  In recent studies on sparse modeling, non-convex penalties have received
considerable attentions due to their superiorities on sparsity-inducing over
the convex counterparts. Compared with the convex optimization approaches,
however, the non-convex approaches have more challenging convergence analysis.
In this paper, we study the convergence of a non-convex iterative thresholding
algorithm for solving sparse recovery problems with a certain class of
non-convex penalties, whose corresponding thresholding functions are
discontinuous with jump discontinuities. Therefore, we call the algorithm the
iterative jumping thresholding (IJT) algorithm. The finite support and sign
convergence of IJT algorithm is firstly verified via taking advantage of such
jump discontinuity. Together with the assumption of the introduced restricted
Kurdyka-{\L}ojasiewicz (rKL) property, then the strong convergence of IJT
algorithm can be proved.Furthermore, we can show that IJT algorithm converges
to a local minimizer at an asymptotically linear rate under some additional
conditions. Moreover, we derive a posteriori computable error estimate, which
can be used to design practical terminal rules for the algorithm. It should be
pointed out that the $l_q$ quasi-norm ($0<q<1$) is an important subclass of the
class of non-convex penalties studied in this paper. In particular, when
applied to the $l_q$ regularization, IJT algorithm can converge to a local
minimizer with an asymptotically linear rate under certain concentration
conditions. We provide also a set of simulations to support the correctness of
theoretical assertions and compare the time efficiency of IJT algorithm for the
$l_{q}$ regularization ($q=1/2, 2/3$) with other known typical algorithms like
the iterative reweighted least squares (IRLS) algorithm and the iterative
reweighted $l_{1}$ minimization (IRL1) algorithm.
