  One-bit measurements widely exist in the real world, and they can be used to
recover sparse signals. This task is known as the problem of learning
halfspaces in learning theory and one-bit compressive sensing (1bit-CS) in
signal processing. In this paper, we propose novel algorithms based on both
convex and nonconvex sparsity-inducing penalties for robust 1bit-CS. We provide
a sufficient condition to verify whether a solution is globally optimal or not.
Then we show that the globally optimal solution for positive homogeneous
penalties can be obtained in two steps: a proximal operator and a normalization
step. For several nonconvex penalties, including minimax concave penalty (MCP),
$\ell_0$ norm, and sorted $\ell_1$ penalty, we provide fast algorithms for
finding the analytical solutions by solving the dual problem. Specifically, our
algorithm is more than $200$ times faster than the existing algorithm for MCP.
Its efficiency is comparable to the algorithm for the $\ell_1$ penalty in time,
while its performance is much better. Among these penalties, the sorted
$\ell_1$ penalty is most robust to noise in different settings.
