Nonconvex penalties with analytical solutions for one-bit compressive sensing

One-bit measurements widely exist in the real world and can be used to recover sparse signals. This task is known as one-bit compressive sensing (1bit-CS). In this paper, we propose novel algorithms based on both convex and non-convex sparsity-inducing penalties for robust 1bit-CS. We consider the dual problem, which has only one variable and provides a sufficient condition to verify whether a solution is globally optimal or not. For positive homogeneous penalties, a globally optimal solution can be obtained in two steps: a proximal operator and a normalization step. For other penalties, we solve the dual problem, and it needs to evaluate the proximal operators for many times. Then we provide fast algorithms for finding analytical solutions for three penalties: minimax concave penalty (MCP), ℓ0 norm, and sorted ℓ1 penalty. Specifically, our algorithm is more than 200 times faster than the existing algorithm for MCP. Its efficiency is comparable to the algorithm for the ℓ1 penalty in time, while its performance is much better than ℓ1. Among these penalties, sorted ℓ1 is most robust to noise in different settings.