Estimating high-dimensional directed acyclic graphs with the PC-algorithm

TL;DR

This paper proves the consistency of the PC-algorithm for high-dimensional sparse DAGs with Gaussian distributions, demonstrating its efficiency and robustness in large-scale, sparse settings.

Contribution

It establishes the theoretical consistency of the PC-algorithm in very high-dimensional sparse DAG estimation, allowing the number of nodes to grow rapidly with sample size.

Findings

The PC-algorithm is consistent for high-dimensional sparse DAGs.

It is computationally feasible for large, sparse graphs.

The algorithm shows robustness to tuning parameter choices.

Abstract

We consider the PC-algorithm Spirtes et. al. (2000) for estimating the skeleton of a very high-dimensional acyclic directed graph (DAG) with corresponding Gaussian distribution. The PC-algorithm is computationally feasible for sparse problems with many nodes, i.e. variables, and it has the attractive property to automatically achieve high computational efficiency as a function of sparseness of the true underlying DAG. We prove consistency of the algorithm for very high-dimensional, sparse DAGs where the number of nodes is allowed to quickly grow with sample size n, as fast as O(n^a) for any 0<a<infinity. The sparseness assumption is rather minimal requiring only that the neighborhoods in the DAG are of lower order than sample size n. We empirically demonstrate the PC-algorithm for simulated data and argue that the algorithm is rather insensitive to the choice of its single tuning parameter.