[Paper Review] Learning linear non-Gaussian directed acyclic graph with diverging number of nodes

Ruixuan Zhao, Xin He+1 more|arXiv (Cornell University)|Nov 1, 2021

Bayesian Modeling and Causal Inference33 references4 citations

TL;DR

This paper proposes a novel, efficient method for learning linear non-Gaussian directed acyclic graphs (DAGs) in high-dimensional settings with a diverging number of nodes. By introducing a topological layer framework, the method enables exact, bottom-up reconstruction of DAG structure via precision matrix estimation and independence testing, achieving consistent causal structure recovery without requiring faithfulness or parental faithfulness assumptions.

ABSTRACT

Acyclic model, often depicted as a directed acyclic graph (DAG), has been widely employed to represent directional causal relations among collected nodes. In this article, we propose an efficient method to learn linear non-Gaussian DAG in high dimensional cases, where the noises can be of any continuous non-Gaussian distribution. This is in sharp contrast to most existing DAG learning methods assuming Gaussian noise with additional variance assumptions to attain exact DAG recovery. The proposed method leverages a novel concept of topological layer to facilitate the DAG learning. Particularly, we show that the topological layers can be exactly reconstructed in a bottom-up fashion, and the parent-child relations among nodes in each layer can also be consistently established. More importantly, the proposed method does not require the faithfulness or parental faithfulness assumption which has been widely assumed in the literature of DAG learning. Its advantage is also supported by the numerical comparison against some popular competitors in various simulated examples as well as a real application on the global spread of COVID-19.

Motivation & Objective

• Address the challenge of learning high-dimensional linear non-Gaussian DAGs where the number of nodes grows with sample size.
• Overcome the limitations of existing methods that rely on strong assumptions such as Gaussian noise, equal variance, or faithfulness.
• Develop a computationally efficient algorithm for exact DAG recovery in high-dimensional settings.
• Provide theoretical guarantees for consistent recovery of topological layers and parent–child relationships without faithfulness assumptions.

Proposed method

• Introduce a novel topological layer concept that partitions nodes into layers such that parents of any node lie in higher layers, ensuring acyclicity by construction.
• Reconstruct topological layers in a bottom-up fashion using precision matrix estimation and conditional independence testing.
• Use the estimated precision matrix to directly identify parent–child relationships within each layer.
• Leverage graphical Lasso for high-dimensional precision matrix estimation under sparsity assumptions.
• Apply a hard-thresholding procedure to refine edge detection based on estimated partial correlations.
• Ensure computational efficiency by avoiding exhaustive search over all possible DAGs, reducing complexity significantly compared to prior methods.

Experimental results

Research questions

• RQ1 Can topological layers in a linear non-Gaussian DAG be exactly reconstructed in high-dimensional settings with a diverging number of nodes?
• RQ2 Does the proposed method achieve consistent recovery of the true DAG structure without relying on faithfulness or parental faithfulness assumptions?
• RQ3 How does the method’s performance compare to existing state-of-the-art DAG learning algorithms in terms of accuracy and computational efficiency?
• RQ4 Can the method successfully recover causal structures in real-world data with complex dependencies, such as the global spread of COVID-19?
• RQ5 What is the impact of tuning parameters and data-generating mechanisms on the method’s performance in finite samples?

Key findings

• The proposed method, named TL (Topological Layer), achieves exact reconstruction of topological layers in high-dimensional settings, as proven in Theorem 1 and Corollary 1.
• Parent–child relationships are consistently recovered alongside layer reconstruction, as shown in Corollary 2, using only the estimated precision matrix.
• The method does not require the faithfulness or parental faithfulness assumption, making it applicable under milder statistical conditions than most existing approaches.
• In simulated experiments, TL outperforms competitors like MDirect and LiNGAM in terms of structural Hamming distance and F1 score, especially in high-dimensional and sparse settings.
• In the real-world application to the global spread of COVID-19, TL identified 736 directed edges among 99 countries, with major hubs including Italy, the UK, South Korea, and the US—consistent with observed early transmission patterns.
• The method’s computational complexity is significantly lower than exponential-order methods, enabling scalable learning in high-dimensional settings.