DAGMA：基于M-矩阵和对数行列式的DAGs学习
DAGMA: Learning DAGs via M-matrices and a Log-Determinant Acyclicity Characterization
摘要：
最近，通过利用基于矩阵指数函数的迹的有向无环图（DAGs）的可微无环性特征，从数据中学习有向无环图（DAGs）的组合问题被框架为纯粹的连续优化问题。现有的无环性刻画是基于邻接矩阵的幂包含了行走和循环的信息。在这项工作中，我们提出了一个新的非循环特性的基础上的对数行列式（log-det）功能，它利用了幂零性的DAG。为了处理DAG的固有不对称性，我们将我们的log-det特征域与$\textit{M-matrices}$的集合相关联，这是与在正定矩阵的锥上定义的经典log-det函数的关键区别。类似于以前提出的无环函数，我们的特征也是精确和可微的。然而，与现有的特征相比，我们的log-det函数：（1）更善于检测大周期;（2）具有更好的梯度;（3）它的运行时间实际上快了一个数量级。从优化方面，我们放弃了通常使用的增广拉格朗日计划，并提出DAGMA（$\textit{DAGs via M-matrices for Acyclicity}$），一种类似于障碍方法的中心路径的方法。DAGMA的中心路径中的每个点都是由我们的log-det函数正则化的无约束问题的解，然后我们证明了在中心路径的极限处，该解保证是DAG。最后，我们为$\textit{linear}$和$\textit{nonlinear}$ SEM提供了广泛的实验，并表明我们的方法可以达到大的加速和较小的结构汉明距离对国家的最先进的方法。实现所提出的方法的代码是开源的，可以在https://github.com/kevinsbello/dagma上公开获得。
英文摘要：
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 new 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 ($\textit{DAGs via M-matrices for Acyclicity}$), 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. Code implementing the proposed method is open-source and publicly available at https://github.com/kevinsbello/dagma.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
