  In this paper, we reformulate the non-convex $\ell_q$-norm minimization
problem with $q\in(0,1)$ into a 2-step problem, which consists of one convex
and one non-convex subproblems, and propose a novel iterative algorithm called
QISTA ($\ell_q$-ISTA) to solve the $\left(\ell_q\right)$-problem. By taking
advantage of deep learning in accelerating optimization algorithms, together
with the speedup strategy that using the momentum from all previous layers in
the network, we propose a learning-based method, called QISTA-Net-s, to solve
the sparse signal reconstruction problem. Extensive experimental comparisons
demonstrate that the QISTA-Net-s yield better reconstruction qualities than
state-of-the-art $\ell_1$-norm optimization (plus learning) algorithms even if
the original sparse signal is noisy. On the other hand, based on the network
architecture associated with QISTA, with considering the use of convolution
layers, we proposed the QISTA-Net-n for solving the image CS problem, and the
performance of the reconstruction still outperforms most of the
state-of-the-art natural images reconstruction methods. QISTA-Net-n is designed
in unfolding QISTA and adding the convolutional operator as the dictionary.
This makes QISTA-Net-s interpretable. We provide complete experimental results
that QISTA-Net-s and QISTA-Net-n contribute the better reconstruction
performance than the competing.
