We propose the Group Orthogonal Matching Pursuit (GOMP) algorithm to recover group sparse signals from noisy measurements. Under the group restricted isometry property (GRIP), we prove the instance optimality of the GOMP algorithm for any decomposable approximation norm. Meanwhile, we show the robustness of the GOMP under the measurement error. Compared with the P-norm minimization approach, the GOMP is easier to implement, and the assumption of $\gamma$-decomposability is not required. The simulation results show that the GOMP is very efficient for group sparse signal recovery and significantly outperforms Basis Pursuit in both scalability and solution quality.

We study the approximation capability of the orthogonal super greedy algorithm (OSGA) with respect to $\mu$-coherent dictionaries in Hilbert spaces. We establish the Lebesgue-type inequalities for OSGA, which show that the OSGA provides an almost optimal approximation on the first $[1/(18\mathsf{\mu }\mathrm{s}\left)\right]$ steps. Moreover, we improve the asymptotic constant in the Lebesgue-type inequality of OGA obtained by Livshitz E D.

We investigate the efficiency of Chebyshev Thresholding Greedy Algorithm (CTGA) for an n-term approximation with respect to general bases in a Banach space. We show that the convergence property of CTGA is better than TGA for non-quasi-greedy bases. Then we determine the exact rate of the Lebesgue constants \(L_{n}^{\mathrm{ch}}\) for two examples of such bases: the trigonometric system and the summing basis. We also establish the upper estimates for \(L_{n}^{\mathrm{ch}}\) with respect to general bases in terms of quasi-greedy parameter, democracy parameter and A-property parameter. These estimates do not involve an unconditionality parameter, therefore they are better than those of TGA. In particular, for conditional quasi-greedy bases, a faster convergence rate is obtained.