The edges of the characteristic imset polytope, $\operatorname{CIM}_p$, were recently shown to have strong connections to causal discovery as many algorithms could be interpreted as greedy restricted edge-walks, even though only a strict subset of the edges are known. To better understand the general edge structure of the polytope we describe the edge structure of faces with a clear combinatorial interpretation: for any undirected graph $G$ we have the face $\operatorname{CIM}_G$, the convex hull of the characteristic imsets of DAGs with skeleton $G$. We give a full edge-description of $\operatorname{CIM}_G$ when $G$ is a tree, leading to interesting connections to other polytopes. In particular the well-studied stable set polytope can be recovered as a face of $\operatorname{CIM}_G$ when $G$ is a tree. Building on this connection we are also able to give a description of all edges of $\operatorname{CIM}_G$ when $G$ is a cycle, suggesting possible inroads for generalization. We then introduce an algorithm for learning directed trees from data, utilizing our newly discovered edges, that outperforms classical methods on simulated Gaussian data.
翻译:显性 聚点的边缘, $\ operatorname{ CIM ⁇ p$, 最近被证明与因果关系发现有着密切的联系, 因为许多算法可以被解释为贪婪的边行限制边行, 即使只知道边缘的严格子集。 为了更好地了解多点的普通边缘结构, 我们用清晰的组合解释来描述面的边缘结构: 对于任何未引导的图形 $G$, 我们拥有面值$\ operatorname{ CIM ⁇ G$, 与骨架 DAG 特征的顶层相近。 当$G$是一棵树时, 我们给出了$\ operatorname{ CIM ⁇ G$的完全边界描述, 当$G$是一棵树树时, 我们的稳定的多点结构可以恢复为$@CIM ⁇ G$的面值。 在此连接上, 我们也可以描述 $$Operatorname{C} G$ 完全的边框。 当我们从 Sqolentalal 中学习了一种新数据时, 我们的G 正在学习一个 Grodeal_ 。