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

In this note, we comprehensively characterize the proximal operator of the $q$-th power of the ${\mathrm{\ell }}_{1,q}$-norm (denoted by ${\mathrm{\ell }}_{1,q}^q$) with $0\text{\hspace{0.17em}⁣}<\text{\hspace{0.17em}⁣}q\text{\hspace{0.17em}⁣}<\text{\hspace{0.17em}⁣}1$ by exploiting the well-known proximal operator of $\mathrm{\mid }\cdot {\mathrm{\mid }}^q$ on the real line. In particular, much more explicit characterizations can be obtained whenever $q\text{\hspace{0.17em}⁣}=\text{\hspace{0.17em}⁣}1\mathrm{/}2$ and $q\text{\hspace{0.17em}⁣}=\text{\hspace{0.17em}⁣}2\mathrm{/}3$ due to the existence of closed-form expressions for the proximal operators of $\mathrm{\mid }\cdot {\mathrm{\mid }}^{1\mathrm{/}2}$ and $\mathrm{\mid }\cdot {\mathrm{\mid }}^{2\mathrm{/}3}$. Numerical experiments demonstrate potential advantages of the ${\mathrm{\ell }}_{1,q}^q$ regularization in the }inter-group and intra-group sparse vector recovery.