One-bit compressed sensing has found broad applications. Due to the constraint on the unit sphere, the classic $\ell_{1}$ minimization frequently returns a signal which is not sparse enough. In this paper, $\ell_{1}-\ell_{q} (1<q\leq 2)$ nonconvex minimization method is developed for one-bit compressed sensing. We demonstrate that $\ell_{1}-\ell_{q}$ minimization does return a much sparser signal than $\ell_{1}$ minimization. Furthermore, we establish $\ell_{1}-\ell_{q}$ optimization iterative algorithm inspired by difference-of-convex algorithm (DCA), and prove the convergence of the new algorithm. Theoretical proofs show that $\ell_{1}-\ell_{q}$ minimization method performs better when $q$ is closer to $1$ , and our algorithm converges in finite steps in normal cases. Numerical experiments display the advantage of this new algorithm compared to the existing ones.

中文翻译：

---

$l_{1}-l_{q} (1 \lt q \leq 2 )$一位压缩感知的最小化