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. We use the new lower bounds to give high-probability guarantees on the maximum likelihood thresholds of sparse Erd{\"o}s-R\'enyi random graphs in terms of their average density. These examples show that the new lower bounds are within a polylog factor of tight, where, on the same graph families, all known lower bounds are trivial. Based on computational experiments made possible by our methods, we conjecture that the MLT of an Erd{\"o}s-R\'enyi random graph is equal to its generic completion rank with high probability. Using structural results on rigid graphs in low dimension, we can prove the conjecture for graphs with MLT at most $4$ and describe the threshold probability for the MLT to switch from $3$ to $4$. We also give a 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.

翻译：图$G$的最大似然阈值（MLT）是在相应高斯图模型中几乎必然保证最大似然估计存在所需的最小样本数。我们基于$G$的刚性理论性质给出了MLT的新刻画，并利用此刻画为任意图的MLT提供了新的组合下界。我们运用这些新下界，根据稀疏埃尔德什-雷尼随机图的平均密度，给出了其最大似然阈值的高概率保证。这些例子表明，新下界与紧性相差至多一个多对数因子，而在相同图族上，所有已知下界均为平凡结果。基于我们的方法所实现的计算实验，我们猜想埃尔德什-雷尼随机图的MLT以高概率等于其泛型完备秩。利用低维刚性图的结构结果，我们可证明MLT不超过4的图符合该猜想，并描述了MLT从3变为4的阈值概率。此外，我们通过一种值得独立关注的框架“提升”问题，给出了图MLT的几何刻画。提升视角揭示了弱MLT（此时最大似然估计仅以正概率存在）与经典哈德维格-纳尔逊问题之间的新联系。