Asymptotic Theory for Graphical SLOPE: Precision Estimation and Pattern Convergence

TL;DR

This paper develops an asymptotic theory for Graphical SLOPE, demonstrating its ability to recover sparsity and clustering in precision matrices, and compares its performance under Gaussian and heavy-tailed distributions.

Contribution

It provides the first asymptotic analysis of Graphical SLOPE, including convergence properties, pattern recovery, and robustness to non-Gaussian data, with practical advantages over GLASSO.

Findings

Root-n scaled estimation error converges to a unique minimizer.

SLOPE pattern convergence characterizes clustering structure.

TSLOPE outperforms GSLOPE under heavy-tailed distributions.

Abstract

This paper studies Graphical SLOPE for precision matrix estimation, with emphasis on its ability to recover both sparsity and clusters of edges with equal or similar strength. In a fixed-dimensional regime, we establish that the root-$n$ scaled estimation error converges to the unique minimizer of a strictly convex optimization problem defined through the directional derivative of the SLOPE penalty. We also establish convergence of the induced SLOPE pattern, thereby obtaining an asymptotic characterization of the clustering structure selected by the estimator. A comparison with GLASSO shows that the grouping property of SLOPE can substantially improve estimation accuracy when the precision matrix exhibits structured edge patterns. To assess the effect of departures from Gaussianity, we then analyze Gaussian-loss precision matrix estimation under elliptical distributions. In this setting, we derive the limiting distribution and quantify the inflation in variability induced by heavy tails relative to the Gaussian benchmark. We also study TSLOPE, based on the multivariate $t$-loss, and derive its limiting distribution. The results show that TSLOPE offers clear advantages over GSLOPE under heavy-tailed data-generating mechanisms. Simulation evidence suggests that these qualitative conclusions persist in high-dimensional settings, and an empirical application shows that SLOPE-based estimators, especially TSLOPE, can uncover economically meaningful clustered dependence structures.