High-Dimensional Poisson Structural Equation Model Learning via $\ell_1$-Regularized Regression

Abstract

In this paper, we develop a new approach to learning high-dimensional Poisson structural equation models from only observational data without strong assumptions such as faithfulness and a sparse moralized graph. 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 moments relation. We show that sample size $n = \Omega( d^{2} \log^{9} p)$ is sufficient for our polynomial time Moments 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.