Covariance of Large-Scale and Sparse of Matrix Estimation

Estimating large-scale covariance matrices with sparse structures is essential in modern multivariate analysis, particularly when the number of variables exceeds the sample size. This paper investigates the performance of various sparse covariance estimation techniques—namely Lasso, Banding, and Adaptive Banding with J1 and J2 penalties applied to simulated multivariate data with different Cholesky sparsity structures. Using both normal and heavy-tailed (t-distribution) data, we evaluate estimation accuracy via Kullback-Leibler Loss and analyze the resulting sparsity in the Cholesky factor and its inverse. The results demonstrate that adaptive methods (J1, J2) and Banding consistently outperform Lasso in both estimation accuracy and structural sparsity, even under challenging conditions such as heavy-tailed distributions and irregular sparsity patterns. These findings highlight the robustness and efficiency of structurally penalized estimators in high-dimensional statistical learning and their relevance to applications in dynamic system stability.