The Geometry of Uniqueness and Model Selection of Penalized Estimators including SLOPE, LASSO, and Basis Pursuit

We provide a necessary and sufficient condition for the uniqueness of penalized least-squares estimators with a penalty term consisting a norm whose unit ball is given by a polytope. The condition is given by a geometric criterion involving how the row span of the design matrix intersects the faces of the dual norm unit cube. This criterion also provides information about the model selection properties of the corresponding estimation method. Our analyses cover LASSO, the related method of basis pursuit, as well as the SLOPE estimator.