High-Dimensional Poisson DAG Model Learning Using $\ell_1$-Regularized Regression
This addresses the problem of estimating DAGs for multivariate count data in high-dimensional settings, offering a method with weaker assumptions, but it is incremental as it builds on existing regression-based approaches.
The paper tackles learning high-dimensional Poisson directed acyclic graphical models from observational data without strong assumptions, achieving recovery with a sample size of n = Ω(d^2 log^9 p) and showing competitive performance against state-of-the-art algorithms in simulations and real data.
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 $n = Ω( d^{2} \log^{9} p)$ 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.