On Grouping Effect of Sparse Stable Outlier-Robust Regression

This paper elucidates the grouping effect of the sparse stable outlier-robust regression (S-SORR) estimator which exploits the minimax concave (MC) penalty and the Tikhonov regularization simultaneously together with the MC loss. The main theoretical result is the following: where μ 1 > 0$ is the regularization parameter, and ai and aj are the unit vectors with their associated coefficients hat xi and hat xj. Remarkably, the bound is independent of possible outliers which may be contained in the observation vector y, whereas the bound for the popular elastic net estimator increases in proportion to the norm of y which is largely affected by outliers. Numerical examples show that S-SORR extracts the group structure correctly under huge outliers.