Computing the proximal operator of the q-th power of the ℓ1,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.