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

TL;DR

This paper characterizes the proximal operator of the $q$-th power of the $\,ell_{1,q}$-norm for $0<q<1$, providing explicit formulas for specific $q$ values and demonstrating benefits in sparse vector recovery.

Contribution

It offers a comprehensive characterization of the proximal operator for the $q$-th power of the $\,ell_{1,q}$-norm, including explicit solutions for certain $q$ values, advancing sparse regularization techniques.

Findings

Explicit proximal operators for $q=1/2$ and $q=2/3$.

Numerical results show improved sparse recovery performance.

Potential advantages in inter-group and intra-group sparsity.

Abstract

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.