Surveying and comparing simultaneous sparse approximation (or group-lasso) algorithms

In this paper, we survey and compare different algorithms that, given an overcomplete dictionary of elementary functions, solve the problem of simultaneous sparse signal approximation, with common sparsity profile induced by a ℓp−ℓqℓp−ℓq mixed-norm. Such a problem is also known in the statistical learning community as the group lasso problem. We have gathered and detailed different algorithmic results concerning these two equivalent approximation problems. We have also enriched the discussion by providing relations between several algorithms. Experimental comparisons of the detailed algorithms have also been carried out. The main lesson learned from these experiments is that depending on the performance measure, greedy approaches and iterative reweighted algorithms are the most efficient algorithms either in term of computational complexities, sparsity recovery or mean-square error.