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The problem of finding sparse solutions to underdetermined systems of linearequations arises in several applications (e.g. signal and image processing,compressive sensing, statistical inference). A standard tool for dealing withsparse recovery is the $\ell_1$-regularized least-squares approach that hasbeen recently attracting the attention of many researchers. In this paper, wedescribe an active set estimate (i.e. an estimate of the indices of the zerovariables in the optimal solution) for the considered problem that tries toquickly identify as many active variables as possible at a given point, whileguaranteeing that some approximate optimality conditions are satisfied. Arelevant feature of the estimate is that it gives a significant reduction ofthe objective function when setting to zero all those variables estimatedactive. This enables to easily embed it into a given globally convergingalgorithmic framework. In particular, we include our estimate into a blockcoordinate descent algorithm for $\ell_1$-regularized least squares, analyzethe convergence properties of this new active set method, and prove that itsbasic version converges with linear rate. Finally, we report some numericalresults showing the effectiveness of the approach.

A Fast Active Set Block Coordinate Descent Algorithm for $\ell_1$-regularized least squares