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

Ulrike Schneider (1) , Patrick Tardivel (2)

1 TU Wien - Vienna University of Technology = Technische Universität Wien
2 UWr - University of Wrocław [Poland]

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.

Domaines: Mathématiques [math]; Statistiques [math.ST]

Ulrike Schneider, Patrick Tardivel. The Geometry of Uniqueness and Model Selection of Penalized Estimators including SLOPE, LASSO and Basis Pursuit. 2020. ⟨hal-02548350v1⟩