Improved RIP-based bounds for guaranteed performance of two compressed sensing algorithms

Abstract

Iterative hard thresholding (IHT) and compressive sampling matching pursuit (CoSaMP) are two mainstream compressed sensing algorithms using the hard thresholding operator. The guaranteed performance of the two algorithms for signal recovery was mainly analyzed in terms of the restricted isometry property (RIP) of sensing matrices. At present, the best known bound using the RIP of order 3k for guaranteed performance of IHT (with the unit stepsize) is \({\delta _{3k}} < 1/\sqrt 3 \approx 0.5774\), and the bound for CoSaMP using the RIP of order 4k is δ4k < 0.4782. A fundamental question in this area is whether such theoretical results can be further improved. The purpose of this paper is to affirmatively answer this question and to rigorously show that the above-mentioned RIP bound for guaranteed performance of IHT can be significantly improved to \({\delta _{3k}} < \left( {\sqrt 5 - 1} \right)/2 \approx 0.618\), and the bound for CoSaMP can be improved to δ4k < 0.5102.