High-Dimensional Poisson DAG Model Learning Using $\ell_1$-Regularized Regression

TL;DR

This paper introduces a new polynomial-time method for learning high-dimensional Poisson DAG models from observational data, using $ ext{l}_1$-regularized regression and mean-variance relationships, effective even with limited sample sizes.

Contribution

It proposes the Mean-variance Ratio Scoring (MRS) algorithm that relaxes strong assumptions and efficiently recovers DAG structures in high-dimensional Poisson models.

Findings

MRS recovers true DAG with sample size $n = ext{Omega}(d^2 ext{log}^9 p)$

Algorithm is statistically consistent in high-dimensional $p > n$ settings

Performs well on multivariate count data compared to existing methods

Abstract

In this paper, we develop a new approach to learning high-dimensional Poisson directed acyclic graphical (DAG) models from only observational data without strong assumptions such as faithfulness and strong sparsity. A key component of our method is to decouple the ordering estimation or parent search where the problems can be efficiently addressed using $\ell_1$-regularized regression and the mean-variance relationship. We show that sample size $n = \Omega( d^{2} \log^{9} p)$ is sufficient for our polynomial time Mean-variance Ratio Scoring (MRS) algorithm to recover the true directed graph, where $p$ is the number of nodes and $d$ is the maximum indegree. We verify through simulations that our algorithm is statistically consistent in the high-dimensional $p>n$ setting, and performs well compared to state-of-the-art ODS, GES, and MMHC algorithms. We also demonstrate through multivariate real count data that our MRS algorithm is well-suited to estimating DAG models for multivariate count data in comparison to other methods used for discrete data.