# Nonconvex penalties with analytical solutions for one-bit compressive sensing

## TL;DR

This paper introduces fast algorithms with analytical solutions for one-bit compressive sensing using nonconvex penalties, improving speed and robustness in sparse signal recovery from binary measurements.

## Contribution

It provides a novel framework for solving 1bit-CS with nonconvex penalties, including analytical solutions and efficiency improvements over existing methods.

## Key findings

- Algorithms for MCP, $ ext{l}_0$, and sorted $ ext{l}_1$ penalties are significantly faster.
- The proposed methods outperform existing algorithms in speed, especially for MCP.
- Sorted $ ext{l}_1$ penalty shows superior robustness to noise.

## Abstract

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.