text: Several exact recovery criteria (ERC) ensuring that orthogonal matching pursuit (OMP) identifies the correct support of sparse signals have been developed in the last few years. These ERC rely on the restricted isometry property (RIP), the associated restricted isometry constant (RIC) and sometimes the restricted orthogonality constant (ROC). In this paper, three of the most recent ERC for OMP are examined. The contribution is to show that these ERC remain valid for a generalization of OMP, entitled simultaneous orthogonal matching pursuit (SOMP), that is capable to process several measurement vectors simultaneously and return a common support estimate for the underlying sparse vectors. The sharpness of the bounds is also briefly discussed in light of previous works focusing on OMP.
  @cite_22: Orthogonal matching pursuit (OMP) algorithm is an efficient method for the recovery of a sparse signal in compressed sensing, due to its ease implementation and low complexity. In this paper, the robustness of the OMP algorithm under the restricted isometry property (RIP) is presented. It is shown that ( _K + K _ K,1 < 1 ) is sufficient for the OMP algorithm to recover exactly the support of arbitrary K-sparse signal if its nonzero components are large enough for both l 2 bounded and l ∞ bounded noises.
  @cite_0: As a paradigm for reconstructing sparse signals using a set of under sampled measurements, compressed sensing has received much attention in recent years. In identifying the sufficient condition under which the perfect recovery of sparse signals is ensured, a property of the sensing matrix referred to as the restricted isometry property (RIP) is popularly employed. In this article, we propose the RIP based bound of the orthogonal matching pursuit (OMP) algorithm guaranteeing the exact reconstruction of sparse signals. Our proof is built on an observation that the general step of the OMP process is in essence the same as the initial step in the sense that the residual is considered as a new measurement preserving the sparsity level of an input vector. Our main conclusion is that if the restricted isometry constant δ K of the sensing matrix satisfies δ K < K - 1 K - 1 + K then the OMP algorithm can perfectly recover K(> 1)-sparse signals from measurements. We show that our bound is sharp and indeed close to the limit conjectured by Dai and Milenkovic.
  @cite_19: Orthogonal matching pursuit (OMP) is a greedy search algorithm popularly being used for the recovery of compressive sensed sparse signals. In this correspondence, we show that if the isometry constant δK+1 of the sensing matrix Φ satisfies δK+1 <; 1 (1 √K+1) then the OMP algorithm can perfectly recover K-sparse signals from the compressed measurements y=Φx. Our bound offers a substantial improvement over the recent result of Davenport and Wakin and also closes gap between the recovery bound and fundamental limit over which the perfect recovery of the OMP cannot be guaranteed.
  @cite_23: This paper demonstrates that if the restricted isometry constant δK+1 of the measurement matrix A satisfies [δK+1 <; 1 √K+1] then a greedy algorithm called Orthogonal Matching Pursuit (OMP) can recover every K-sparse signal x in K iterations from Ax. By contrast, a matrix is also constructed with the restricted isometry constant [δK+1 = 1 √K] such that OMP can not recover some K-sparse signal x in K iterations. This result positively verifies the conjecture given by Dai and Milenkovic in 2009.
  @cite_11: 
summary: As demonstrated in , Theorem is a straightforward consequence of Lemma and Lemma . The authors of @cite_11 and @cite_22 independently obtained ) for OMP. To the best of the authors' knowledge, the second ERC was first obtained simultaneously in @cite_23 and @cite_19 while ) was initially published in @cite_0 , both ERC being derived for OMP.