation="LaTeX">$\tau$</tex-math></inline-formula>-stationarity, based on which, a gradient projection subspace pursuit (GPSP) algorithm is developed. It is proven that GPSP can converge globally and terminate within finite steps. Numerical experiments have demonstrated its excellent performance in terms of a high order of accuracy with a high computational speed.

Computing One-Bit Compressive Sensing via Double-Sparsity Constrained Optimization