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In recent years, unfolding iterative algorithms as neural networks has becomean empirical success in solving sparse recovery problems. However, itstheoretical understanding is still immature, which prevents us from fullyutilizing the power of neural networks. In this work, we study unfolded ISTA(Iterative Shrinkage Thresholding Algorithm) for sparse signal recovery. Weintroduce a weight structure that is necessary for asymptotic convergence tothe true sparse signal. With this structure, unfolded ISTA can attain a linearconvergence, which is better than the sublinear convergence of ISTA/FISTA ingeneral cases. Furthermore, we propose to incorporate thresholding in thenetwork to perform support selection, which is easy to implement and able toboost the convergence rate both theoretically and empirically. Extensivesimulations, including sparse vector recovery and a compressive sensingexperiment on real image data, corroborate our theoretical results anddemonstrate their practical usefulness. We have made our codes publiclyavailable: https://github.com/xchen-tamu/linear-lista-cpss.

Theoretical Linear Convergence of Unfolded ISTA and its Practical Weights and Thresholds