Uniform RIP Conditions for Recovery of Sparse Signals by $\ell _p\,(0< p\leq 1)$ Minimization

Abstract

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 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} , 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.