  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
$\tau$-stationarity, based on which, a gradient projection subspace pursuit
(\texttt{GPSP}) algorithm is developed. It is proven that \texttt{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 fast computational speed.
