A2034
Title: On an essential extension of the G-LASSO
Authors: Piotr Graczyk - Universite Angers (France) [presenting]
Bartosz Kolodziejek - Politechnika Warszawska (Poland)
Hideto Nakashima - Tokai University (Japan)
Maciej Wilczynski - Politechnika Wroclawska (Poland)
Abstract: The Graphical LASSO is improved by addressing the estimation of high-dimensional precision matrices, a fundamental problem in modern statistics. The Graphical LASSO with its L1-penalty is a standard approach for recovering sparsity patterns of K, with mathematical properties presented in previous work. However, many statistical models exhibit richer patterns, such as colored Graphical models with equality constraints in K, which the Graphical LASSO cannot capture. Recovering these richer structures through high-dimensional estimation of the precision matrix with atomic norm penalties is addressed. The unit balls of these penalties are polytopes, and induced patterns correspond to the polytope's facial structure by belonging to a cone in the related normal cone partition of $\mathbb{R}^p$. Theoretical guarantees are established for recovering the true pattern of K. The methods extend the primal-dual witness methodology. The analysis provides conditions on the deviation between sample and true covariance matrices for successful pattern recovery, given a novel Irrepresentability Condition (IR) for any atomic penalty. When specialized to Graphical LASSO, the results improve it by requiring weaker deviation conditions and a less restrictive IR, leading to tighter bounds and better asymptotic performance than prior work. The proposed general IR, based on a new thresholding concept, provides a unified perspective on model selection consistency.
