  This paper studies ordered weighted L1 (OWL) norm regularization for sparse
estimation problems with strongly correlated variables. We prove sufficient
conditions for clustering based on the correlation/colinearity of variables
using the OWL norm, of which the so-called OSCAR is a particular case. Our
results extend previous ones for OSCAR in several ways: for the squared error
loss, our conditions hold for the more general OWL norm and under weaker
assumptions; we also establish clustering conditions for the absolute error
loss, which is, as far as we know, a novel result. Furthermore, we characterize
the statistical performance of OWL norm regularization for generative models in
which certain clusters of regression variables are strongly (even perfectly)
correlated, but variables in different clusters are uncorrelated. We show that
if the true p-dimensional signal generating the data involves only s of the
clusters, then O(s log p) samples suffice to accurately estimate the signal,
regardless of the number of coefficients within the clusters. The estimation of
s-sparse signals with completely independent variables requires just as many
measurements. In other words, using the OWL we pay no price (in terms of the
number of measurements) for the presence of strongly correlated variables.
