Theoretical Linear Convergence of Unfolded ISTA and its Practical Weights and Thresholds
This work addresses the theoretical gap in unfolded iterative algorithms for sparse recovery, which is important for researchers in signal processing and machine learning, though it is incremental as it builds on existing ISTA methods.
The authors tackled the theoretical understanding of unfolded ISTA for sparse signal recovery, showing that with a specific weight structure, it achieves linear convergence, outperforming the sublinear convergence of ISTA/FISTA, and incorporating thresholding further boosts the convergence rate both theoretically and empirically.
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.