  One-bit compressive sensing is popular in signal processing and
communications due to the advantage of its low storage costs and hardware
complexity. However, it has been a challenging task all along since only the
one-bit (the sign) information is available to recover the signal. In this
paper, we appropriately formulate the one-bit compressed sensing by a
double-sparsity constrained optimization problem. The first-order optimality
conditions via the newly introduced $\tau$-stationarity for this nonconvex and
discontinuous problem are established, based on which, a gradient projection
subspace pursuit (GPSP) approach with global convergence and fast convergence
rate is proposed. Numerical experiments against other leading solvers
illustrate the high efficiency of our proposed algorithm in terms of the
computation time and the quality of the signal recovery as well.
