  In recent years, unfolding iterative algorithms as neural networks has become
an empirical success in solving sparse recovery problems. However, its
theoretical understanding is still immature, which prevents us from fully
utilizing the power of neural networks. In this work, we study unfolded ISTA
(Iterative Shrinkage Thresholding Algorithm) for sparse signal recovery. We
introduce a weight structure that is necessary for asymptotic convergence to
the true sparse signal. With this structure, unfolded ISTA can attain a linear
convergence, which is better than the sublinear convergence of ISTA/FISTA in
general cases. Furthermore, we propose to incorporate thresholding in the
network to perform support selection, which is easy to implement and able to
boost the convergence rate both theoretically and empirically. Extensive
simulations, including sparse vector recovery and a compressive sensing
experiment on real image data, corroborate our theoretical results and
demonstrate their practical usefulness. We have made our codes publicly
available: https://github.com/xchen-tamu/linear-lista-cpss.
