Computing maximum likelihood thresholds using graph rigidity

TL;DR

This paper advances the understanding of the maximum likelihood threshold (MLT) in Gaussian graphical models by leveraging graph rigidity theory to compute exact MLT values for various graph families.

Contribution

It introduces methods to precisely compute MLT for specific graph classes using combinatorial rigidity results, extending prior bounds.

Findings

Exact MLT values for graphs with up to 9 vertices

MLT computation for graphs with up to 24 edges

Determination of MLT for graphs close to complete graphs and with bounded degrees

Abstract

The maximum likelihood threshold (MLT) of a graph $G$ is the minimum number of samples to almost surely guarantee existence of the maximum likelihood estimate in the corresponding Gaussian graphical model. Recently a new characterization of the MLT in terms of rigidity-theoretic properties of $G$ was proved \cite{Betal}. This characterization was then used to give new combinatorial lower bounds on the MLT of any graph. We continue this line of research by exploiting combinatorial rigidity results to compute the MLT precisely for several families of graphs. These include graphs with at most $9$ vertices, graphs with at most 24 edges, every graph sufficiently close to a complete graph and graphs with bounded degrees.