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U. Schneider, P. Tradivel:
"On the Geometry of Uniqueness, Sparsity and Clustering of LASSO, SLOPE and Related Estimators";
Talk: Summer Term 2021, Wien WU (invited); 2021-05-07.

This talk follows a recent trend in the statistics literature, where geometric properties are exploited to derive results for statistical procedures in the context of high-dimensional regression models. We consider estimation methods such as the Lasso and SLOPE, which are defined as solutions to a penalized optimization problem. We provide a geometric condition for uniqueness of the estimator -- in contrast to previously known conditions in the literature, our approach provides a criterion that is both necessary and sufficient. Moreover, the geometric considerations also give insights into which models are accessible for the corresponding estimation method. This can be determined by investigating which faces of a certain polytope (depending on the estimator) are intersected by the row span of the regressor matrix. We illustrate this approach for the SLOPE estimator using the sign permutahedron.

Joint work with Patrick Tardivel (Université Bourgogne).