Estimation When Both Covariance And Precision Matrices Are Sparse

TL;DR

This paper introduces a maximum likelihood estimation method for covariance matrices where both the matrix and its inverse are sparse, leveraging a chordal graph structure to improve computational efficiency and address a novel double sparsity constraint.

Contribution

It proposes the first method to estimate pairs of covariance and precision matrices that are simultaneously sparse using a chordal graph-based approach.

Findings

Utilizes a local inverse formula for faster computations.

Addresses the double sparsity constraint in covariance estimation.

First to identify such pairs of sparse covariance and precision matrices.

Abstract

We offer a method to estimate a covariance matrix in the special case that \textit{both} the covariance matrix and the precision matrix are sparse --- a constraint we call double sparsity. The estimation method is maximum likelihood, subject to the double sparsity constraint. In our method, only a particular class of sparsity pattern is allowed: both the matrix and its inverse must be subordinate to the same chordal graph. Compared to a naive enforcement of double sparsity, our chordal graph approach exploits a special algebraic local inverse formula. This local inverse property makes computations that would usually involve an inverse (of either precision matrix or covariance matrix) much faster. In the context of estimation of covariance matrices, our proposal appears to be the first to find such special pairs of covariance and precision matrices.