Maximum likelihood thresholds via graph rigidity

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. We give a new characterization of the MLT in terms of rigidity-theoretic properties of $G$ and use this characterization to give new combinatorial lower bounds on the MLT of any graph. Our bounds, based on global rigidity, generalize existing bounds and are considerably sharper. We classify the graphs with MLT at most three, and compute the MLT of every graph with at most $9$ vertices. Additionally, for each $k$ and $n\ge k$, we describe graphs with $n$ vertices and MLT $k$, adding substantially to a previously small list of graphs with known MLT. We also give a purely geometric characterization of the MLT of a graph in terms of a new "lifting" problem for frameworks that is interesting in its own right. The lifting perspective yields a new connection between the weak MLT (where the maximum likelihood estimate exists only with positive probability) and the classical Hadwiger-Nelson problem.