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The Sorted L-One Estimator (SLOPE) is a popular regularization method inregression, which induces clustering of the estimated coefficients. That is,the estimator can have coefficients of identical magnitude. In this paper, wederive an asymptotic distribution of SLOPE for the ordinary least squares,Huber, and Quantile loss functions, and use it to study the clustering behaviorin the limit. This requires a stronger type of convergence since clusteringproperties do not follow merely from the classical weak convergence. For thisaim, we utilize the Hausdorff distance, which provides a suitable notion ofconvergence for the penalty subdifferentials and a bridge toward weakconvergence of the clustering pattern. We establish asymptotic control of thefalse discovery rate for the asymptotic orthogonal design of the regressor. Wealso show how to extend the framework to a broader class of regularizers otherthan SLOPE.

Weak pattern convergence for SLOPE and its robust versions