Exact covariance thresholding into connected components for large-scale Graphical Lasso

TL;DR

This paper introduces a method that leverages covariance thresholding to decompose large-scale graphical lasso problems into smaller, manageable subproblems, significantly improving computational efficiency.

Contribution

It proves that the vertex-partition from thresholded covariance graphs matches the estimated graph, enabling a simple yet powerful decomposition approach for large-scale problems.

Findings

Enables decomposition of large graphical lasso problems into smaller components.

Significantly improves computational efficiency for large-scale graphical models.

Applicable to high regularization parameters, facilitating scalable solutions.

Abstract

We consider the sparse inverse covariance regularization problem or graphical lasso with regularization parameter $\rho$. Suppose the co- variance graph formed by thresholding the entries of the sample covariance matrix at $\rho$ is decomposed into connected components. We show that the vertex-partition induced by the thresholded covariance graph is exactly equal to that induced by the estimated concentration graph. This simple rule, when used as a wrapper around existing algorithms, leads to enormous performance gains. For large values of $\rho$, our proposal splits a large graphical lasso problem into smaller tractable problems, making it possible to solve an otherwise infeasible large scale graphical lasso problem.