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

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.

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