DAG-Informed Structure Learning from Multi-Dimensional Point Processes

Abstract

Motivated by inferring causal relationships among neurons using ensemble spike train data, this paper introduces a new technique for learning the structure of a directed acyclic graph (DAG) within a large network of events, applicable to diverse multi-dimensional temporal point process (MuTPP) data. At the core of MuTPP lie the conditional intensity functions, for which we construct a generative model parameterized by the graph parameters of a DAG and develop an equality-constrained estimator, departing from exhaustive search-based methods. We present a novel, flexible augmented Lagrangian (Flex-AL) optimization scheme that ensures provable global convergence and computational efficiency gains over the classical AL algorithm. Additionally, we explore causal structure learning by integrating acyclicity-constraints and sparsity-regularization. We demonstrate: (i) in cases without regularization, the incorporation of the acyclicity constraint is essential for ensuring DAG recovery consistency; (ii) with suitable regularization, the DAG-constrained estimator achieves both parameter estimation and DAG reconstruction consistencies similar to the unconstrained counterpart, but significantly enhances empirical performance. Furthermore, simulation studies indicate that our proposed DAG-constrained estimator, when appropriately penalized, yields more accurate graphs compared to unconstrained or unregularized estimators. Finally, we apply the proposed method to two real MuTPP datasets.