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_1: In this paper, we consider orthogonal matching pursuit (OMP) algorithm for multiple measurement vectors (MMV) problem. The robustness of OMPMMV is studied under general perturbations—when the measurement vectors as well as the sensing matrix are incorporated with additive noise. The main result shows that although exact recovery of the sparse solutions is unrealistic in noisy scenario, recovery of the support set of the solutions is guaranteed under suitable conditions. Specifically, a sufficient condition is derived that guarantees exact recovery of the sparse solutions in noiseless scenario.
  @cite_8: We shall show that if the restricted isometry constant (RIC) @math of the measurement matrix @math satisfies @math then the greedy algorithm Orthogonal Matching Pursuit(OMP) will succeed. That is, OMP can recover every @math -sparse signal @math in @math iterations from @math . Moreover, we shall show the upper bound of RIC is sharp in the following sense. For any given @math , we shall construct a matrix @math with the RIC @math such that OMP may not recover some @math -sparse signal @math in @math iterations.
summary: Regarding older works, it is also worth pointing out that the ERC @math , first obtained in [Theorem 5.2] liu2012orthogonal for OMP, has been shown to remain valid for SOMP in [Corollary 1] ding2012robustness . Thereby, the authors of @cite_1 also proved that the older ERC @math , initially derived in [Theorem 3.1] davenport2010analysis for OMP, remains correct for SOMP as @math is implied by @math . Very recently, ) was extended to SOMP in [Remark 1] xu2015perturbation . However, the extension to SOMP of both ) and ) is a novel result. In @cite_8 , the author has derived the ERC @math , which is sharper than ). Combining the ideas developed in @cite_8 and our paper could possibly extend this ERC to SOMP.