BUILD with Precision: Bottom-Up Inference of Linear DAGs

TL;DR

BUILD is a deterministic, bottom-up algorithm that accurately reconstructs linear DAGs from observational data by exploiting the structure of the precision matrix, with robustness enhancements for practical data estimation.

Contribution

The paper introduces BUILD, a novel stepwise method for DAG recovery that leverages precision matrix structure and improves robustness through iterative re-estimation.

Findings

BUILD outperforms existing algorithms on synthetic benchmarks.

The method provides explicit complexity control.

Periodic re-estimation enhances robustness in finite data scenarios.

Abstract

Learning the structure of directed acyclic graphs (DAGs) from observational data is a central problem in causal discovery, statistical signal processing, and machine learning. Under a linear Gaussian structural equation model (SEM) with equal noise variances, the problem is identifiable and we show that the ensemble precision matrix of the observations exhibits a distinctive structure that facilitates DAG recovery. Exploiting this property, we propose BUILD (Bottom-Up Inference of Linear DAGs), a deterministic stepwise algorithm that identifies leaf nodes and their parents, then prunes the leaves by removing incident edges to proceed to the next step, exactly reconstructing the DAG from the true precision matrix. In practice, precision matrices must be estimated from finite data, and ill-conditioning may lead to error accumulation across BUILD steps. As a mitigation strategy, we periodically re-estimate the precision matrix (with less variables as leaves are pruned), trading off runtime for enhanced robustness. Reproducible results on challenging synthetic benchmarks demonstrate that BUILD compares favorably to state-of-the-art DAG learning algorithms, while offering an explicit handle on complexity.