Recently, the task of acquiring causal directed acyclic graphs (DAGs) from empirical data has been modeled as an iterative process within the framework of continuous optimization with a differentiable acyclicity characterization. However, learning DAGs from data is an NP-hard problem since the DAG space increases super-exponentially with the number of variables. In this work, we introduce the graph topological sorts in solving the continuous optimization problem, which is substantially smaller than the DAG space and beneficial in avoiding local optima. Moreover, the topological sorts space does not require consideration of acyclicity, which can significantly reduce the computational cost. To further deal with the inherent asymmetries of DAGs, we investigate the acyclicity characterization and propose a new DAGs learning optimization strategy based on QR factorization, named DAGOR. First, using the matrix congruent transformation, the adjacency matrix of the DAG is transformed into an upper triangular matrix with a topological sort. Next, using the QR factorization as a basis, we construct a least-square penalty function as constraints for optimization in the graph autoencoder framework. Numerical experiments are performed to further validate our theoretical results and demonstrate the competitive performance of our method.

中文翻译：

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DAGOR：通过拓扑排序和 QR 分解学习 DAG

最近，从经验数据获取因果有向无环图（DAG）的任务已被建模为具有可微分无环特征的连续优化框架内的迭代过程。然而，从数据中学习 DAG 是一个 NP 难题，因为 DAG 空间随着变量数量呈超指数增长。在这项工作中，我们引入了图拓扑排序来解决连续优化问题，它比 DAG 空间小得多，有利于避免局部最优。而且，拓扑排序空间不需要考虑非循环性，可以显着降低计算成本。为了进一步处理 DAG 固有的不对称性，我们研究了非循环性表征，并提出了一种基于 QR 分解的新 DAG 学习优化策略，称为 DAGOR。首先，利用矩阵全等变换，将DAG的邻接矩阵变换为拓扑排序的上三角矩阵。接下来，以 QR 分解为基础，我们构造一个最小二乘罚函数作为图自动编码器框架中优化的约束。进行数值实验以进一步验证我们的理论结果并证明我们的方法的竞争性能。