Uniform RIP Conditions for Recovery of Sparse Signals by $\ell _p\,(0< p\leq 1)$ Minimization
Anhua Wan
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 ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> (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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> sparse recovery will occur.