Greedy Causal Discovery is Geometric

Finding a directed acyclic graph (DAG) that best encodes the conditional independence statements observable from data is a central question within causality. Algorithms that greedily transform one candidate DAG into another given a fixed set of moves have been particularly successful, for example the GES, GIES, and MMHC algorithms. In 2010, Studen\'y, Hemmecke and Lindner introduced the characteristic imset polytope, $\operatorname{CIM}_p$, whose vertices correspond to Markov equivalence classes, as a way of transforming causal discovery into a linear optimization problem. We show that the moves of the aforementioned algorithms are included within classes of edges of $\operatorname{CIM}_p$ and that restrictions placed on the skeleton of the candidate DAGs correspond to faces of $\operatorname{CIM}_p$. Thus, we observe that GES, GIES, and MMHC all have geometric realizations as greedy edge-walks along $\operatorname{CIM}_p$. Furthermore, the identified edges of $\operatorname{CIM}_p$ strictly generalize the moves of these algorithms. Exploiting this generalization, we introduce a greedy simplex-type algorithm called \emph{greedy CIM}, and a hybrid variant, \emph{skeletal greedy CIM}, that outperforms current competitors among hybrid and constraint-based algorithms.