Compressed sensing in both noiseless, and noisy cases is considered in this article, and uniform restricted isometry property (RIP) conditions for sparse signal recovery are established via $\ell _p\,(0< p\leq 1)$ minimization. It is shown that if the measurement matrix satisfies the sharp condition $\Phi (p,t)>0$ for any given constant $t>1$, where $\Phi (p,t)$ concerning the restricted isometry constants $\delta _{tk}$, and $\delta _{2(t-1)k}$ is specified in the context, then all $k$-sparse signals can be exactly recovered by the constrained $\ell _p$ minimization. This uniform RIP framework with general $p$, and $t$ includes three state-of-the-art results concerning $p=1$, $t=2$, and $t\in [\frac{4}{2+p},2]$ as special cases. Utilizing higher-order RIP conditions can result in a milder sufficient condition for sparse recovery. For $t\geq 2$, the RIP condition $\delta _{tk}< \delta (p,t)$, where the upper bound $\delta (p,t)$ is defined in the context, is shown to be sufficient to guarantee both the exact recovery of all $k$-sparse signals in the noiseless case, and the stable recovery of approximately $k$-sparse signals in noisy cases. Moreover, we establish a threshold of the restricted isometry constant $\delta _{tk}$ where the failure of $\ell _p$ sparse recovery will occur.

中文翻译：

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通过 $\ell_p\,(0

本文考虑了无噪声和有噪声情况下的压缩感知，并通过以下方式建立了用于稀疏信号恢复的统一受限等距特性 (RIP) 条件 $\ell _p\,(0< p\leq 1)$最小化。表明如果测量矩阵满足锐化条件$\Phi (p,t)>0$ 对于任何给定的常数 $t>1$， 在哪里 $\Phi (p,t)$ 关于受限等距常数 $\delta _{tk}$， 和 $\delta _{2(t-1)k}$ 在上下文中指定，那么所有 $千$- 稀疏信号可以被约束精确恢复 $\ell _p$最小化。这种统一的 RIP 框架具有通用性$p$， 和 $t$ 包括三个最先进的结果，涉及 $p=1$, $t=2$， 和 $t\in [\frac{4}{2+p},2]$作为特殊情况。利用更高阶的 RIP 条件可以为稀疏恢复提供更温和的充分条件。为了$t\geq 2$, RIP 条件 $\delta _{tk}< \delta (p,t)$，其中上界 $\delta (p,t)$ 在上下文中定义，显示足以保证所有 $千$- 无噪声情况下的稀疏信号，稳定恢复约 $千$- 嘈杂情况下的稀疏信号。此外，我们建立了限制等距常数的阈值$\delta _{tk}$ 失败的地方 $\ell _p$ 将发生稀疏恢复。