通过图刚性求解最大似然阈值
Maximum likelihood thresholds via graph rigidity
AI总结：
本文用图刚性理论刻画 Gaussian 图模型的最大似然阈值，提出新的组合下界并分析稀疏 Erdős-Rényi 随机图，还建立其与泛型补全秩及 Hadwiger-Nelson 问题的联系。
摘要：
图 $G$ 的最大似然阈值（MLT）是指在相应的 Gaussian 图模型中，几乎必然保证最大似然估计存在所需的最少样本数。我们利用 $G$ 的刚性理论性质给出了 MLT 的一种新刻画，并利用该刻画对任意图的 MLT 给出了新的组合下界。我们利用这些新下界，依据稀疏 Erdős-Rényi 随机图的平均密度，对其最大似然阈值给出了高概率保证。这些例子表明，新下界与紧界之间仅相差一个 polylog 因子；而在相同的图族上，所有已知下界都是平凡的。基于由我们的方法所促成的计算实验，我们猜想 Erdős-Rényi 随机图的 MLT 以高概率等于其泛型补全秩。利用低维刚性图的结构性结果，我们能够对 MLT 不超过 $4$ 的图证明该猜想，并描述 MLT 从 $3$ 转变为 $4$ 的阈值概率。我们还通过框架的一个新的“提升”问题给出了图 MLT 的几何刻画，该问题本身也很有意义。提升视角揭示了弱 MLT（即最大似然估计仅以正概率存在）与经典 Hadwiger-Nelson 问题之间的一种新联系。
英文摘要：
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{ö}s-Rényi 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{ö}s-Rényi 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
