Proximal operator for the sorted <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e20" altimg="si7.svg"><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> norm: Application to testing procedures based on SLOPE

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