Improved RIP-Based Bounds for Guaranteed Performance of two Compressed Sensing Algorithms

TL;DR

This paper improves the theoretical RIP-based bounds for guaranteed signal recovery in two popular compressed sensing algorithms, IHT and CoSaMP, using a novel analysis of the hard thresholding operator.

Contribution

The paper provides tighter RIP-based bounds for IHT and CoSaMP, enhancing the theoretical guarantees of their performance in compressed sensing.

Findings

IHT bound improved to δ_{3k} < 0.618

CoSaMP bound improved to δ_{4k} < 0.5102

Utilizes a deep property of the hard thresholding operator

Abstract

Iterative hard thresholding (IHT) and compressive sampling matching pursuit (CoSaMP) are two types of mainstream compressed sensing algorithms using hard thresholding operators for signal recovery and approximation. The guaranteed performance for signal recovery via these algorithms has mainly been analyzed under the condition that the restricted isometry constant of a sensing matrix, denoted by $ \delta_K$ (where $K$ is an integer number), is smaller than a certain threshold value in the interval $(0,1).$ The condition $ \delta_{K}< \delta^*$ for some constant $ \delta^* \leq 1 $ ensuring the success of signal recovery with a specific algorithm is called the restricted-isometry-property-based (RIP-based) bound for guaranteed performance of the algorithm. At the moment, the best known RIP-based bound for the guaranteed recovery of $k$-sparse signals via IHT is $\delta_{3k}< 1/\sqrt{3}\approx 0.5774,$ and the bound for guaranteed recovery via CoSaMP is $\delta_{4k} < 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 rigorously show that the RIP-based bounds for guaranteed performance of IHT can be significantly improved to $ \delta_{3k} < (\sqrt{5}-1)/2 \approx 0.618, $ and the bound for CoSaMP can be improved and pushed to $ \delta_{4k}< 0.5102. $ These improvements are achieved through a deep property of the hard thresholding operator.