  The ordered weighted $\ell_1$ norm (OWL) was recently proposed, with two
different motivations: its good statistical properties as a sparsity promoting
regularizer; the fact that it generalizes the so-called {\it octagonal
shrinkage and clustering algorithm for regression} (OSCAR), which has the
ability to cluster/group regression variables that are highly correlated. This
paper contains several contributions to the study and application of OWL
regularization: the derivation of the atomic formulation of the OWL norm; the
derivation of the dual of the OWL norm, based on its atomic formulation; a new
and simpler derivation of the proximity operator of the OWL norm; an efficient
scheme to compute the Euclidean projection onto an OWL ball; the instantiation
of the conditional gradient (CG, also known as Frank-Wolfe) algorithm for
linear regression problems under OWL regularization; the instantiation of
accelerated projected gradient algorithms for the same class of problems.
Finally, a set of experiments give evidence that accelerated projected gradient
algorithms are considerably faster than CG, for the class of problems
considered.
