Decreasing Weighted Sorted ${\ell_1}$ Regularization

We consider a new family of regularizers, termed weighted sorted
${\ell_1}$
norms (WSL1), which generalizes the recently introduced octagonal shrinkage and clustering algorithm for regression (OSCAR) and also contains the
${\ell_1}$
and
${\ell_\infty}$
norms as particular instances. We focus on a special case of the WSL1, the decreasing WSL1 (DWSL1), where the elements of the argument vector are sorted in non-increasing order and the weights are also non-increasing. In this letter, after showing that the DWSL1 is indeed a norm, we derive two key tools for its use as a regularizer: the dual norm and the Moreau proximity operator.