$\ell_{1}-\ell_{q}$ ($1<q\leq 2$) Minimization for One-Bit Compressed Sensing

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.