Incorporating structural constraints into continuous optimization for causal discovery

Directed Acyclic Graphs (DAGs) provide an efficient framework to describe the causal relations in actual applications, and it appears more and more important to learn a DAG from training data in causal discovery. Recently, a novel methodology, which projects the acyclic constraints by an algebraic characterization and employs continuous optimization to carry the causal discovery, gradually became the mainstream. However, such methods focus on a best-fitting to the training data and cannot utilize the prior knowledge in an efficient way. To resolve this problem, we suggest incorporating structural constraints into continuous optimization. For edge constraints, we regard the activation value of the difference between the constraint matrix after thresholding and the weight matrix as the optimization goal. For path constraints, we use the deviation concluded from the power matrix on kth path graphs to design the penalty functions. For ordering constraints, we exploit the representation based on the negative edge/path constraints. The mathematical derivations prove that equality constraint program (ECP), in which proposed equality constraints powerfully embody the required structural restrictions, are solvable. Furthermore, the experimental evaluations indicate that the proposed method develops higher scalability and accuracy against state-of-the-art algorithms.