Proximal operator for the sorted l 1 norm with application to testing procedures based on SLOPE

Xavier Dupuis (1) , Patrick J C Tardivel (1)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]

A decade ago OSCAR was introduced as a penalized estimator where the penalty term, the sorted l1 norm, allows to perform clustering selection. More recently, SLOPE was introduced as a penalized estimator controlling the False Discovery Rate (FDR) as soon as the hyper-parameter of the sorted l1 norm is properly selected. For both, OSCAR and SLOPE, numerical schemes to compute these estimators are based on the proximal operator of the sorted l1 norm. The main goal of this note is to provide a short and simple formula for this operator. Based on this formula one may observe that the output of the proximal operator has some components equal and thus this formula corroborate that SLOPE as well as OSCAR perform clustering selection. Moreover, our geometric approach to prove the formula for the proximal operator provides insight to show that testing procedures based on SLOPE are more powerful than step-down testing procedures but less powerful than step-up testing procedures.

Domaines: Mathématiques [math]

Xavier Dupuis, Patrick J C Tardivel. Proximal operator for the sorted l 1 norm with application to testing procedures based on SLOPE. 2021. ⟨hal-03177108v1⟩