Σεμινάριο: "Pattern Recovery In Linear Regression With Penalized Estimators: Lasso, Slope And Others"

ΚΥΚΛΟΣ ΣΕΜΙΝΑΡΙΩΝ ΣΤΑΤΙΣΤΙΚΗΣ 2024-2025

Ομιλητής: Piotr Graczyk, Université d'Angers

Pattern Recovery In Linear Regression With Penalized Estimators: Lasso, Slope And Others

ΑΙΘΟΥΣΑ 701

ΠΕΡΙΛΗΨΗ

LASSO (Least Absolute Shrinkage and Selection Operator ) is a famous method for dimensionality reduction in the high-dimensional regression, by assigning the 0 value to numerous linear regression coefficients. SLOPE (Sorted L-One Penalized Estimator) is another method for dimensionality reduction in the high- dimensional regression, encompassing the LASSO estimator and many other regularizors : grouped LASSO, D-LASSO, L-infinity… Some regression coefficient estimates of SLOPE can be null (sparsity) or can be equal in absolute value (clustering). Consequently, SLOPE may eliminate irrelevant predictors and may identify groups of predictors having the same influence on the vector of responses. The notion of SLOPE pattern allows to derive theoretical properties on sparsity and clustering by SLOPE. Specifically, the SLOPE pattern of a vector provides: the sign of its components (positive, negative or
null), the clusters (indices of components equal in absolute value) and clusters ranking. 

[1] is the first paper where the SLOPE pattern consistency is studied thoroughly. In this article, we provide necessary and sufficient conditions for SLOPE pattern recovery of an unknown vector of regression coefficients. This powerful result is new compared to previous work on the topic. It also enables the derivation of the SLOPE irrepresentability (IR) condition, which is crucial for establishing the pattern consistency of the SLOPE estimator. Even in the special case of the LASSO, we provide a novel sign recovery characterization that could simplify the proofs of well-known results concerning the LASSO IR condition. In these lectures I will present recent results on penalized estimators for the linear regression coefficients obtained in [1] and [2] jointly with M. Bogdan, X. Dupuis, B. Kolodziejek, U. Schneider, T. Skalski, P. Tardivel and M. Wilczynski.

BIBLIOGRAPHY
[1] M. Bogdan, X. Dupuis, P. Graczyk, B. Kolodziejek, T. Skalski, P. Tardivel, M. Wilczynski, Pattern recovery by SLOPE, to appear in ACHA 2025, arXiv: 2203.12086 

[2] P. Graczyk, U. Schneider, T. Skalski, P. Tardivel, A Unified Framework for Pattern Recovery in Penalized and Thresholded Estimation and its Geometry, to appear in JOTA 2025, arXiv:2307.10158