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The problems of outliers detection and robust regression in ahigh-dimensional setting are fundamental in statistics, and have numerousapplications. Following a recent set of works providing methods forsimultaneous robust regression and outliers detection, we consider in thispaper a model of linear regression with individual intercepts, in ahigh-dimensional setting. We introduce a new procedure for simultaneousestimation of the linear regression coefficients and intercepts, using twodedicated sorted-$\ell_1$ penalizations, also called SLOPE. We develop acomplete theory for this problem: first, we provide sharp upper bounds on thestatistical estimation error of both the vector of individual intercepts andregression coefficients. Second, we give an asymptotic control on the FalseDiscovery Rate (FDR) and statistical power for support selection of theindividual intercepts. As a consequence, this paper is the first to introduce aprocedure with guaranteed FDR and statistical power control for outliersdetection under the mean-shift model. Numerical illustrations, with acomparison to recent alternative approaches, are provided on both simulated andseveral real-world datasets. Experiments are conducted using an open-sourcesoftware written in Python and C++.

High-dimensional robust regression and outliers detection with SLOPE