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.
翻译:从数据 { 中导出循环图( DAGs) 的组合问题最近被定义为一个纯粹的持续优化问题,它利用了基于矩阵指数函数的跟踪对 DAG 进行不同的周期性描述。 现有的循环性描述是基于一个想法, 即对称矩阵的功率包含关于行走和周期的信息。 在这项工作中, 我们提议了一个 $\ textit{ 基础不同} 美元周期性描述 。 基于对数确定( log- deit) 函数的周期性描述, 从而利用 DAGs 的无效性属性。 要处理 DAG 的内在偏差性速度性描述。 我们将日- 检测域特性与 $\ text{ M- 数学} 的设置联系起来, 这是对正确定矩阵定义定义的经典日志偏差的关键。 与先前提议的周期性函数相似, 我们的直径偏差性描述也是精确和不同的。 但是, 与现有的侧描述相比, 我们的逻辑- deal deal deal dreal 方法比, 我们的对 had- mal deal- male lialalalalalalalal 表示 a a a rode rode rodeal deal deal deal deal deal deal deal deal deal deal deal deal deal madal mas mas to the a ma a ma ma ma ma ma ma ma ma la la la la la ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma la ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma ma