The optimal $k$-thresholding (OT) and optimal $k$-thresholding pursuit (OTP) are newly introduced frameworks of thresholding techniques for compressed sensing and signal approximation. Such frameworks motivate the practical and efficient algorithms called relaxed optimal $k$-thresholding ($\text{ROT}\omega$) and relaxed optimal $k$-thresholding pursuit ($\text{ROTP}\omega$) which are developed through the tightest convex relaxations of OT and OTP, where $\omega$ is a prescribed integer number. The preliminary numerical results demonstrated in Zhao (2020) indicate that these approaches can stably reconstruct signals with a wide range of sparsity levels. However, the guaranteed performance of these algorithms with parameter $\omega \ge 2$ has not yet established in Zhao (2020). The purpose of this paper is to show the guaranteed performance of OT and OTP in terms of the restricted isometry property (RIP) of nearly optimal order for the sensing matrix governing the $k$-sparse signal recovery, and to establish the first guaranteed performance result for $\text{ROT}\omega$ and $\text{ROTP}\omega$ with $\omega \ge 2.$ In the meantime, we provide a numerical comparison between ROTP$\omega$ and several existing thresholding methods.

最优的 $ķ$阈值（OT）和最佳 $ķ$阈值追踪（OTP）是新引入的用于压缩感测和信号逼近的阈值技术框架。这样的框架激发了称为轻松最优的实用高效算法$ķ$-阈值（$\text{腐烂}\omega$）和轻松的最佳 $ķ$阈值追求（$\text{ROTP}\omega$）是通过OT和OTP的最紧密的凸松弛而开发的，其中 $\omega$是规定的整数。Zhao（2020）证明的初步数值结果表明，这些方法可以稳定地重建具有广泛稀疏性水平的信号。但是，这些算法的带参数保证性能$\omega \ge \mathrm{2个}$在Zhao（2020）中尚未建立。本文的目的是根据控制矩阵的传感矩阵的接近最佳阶的受限等距特性（RIP）来显示OT和OTP的性能保证。$ķ$-稀疏信号恢复，并建立第一个有保证的性能结果 $\text{腐烂}\omega$ 和 $\text{ROTP}\omega$ 和 $\omega \ge \mathrm{2个}。$ 同时，我们提供了ROTP之间的数值比较$\omega$ 和几种现有的阈值化方法。