Ordered Weighted L1 Regularized Regression with Strongly Correlated Covariates: Theoretical Aspects

This paper studies the ordered weighted `1 (OWL) family of regularizers for sparse linear regression with strongly correlated covariates. We prove sufficient conditions for clustering correlated covariates, extending and qualitatively strengthening previous results for a particular member of the OWL family: OSCAR (octagonal shrinkage and clustering algorithm for regression). We derive error bounds for OWL with correlated Gaussian covariates: for cases in which clusters of covariates are strongly (even perfectly) correlated, but covariates in different clusters are uncorrelated, we show that if the true p-dimensional signal involves only s clusters, thenO(s log p) samples suffice to accurately estimate it, regardless of the number of coefficients within the clusters. Since the estimation of s-sparse signals with completely independent covariates also requires O(s log p) measurements, this shows that by using OWL regularization, we pay no price (in the number of measurements) for the presence of strongly correlated covariates.