Sharper lower and upper bounds for the gaussian rank of a graph

TL;DR

This paper introduces new, tighter bounds for the Gaussian rank of a graph, which determines the minimum observations needed for reliable covariance estimation in Gaussian graphical models.

Contribution

The paper establishes sharper bounds for the Gaussian rank, lying between subgraph connectivity and degeneracy, and shows these bounds are efficiently computable.

Findings

Gaussian rank is strictly between subgraph connectivity and degeneracy.

New bounds are sharper than previous known bounds.

Bounds are computable in polynomial time.

Abstract

An open problem in graphical Gaussian models is to determine the smallest number of observations needed to guarantee the existence of the maximum likelihood estimator of the covariance matrix with probability one. In this paper we formalize a closely related problem in which the existence of the maximum likelihood estimator is guaranteed for all generic observations. We call the number determined by this problem the Gaussian rank of the graph representing the model. We prove that the Gaussian rank is strictly between the subgraph connectivity number and the graph degeneracy number. These bounds are in general much sharper than the best bounds known in the literature and furthermore computable in polynomial time.