On Low-Rank Directed Acyclic Graphs and Causal Structure Learning

Despite several advances in recent years, learning causal structures represented by directed acyclic graphs (DAGs) remains a challenging task in high-dimensional settings when the graphs to be learned are not sparse. In this article, we propose to exploit a low-rank assumption regarding the (weighted) adjacency matrix of a DAG causal model to help address this problem. We utilize existing low-rank techniques to adapt causal structure learning methods to take advantage of this assumption and establish several useful results relating interpretable graphical conditions to the low-rank assumption. Specifically, we show that the maximum rank is highly related to hubs, suggesting that scale-free (SF) networks, which are frequently encountered in practice, tend to be low rank. Our experiments demonstrate the utility of the low-rank adaptations for a variety of data models, especially with relatively large and dense graphs. Moreover, with a validation procedure, the adaptations maintain a superior or comparable performance even when graphs are not restricted to be low rank.