The combinatorial problem of learning directed acyclic graphs (DAGs) from data was recently framed as a purely continuous optimization problem by leveraging a differentiable acyclicity characterization of DAGs based on the trace of a matrix exponential function. Existing acyclicity characterizations are based on the idea that powers of an adjacency matrix contain information about walks and cycles. In this work, we propose a $\textit{fundamentally different}$ acyclicity characterization based on the log-determinant (log-det) function, which leverages the nilpotency property of DAGs. To deal with the inherent asymmetries of a DAG, we relate the domain of our log-det characterization to the set of $\textit{M-matrices}$, which is a key difference to the classical log-det function defined over the cone of positive definite matrices. Similar to acyclicity functions previously proposed, our characterization is also exact and differentiable. However, when compared to existing characterizations, our log-det function: (1) Is better at detecting large cycles; (2) Has better-behaved gradients; and (3) Its runtime is in practice about an order of magnitude faster. From the optimization side, we drop the typically used augmented Lagrangian scheme, and propose DAGMA ($\textit{Directed Acyclic Graphs via M-matrices for Acyclicity}$), a method that resembles the central path for barrier methods. Each point in the central path of DAGMA is a solution to an unconstrained problem regularized by our log-det function, then we show that at the limit of the central path the solution is guaranteed to be a DAG. Finally, we provide extensive experiments for $\textit{linear}$ and $\textit{nonlinear}$ SEMs, and show that our approach can reach large speed-ups and smaller structural Hamming distances against state-of-the-art methods.

该研究提出了一种基于对数行列式函数的新型有向无环图（DAG）的无环性描述，该描述利用DAG的幂零属性将其与正定矩阵锥定义的经典对数行列式函数进行了区分。该方法与现有技术相比，对于检测大型循环性更好，在梯度方面更好，并且实际运行时间大约快一个数量级。在数值实验中表明，该方法能够获得较大的速度提升和较小的结构Hamming距离。

DAGMA: 基于M矩阵和对数行列式无环特征学习DAGs