Learning Linear Polytree Structural Equation Models

TL;DR

This paper investigates the conditions under which linear polytree structures can be accurately learned from data, providing theoretical bounds, algorithmic guarantees, and empirical validation for structure recovery and inverse correlation matrix estimation.

Contribution

It offers sharp sample complexity bounds for learning Gaussian polytree structures using the Chow-Liu algorithm and extends the analysis to group polytrees, with comprehensive theoretical and empirical results.

Findings

Exact recovery conditions for skeleton and CPDAG using Chow-Liu algorithm.

Information-theoretic lower bounds match sufficient conditions, characterizing difficulty.

Estimation error bounds for inverse correlation matrices in linear polytree models.

Abstract

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. Under the Gaussian polytree models, we study sufficient conditions on the sample sizes for the well-known Chow-Liu algorithm to exactly recover both the skeleton and the equivalence class of the polytree, which is uniquely represented by a CPDAG. On the other hand, necessary conditions on the required sample sizes for both skeleton and CPDAG recovery are also derived in terms of information-theoretic lower bounds, which match the respective sufficient conditions and thereby give a sharp characterization of the difficulty of these tasks. We also consider the problem of inverse correlation matrix estimation under the linear polytree models, and establish the estimation error bound in terms of the dimension and the total number of v-structures. We also consider an extension of group linear polytree models, in which each node represents a group of variables. Our theoretical findings are illustrated by comprehensive numerical simulations, and experiments on benchmark data also demonstrate the robustness of polytree learning when the true graphical structures can only be approximated by polytrees.