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

In this note, we comprehensively characterize the proximal operator of the $q$-th power of the $\ell_{1,q}$-norm (denoted by $\ell_{1,q}^{q}$) with $0\!<\!q\!<\!1$ by exploiting the well-known proximal operator of $|\cdot|^q$ 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 $|\cdot|^{1/2}$ and $|\cdot|^{2/3}$. Numerical experiments demonstrate potential advantages of the $\ell_{1,q}^{q}$ regularization in the }inter-group and intra-group sparse vector recovery.