High-dimensional robust regression and outliers detection with SLOPE

TL;DR

This paper introduces a new high-dimensional robust regression method using SLOPE for simultaneous estimation and outlier detection, with theoretical guarantees on error bounds, FDR, and power, validated through simulations and real data.

Contribution

It presents the first procedure with guaranteed FDR and power control for outlier detection in high-dimensional linear regression with individual intercepts.

Findings

Provides sharp bounds on estimation errors.

Achieves asymptotic control of FDR and power.

Outperforms recent methods in simulations and real data.

Abstract

The problems of outliers detection and robust regression in a high-dimensional setting are fundamental in statistics, and have numerous applications. Following a recent set of works providing methods for simultaneous robust regression and outliers detection, we consider in this paper a model of linear regression with individual intercepts, in a high-dimensional setting. We introduce a new procedure for simultaneous estimation of the linear regression coefficients and intercepts, using two dedicated sorted-$\ell_1$ penalizations, also called SLOPE. We develop a complete theory for this problem: first, we provide sharp upper bounds on the statistical estimation error of both the vector of individual intercepts and regression coefficients. Second, we give an asymptotic control on the False Discovery Rate (FDR) and statistical power for support selection of the individual intercepts. As a consequence, this paper is the first to introduce a procedure with guaranteed FDR and statistical power control for outliers detection under the mean-shift model. Numerical illustrations, with a comparison to recent alternative approaches, are provided on both simulated and several real-world datasets. Experiments are conducted using an open-source software written in Python and C++.