Sparse Inverse Covariance Estimation for Chordal Structures

TL;DR

This paper introduces a closed-form solution for the Graphical Lasso problem when the underlying graph has a chordal structure, enabling efficient high-dimensional covariance estimation.

Contribution

It generalizes previous results by deriving a closed-form solution for GL with chordal graphs, significantly improving computational efficiency for large-scale problems.

Findings

The method solves large-scale GL problems with up to 450 million variables in under 2 minutes.

It simplifies the GL problem to a maximum determinant matrix completion problem.

The approach outperforms state-of-the-art methods in speed for high-dimensional datasets.

Abstract

In this paper, we consider the Graphical Lasso (GL), a popular optimization problem for learning the sparse representations of high-dimensional datasets, which is well-known to be computationally expensive for large-scale problems. Recently, we have shown that the sparsity pattern of the optimal solution of GL is equivalent to the one obtained from simply thresholding the sample covariance matrix, for sparse graphs under different conditions. We have also derived a closed-form solution that is optimal when the thresholded sample covariance matrix has an acyclic structure. As a major generalization of the previous result, in this paper we derive a closed-form solution for the GL for graphs with chordal structures. We show that the GL and thresholding equivalence conditions can significantly be simplified and are expected to hold for high-dimensional problems if the thresholded sample covariance matrix has a chordal structure. We then show that the GL and thresholding equivalence is enough to reduce the GL to a maximum determinant matrix completion problem and drive a recursive closed-form solution for the GL when the thresholded sample covariance matrix has a chordal structure. For large-scale problems with up to 450 million variables, the proposed method can solve the GL problem in less than 2 minutes, while the state-of-the-art methods converge in more than 2 hours.