Partial correlation hypersurfaces in Gaussian graphical models

TL;DR

This paper establishes a combinatorial condition ensuring the nonsingularity of partial correlation hypersurfaces in Gaussian graphical models, with implications for graph learning algorithms and confirming a prior conjecture for complete DAGs.

Contribution

It introduces a new combinatorial criterion for nonsingularity of hypersurfaces in Gaussian models and explores its application in learning algorithms, confirming a conjecture for complete DAGs.

Findings

Condition is fulfilled for complete DAGs of any size

Supports potential use in graph learning algorithms

Confirms a conjecture by Lin-Uhler-Sturmfels-Bühlmann

Abstract

We derive a combinatorial sufficient condition for a partial correlation hypersurface in the parameter space of a directed Gaussian graphical model to be nonsingular, and speculate on whether this condition can be used in algorithms for learning the graph. Since the condition is fulfilled in the case of a complete DAG on any number of vertices, the result implies an affirmative answer to a question raised by Lin-Uhler-Sturmfels-B\"uhlmann.