Computing the Proximal Operator of the $\ell_{1,q}$-norm for Group Sparsity

In this note, we comprehensively characterize the proximal operator of the $\ell_{1,q}$-norm with $0\!<\!q\!<\!1$ by exploiting the well-known proximal operator of the $\ell_q$-norm on the real line. In particular, much more explicit characterizations can be obtained whenever $q\!=\!1/2$ and $q\!=\!2/3$ due to the existence of closed-form expressions for the proximal operators of the $\ell_{1/2}$- and $\ell_{2/3}$-norms. Numerical experiments demonstrate potential advantages of the $\ell_{1,q}$-regularization in the group sparse vector recovery.