An Improved Modified Cholesky Decomposition Method for Precision Matrix Estimation

TL;DR

This paper introduces a new approach for estimating sparse precision matrices that addresses variable order uncertainty by combining multiple permutation-based estimates and applying thresholding, improving accuracy and applicability.

Contribution

It proposes an ensemble method for precision matrix estimation that effectively handles unknown variable order, with theoretical consistency and practical validation.

Findings

The method achieves competitive or superior estimation accuracy in simulations.

It demonstrates improved classification performance in real data analysis.

The approach is computationally efficient and theoretically justified.

Abstract

The modified Cholesky decomposition is commonly used for precision matrix estimation given a specified order of random variables. However, the order of variables is often not available or cannot be pre-determined. In this work, we propose to address the variable order issue in the modified Cholesky decomposition for sparse precision matrix estimation. The key idea is to effectively combine a set of estimates obtained from multiple permutations of variable orders, and to efficiently encourage the sparse structure for the resultant estimate by the thresholding technique on the ensemble Cholesky factor matrix. The consistent property of the proposed estimate is established under some weak regularity conditions. Simulation studies are conducted to evaluate the performance of the proposed method in comparison with several existing approaches. The proposed method is also applied into linear discriminant analysis of real data for classification.