On Block Cholesky Decomposition for Sparse Inverse Covariance Estimation

TL;DR

This paper introduces a block Cholesky decomposition method for estimating inverse covariance matrices that leverages partial variable ordering information, ensuring positive definiteness and unifying existing approaches.

Contribution

It proposes a novel block Cholesky decomposition framework that handles partial ordering, extending and unifying existing inverse covariance estimation methods.

Findings

Guarantees positive definiteness of estimates.

Provides a unified framework for existing methods.

Validated through simulations and case studies.

Abstract

The modified Cholesky decomposition is popular for inverse covariance estimation, but often needs pre-specification on the full information of variable ordering. In this work, we propose a block Cholesky decomposition (BCD) for estimating inverse covariance matrix under the partial information of variable ordering, in the sense that the variables can be divided into several groups with available ordering among groups, but variables within each group have no orderings. The proposed BCD model provides a unified framework for several existing methods including the modified Cholesky decomposition and the Graphical lasso. By utilizing the partial information on variable ordering, the proposed BCD model guarantees the positive definiteness of the estimated matrix with statistically meaningful interpretation. Theoretical results are established under regularity conditions. Simulation and case studies are conducted to evaluate the proposed BCD model.