The Ordered Weighted ℓ_1 Norm: Atomic Formulation, Projections, and Algorithms
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The ordered weighted ℓ_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 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.
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