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

Sparse Estimation with Strongly Correlated Variables using Ordered Weighted L1 Regularization