Rigidity theory in statistical inference

TL;DR

This work analyzes the maximum likelihood threshold (MLT) for Gaussian statistical models and reveals deep connections to rigidity theory. By recasting ML estimation with inverse-covariance constraints, it shows that the MLT collapses to a combinatorial question about completing partial matrices, with foundational bounds linking MLT to graph invariants like clique number and treewidth. The rigidity perspective yields sharp, dimension-driven bounds in several settings, including chordal graphs, random graphs, and linear concentration models, and it connects to practical methods such as graphical lasso for model selection. The results illuminate when high-dimensional Gaussians can be fit from limited data and how typical graphs arising in applications influence the feasibility of maximum likelihood fitting. Overall, the paper bridges statistical inference, algebraic geometry, and rigidity to articulate when Gaussian models admit ML solutions under structural constraints.

Abstract

In this expository article, we summarize what is known about maximum likelihood thresholds of Gaussian models, paying special attention to connections with rigidity theory.

Figures (4)

• Figure 1: A graph and its corresponding Gaussian graphical model
• Figure 2: The two frameworks on the left are equivalent but not congruent. All edge lengths are the same, but the distance between the two non-adjacent vertices is different. One can visualize moving between the frameworks by folding the white vertex out of the plane of the screen. Without leaving two dimensions, there is no way to continuously deform one of the frameworks into another equivalent framework, so both are locally rigid. The framework on the right is not locally rigid since it can be continuously deformed in the plane without changing any edge lengths, as shown.
• Figure 3: Generic frameworks on the four-cycle in one and two dimensions. In the framework on the right, the longest edge length is the sum of the shorter three. Any framework satisfying this property must have all vertices lying on a line. In particular, the affine span of the vertices of any equivalent framework is one-dimensional. Moreover, this property is robust with respect to perturbing the framework (while staying on the line). Thus one cannot simply dismiss this obstacle by invoking genericity and so Theorem \ref{['thm: rigidity classification of MLT']} implies that the MLT of the four cycle is strictly greater than $2$. The two-dimensional framework on the right is equivalent to any three-dimensional framework obtained by folding the framework along a diagonal out of the plane of the page (or computer screen). In fact, any generic two-dimensional framework on this graph will satisfy this property. Theorem \ref{['thm: rigidity classification of MLT']} then implies that the MLT of the four cycle is at most 3, and thus exactly 3 in light of the one-dimensional frameworks.
• Figure 4: A graph with vertex partition $\mathcal{P} = \{\{1,2\},\{3,4\}\}$, as indicated by the vertex colors, and edge partition $\mathcal{Q} = \{\{12,23\},\{34,14\}\}$, as indicated by the edge line styles, alongside the corresponding RCON model.

Theorems & Definitions (9)

• Definition 1
• Theorem 2: dempster1972covariance
• Theorem 3: buhl1993existence
• Theorem 4: gross2018maximum
• Theorem 5: bernstein2024maximum
• Theorem 6
• Conjecture 7: bernstein2024maximum
• Theorem 8: bernstein2023maximum
• Theorem 9: drton2019maximum