  This paper considers solving the unconstrained $\ell_q$-norm ($0\leq q<1$)
regularized least squares ($\ell_q$-LS) problem for recovering sparse signals
in compressive sensing. We propose two highly efficient first-order algorithms
via incorporating the proximity operator for nonconvex $\ell_q$-norm functions
into the fast iterative shrinkage/thresholding (FISTA) and the alternative
direction method of multipliers (ADMM) frameworks, respectively. Furthermore,
in solving the nonconvex $\ell_q$-LS problem, a sequential minimization
strategy is adopted in the new algorithms to gain better global convergence
performance. Unlike most existing $\ell_q$-minimization algorithms, the new
algorithms solve the $\ell_q$-minimization problem without smoothing
(approximating) the $\ell_q$-norm. Meanwhile, the new algorithms scale well for
large-scale problems, as often encountered in image processing. We show that
the proposed algorithms are the fastest methods in solving the nonconvex
$\ell_q$-minimization problem, while offering competent performance in
recovering sparse signals and compressible images compared with several
state-of-the-art algorithms.
