Serdecznie zapraszamy na kolejne seminarium z zakresu Statystyki Matematycznej, podczas którego wykład wygłosi prof. Piotr Graczyk z LAREMA, Angers University (Francja).

Termin: środa, 8 lipca, godz. 11:15–13:00, sala 4.1, budynek C-19.

Referat

From Graphical Lasso to Atomic Norms: High-Dimensional Pattern Recovery

Abstract

In this talk we propose an essential extension and improvement of the Graphical LASSO, by using an atomic penalty, in view of a pattern recovery of the inverse covariance matrix (K) of a Gaussian (p)-dimensional vector (N(m,Sigma)). The matrix (K=Sigma^{-1}) is called the precision matrix.

Estimating high-dimensional precision matrices is a fundamental problem in modern statistics, with the Graphical LASSO and its (L^1)-penalty being a standard approach for recovering sparsity patterns of (K). The mathematical qualities of the Graphical LASSO, presented in M. J. Wainwright's book High-Dimensional Statistics (2019), were proven by Ravikumar, Wainwright et al. (2011).

However, many statistical models exhibit richer structures (patterns), e.g. colored graphical models with equality constraints in (K), impossible for the Graphical LASSO to capture.

Our paper addresses the challenge of recovering these richer structures by high-dimensional estimation of the precision matrix with atomic norm penalties. Their unit balls are polytopes. Induced patterns correspond to the polytope's facial structure by belonging to a cone in the related normal cone partition of (mathbb{R}^p). We establish theoretical guarantees for recovering the true pattern of (K).

Our methods extend the primal-dual witness methodology of Ravikumar, Wainwright et al. Our analysis provides conditions on the deviation between sample and true covariance matrices for successful pattern recovery, given a novel Irrepresentability Condition for any atomic penalty. When specialized to the Graphical LASSO, our results improve it by weaker deviation requirements and a less restrictive Irrepresentability Condition, leading to tighter bounds and better asymptotic performance than prior work. The proposed general Irrepresentability Condition, based on a new thresholding concept, provides a unified perspective on model selection consistency. Numerical examples demonstrate the tightness of the derived theoretical bounds.