The finite steps of convergence of the fast thresholding algorithms with feedbacks

TL;DR

This paper analyzes the finite convergence steps of fast thresholding algorithms with feedback for sparse signal recovery, providing new insights, improved convergence conditions, and an adaptive scheme without prior sparsity knowledge.

Contribution

It introduces a new perspective on the convergence analysis, improves convergence conditions, and proposes an adaptive scheme with guaranteed finite convergence.

Findings

Convergence occurs in finitely many steps under restricted isometry conditions.

The number of iterations can be estimated and reduced by exploiting signal or measurement structure.

An adaptive scheme without sparsity knowledge also guarantees finite convergence.

Abstract

Iterative algorithms based on thresholding, feedback and null space tuning (NST+HT+FB) for sparse signal recovery are exceedingly effective and fast, particularly for large scale problems. The core algorithm is shown to converge in finitely many steps under a (preconditioned) restricted isometry condition. In this paper, we present a new perspective to analyze the algorithm, which turns out that the efficiency of the algorithm can be further elaborated by an estimate of the number of iterations for the guaranteed convergence. The convergence condition of NST+HT+FB is also improved. Moreover, an adaptive scheme (AdptNST+HT+FB) without the knowledge of the sparsity level is proposed with its convergence guarantee. The number of iterations for the finite step of convergence of the AdptNST+HT+FB scheme is also derived. It is further shown that the number of iterations can be significantly reduced by exploiting the structure of the specific sparse signal or the random measurement matrix.