Nonconvex penalties with analytical solutions for one-bit compressive sensing
This work addresses the challenge of efficient and robust signal recovery in one-bit compressive sensing, which is incremental but offers specific algorithmic improvements.
The paper tackles the problem of recovering sparse signals from one-bit measurements by proposing novel algorithms based on convex and nonconvex sparsity-inducing penalties, resulting in a solution that is over 200 times faster than existing methods for MCP and shows improved robustness, particularly with the sorted ℓ1 penalty.
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.