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.
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.
@cite_1 @cite_8
2169380424 1557195970
Introduction Preliminaries Special linear groups: @math Orthogonal groups: @math Symplectic groups: @math Unitary groups: @math Proofs of Theorems 1.1 and 1.1, and of corollaries 1.2-1.4 Permutation group algorithms Concluding remarks References. In this paper we study general @math l p regularized unconstrained minimization problems. In particular, we derive lower bounds for nonzero entries of the first- and second-order stationary points and hence also of local minimizers of the @math l p minimization problems. We extend some existing iterative reweighted @math l 1 ( @math IRL 1 ) and @math l 2 ( @math IRL 2 ) minimization methods to solve these problems and propose new variants for them in which each subproblem has a closed-form solution. Also, we provide a unified convergence analysis for these methods. In addition, we propose a novel Lipschitz continuous @math  -approximation to @math ? x ? p p . Using this result, we develop new @math IRL 1 methods for the @math l p minimization problems and show that any accumulation point of the sequence generated by these methods is a first-order stationary point, provided that the approximation parameter @math  is below a computable threshold value. This is a remarkable result since all existing iterative reweighted minimization methods require that @math  be dynamically updated and approach zero. Our computational results demonstrate that the new @math IRL 1 method and the new variants generally outperform the existing @math IRL 1 methods (Chen and Zhou in 2012; Foucart and Lai in Appl Comput Harmon Anal 26:395---407, 2009).