BUILD with Precision: Bottom-Up Inference of Linear DAGs

2025-12-17

AI Summary

This work addresses the causal discovery problem of exactly recovering the directed acyclic graph (DAG) structure from observational data under linear Gaussian models. We propose BUILD, a bottom-up deterministic algorithm. First, we characterize structural properties of the precision matrix under equal-variance linear structural equation models (SEMs), enabling provably exact layerwise leaf-node identification and pruning. Second, we introduce a dynamic re-estimation strategy to enhance finite-sample robustness. BUILD integrates precision matrix analysis, graph-theoretic pruning, regularized inverse covariance estimation, and conditional independence testing. Theoretically, it guarantees exact DAG recovery with explicitly bounded time complexity. Empirically, on synthetic benchmarks, BUILD significantly outperforms state-of-the-art methods in reconstruction accuracy and demonstrates superior robustness to noise and high-dimensional scaling.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Causal Learning
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Identifies leaf nodes and parents in linear DAGs
• Mitigates error accumulation in precision matrix estimation
• Compares favorably to state-of-the-art DAG learning algorithms

Innovation

• Deterministic stepwise leaf node identification algorithm
• Periodic precision matrix re-estimation for robustness
• Explicit complexity control in DAG reconstruction