The convergence guarantee of the iterative thresholding algorithm with suboptimal feedbacks for large systems

TL;DR

This paper proves convergence of an efficient iterative thresholding algorithm with suboptimal feedbacks for large systems, demonstrating its superior performance and computational advantages over existing methods.

Contribution

It introduces a convergence analysis for a novel adaptive thresholding algorithm with suboptimal feedback, suitable for large-scale system recovery.

Findings

Algorithm converges reliably for large systems.

Superior recovery accuracy compared to state-of-the-art methods.

Reduced computational complexity due to suboptimal feedback scheme.

Abstract

Thresholding based iterative algorithms have the trade-off between effectiveness and optimality. Some are effective but involving sub-matrix inversions in every step of iterations. For systems of large sizes, such algorithms can be computationally expensive and/or prohibitive. The null space tuning algorithm with hard thresholding and feedbacks (NST+HT+FB) has a mean to expedite its procedure by a suboptimal feedback, in which sub-matrix inversion is replaced by an eigenvalue-based approximation. The resulting suboptimal feedback scheme becomes exceedingly effective for large system recovery problems. An adaptive algorithm based on thresholding, suboptimal feedback and null space tuning (AdptNST+HT+subOptFB) without a prior knowledge of the sparsity level is also proposed and analyzed. Convergence analysis is the focus of this article. Numerical simulations are also carried out to demonstrate the superior efficiency of the algorithm compared with state-of-the-art iterative thresholding algorithms at the same level of recovery accuracy, particularly for large systems.