Linear Polytree Structural Equation Models: Structural Learning and Inverse Correlation Estimation

We are interested in the problem of learning the directed acyclic graph (DAG) when data are generated from a linear structural equation model (SEM) and the causal structure can be characterized by a polytree. Specially, under both Gaussian and sub-Gaussian models, we study the sample size conditions for the well-known Chow-Liu algorithm to exactly recover the equivalence class of the polytree, which is uniquely represented by a CPDAG. We also study the error rate for the estimation of the inverse correlation matrix under such models. Our theoretical findings are illustrated by comprehensive numerical simulations, and experiments on benchmark data also demonstrate the robustness of the method when the ground truth graphical structure can only be approximated by a polytree.