Decreasing Weighted Sorted ℓ1 Regularization

We consider a new family of regularizers, termed it weighted sorted ℓ1 norms (WSL1), which generalizes the recently introduced it octagonal shrinkage and clustering algorithm for regression (OSCAR) and also contains the ℓ1 and ℓ∞ norms as particular instances. We focus on a special case of the WSL1, the sl 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 paper, 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.