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

TL;DR

This paper provides a theoretical analysis of unfolded ISTA, demonstrating linear convergence with a specific weight structure and support thresholding, supported by simulations on sparse recovery and real image data.

Contribution

It introduces a weight structure ensuring convergence and incorporates thresholding to enhance convergence rate, bridging theory and practice in unfolded ISTA.

Findings

Unfolded ISTA achieves linear convergence with the proposed weight structure.

Thresholding improves convergence speed both theoretically and empirically.

Simulations validate the theoretical results and demonstrate practical effectiveness.

Abstract

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.