  In this work we propose and analyze a novel approach for group sparse
recovery. It is based on regularized least squares with an $\ell^0(\ell^2)$
penalty, which penalizes the number of nonzero groups. One distinct feature of
the approach is that it has the built-in decorrelation mechanism within each
group, and thus can handle challenging strong inner-group correlation. We
provide a complete analysis of the regularized model, e.g., existence of a
global minimizer, invariance property, support recovery, and properties of
block coordinatewise minimizers. Further, the regularized problem admits an
efficient primal dual active set algorithm with a provable finite-step global
convergence. At each iteration, it involves solving a least-squares problem on
the active set only, and exhibits a fast local convergence, which makes the
method extremely efficient for recovering group sparse signals. Extensive
numerical experiments are presented to illustrate salient features of the model
and the efficiency and accuracy of the algorithm. A comparative study indicates
its competitiveness with existing approaches.
