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 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.
%0 Generic
%1 zeng2015ordered
%A Zeng, Xiangrong
%A Figueiredo, Mário A. T.
%D 2015
%K fused-lasso machine_learning owl structured_sparsity fista
%T The Ordered Weighted \$\ell\_1\$ Norm: Atomic Formulation, Projections, and Algorithms
%U http://arxiv.org/abs/1409.4271
%X 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 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.
@misc{zeng2015ordered,
abstract = {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.},
added-at = {2018-12-07T09:10:16.000+0100},
archiveprefix = {arXiv},
author = {Zeng, Xiangrong and Figueiredo, M\'{a}rio A. T.},
biburl = {https://www.bibsonomy.org/bibtex/263f455a1605b484ee39674074e243248/jpvaldes},
citeulike-article-id = {14476559},
citeulike-linkout-0 = {http://arxiv.org/abs/1409.4271},
citeulike-linkout-1 = {http://arxiv.org/pdf/1409.4271},
day = 10,
eprint = {1409.4271},
interhash = {f20f3e63c6b9f0366a80f30891038dcd},
intrahash = {63f455a1605b484ee39674074e243248},
keywords = {fused-lasso machine_learning owl structured_sparsity fista},
month = apr,
posted-at = {2017-11-15 15:02:39},
priority = {3},
timestamp = {2018-12-07T09:42:07.000+0100},
title = {{The Ordered Weighted \$\ell\_1\$ Norm: Atomic Formulation, Projections, and Algorithms}},
url = {http://arxiv.org/abs/1409.4271},
year = 2015
}