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

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 $ell _p,(0< pleq 1)$ 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 $tin [frac{4}{2+p},2]$ as special cases. Utilizing higher-order RIP conditions can result in a milder sufficient condition for sparse recovery. For $tgeq 2$, 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.
Source: IEEE Transactions on Signal Processing - Category: Biomedical Engineering Source Type: research