Sufficient condition based on nearly optimal order RIC for IHT algorithm

Abstract

Iterative hard thresholding (IHT) is one class of sparse signal recovery algorithms widely used in compressed sensing. The well-known restricted isometry constant (RIC) is one of the most popular frameworks used to ensure the convergence of recovery algorithms. In previous literature, sufficient conditions to guarantee the recovery performance of the IHT algorithm have usually been established based on RICs with orders 3 s and 2 s , while no such condition in terms of RIC with order less than 2 s has been reported yet. In this study, based on the theoretical optimal step-length, we develop the RIC condition γ < ( 5 − 1 ) / 8 ≈ 0.1545 for successful recovery via the IHT algorithm. Here, γ = δ s if s is an even number, otherwise γ = δ s + 1 . To the best of our knowledge, this is the first result based on RIC with nearly optimal order for the IHT algorithm.