  We address the task of identifying densely connected subsets of multivariate
Gaussian random variables within a graphical model framework. We propose two
novel estimators based on the Ordered Weighted $\ell_1$ (OWL) norm: 1) The
Graphical OWL (GOWL) is a penalized likelihood method that applies the OWL norm
to the lower triangle components of the precision matrix. 2) The
column-by-column Graphical OWL (ccGOWL) estimates the precision matrix by
performing OWL regularized linear regressions. Both methods can simultaneously
identify highly correlated groups of variables and control the sparsity in the
resulting precision matrix. We formulate GOWL such that it solves a composite
optimization problem and establish that the estimator has a unique global
solution. In addition, we prove sufficient grouping conditions for each column
of the ccGOWL precision matrix estimate. We propose proximal descent algorithms
to find the optimum for both estimators. For synthetic data where group
structure is present, the ccGOWL estimator requires significantly reduced
computation and achieves similar or greater accuracy than state-of-the-art
estimators. Timing comparisons are presented and demonstrates the superior
computational efficiency of the ccGOWL. We illustrate the grouping performance
of the ccGOWL method on a cancer gene expression data set and an equities data
set.
