Uniform RIP Conditions for Recovery of Sparse Signals by $\ell _p\,(0< p\leq 1)$ 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 ℓ
_p
(0 <; p ≤ 1) minimization. It is shown that if the measurement matrix satisfies the sharp condition Φ(p, t) > 0 for any given constant t > 1, where Φ(p, t) concerning the restricted isometry constants δtk, and δ2(t-1)k is specified in the context, then all k-sparse signals can be exactly recovered by the constrained ℓ
_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 E [ 4 2+p, 2] as special cases. Utilizing higher-order RIP conditions can result in a milder sufficient condition for sparse recovery. For t ≥ 2, the RIP condition δtk <; δ(p, t), where the upper bound δ(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 δtk where the failure of ℓ
_p
sparse recovery will occur.