论文标题
改进基于RIP的界限，以确保两种压缩传感算法的性能
Improved RIP-Based Bounds for Guaranteed Performance of two Compressed Sensing Algorithms
论文作者
论文摘要
迭代硬阈值（IHT）和压缩采样匹配追踪（COSAMP）是使用硬阈值操作员进行信号恢复和近似值的两种类型的主流压缩感测算法。通过这些算法恢复信号的保证性能主要是在传感矩阵的受限等轴测常数（用$Δ_K$表示的（$ k $是一个整数数）表示的条件下，在间隔$（0,1）中，$Δ__$Δ$Δ信号恢复使用特定算法的成功称为基于限制的ISMetry-property（基于RIP），以保证算法的性能。目前，通过IHT保证$ k $ -sparse信号的确保恢复$δ__{3K} <1/\ sqrt {3} \ of 0.5774，$和通过COSAMP的保证恢复为$δ__{4K {4K} <0.4782。 $在该领域的一个基本问题是是否可以进一步改善这种理论结果。本文的目的是肯定地回答这个问题，并严格地表明，基于RIP的IHT性能的界限可以显着提高到$δ_{3K} <（\ sqrt {5} -1）/2 \ y约0.618，$，可以改善并可以将其绑定到$Δ__________4k}。 $这些改进是通过硬阈值运算符的深度属性来实现的。
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 $ δ_K$ (where $K$ is an integer number), is smaller than a certain threshold value in the interval $(0,1).$ The condition $ δ_{K}< δ^*$ for some constant $ δ^* \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 $δ_{3k}< 1/\sqrt{3}\approx 0.5774,$ and the bound for guaranteed recovery via CoSaMP is $δ_{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 $ δ_{3k} < (\sqrt{5}-1)/2 \approx 0.618, $ and the bound for CoSaMP can be improved and pushed to $ δ_{4k}< 0.5102. $ These improvements are achieved through a deep property of the hard thresholding operator.
