Orateur
Description
We consider the framework of penalized least-squares estimation where the penalty term is given by a real-valued polyhedral gauge, which encompasses methods such as LASSO (and many variants thereof), SLOPE, OSCAR, PACS and others. Each of these estimators can uncover a different structure or ``pattern'' of the unknown parameter vector. We define a general notion of patterns for a penalized procedure based on subdifferentials and provide a necessary condition for a particular pattern to be detected with positive probability, the so-called accessibility condition. We also introduce the stronger noiseless recovery condition which is shown to be necessary for pattern
recovery with probability larger than $1/2$, thereby generalizing the irrepresentability condition of the LASSO to a general framework. We show that the noiseless recovery condition can be relaxed when turning to thresholded penalized estimators, generalizing the idea of the thresholded LASSO: we prove that the accessibility condition is already sufficient for sure pattern recovery by thresholded penalized estimation provided that the signal of the pattern is large enough. We also discuss how our findings can be interpreted through a geometrical lens.
