The study of the closest point(s) on a statistical model from a given distribution in the probability simplex with respect to a fixed Wasserstein metric gives rise to a polyhedral norm distance optimization problem. There are two components to the complexity of determining the Wasserstein distance from a data point to a model. One is the combinatorial complexity that is governed by the combinatorics of the Lipschitz polytope of the finite metric to be used. Another is the algebraic complexity, which is governed by the polar degrees of the Zariski closure of the model. We find formulas for the polar degrees of rational normal scrolls and graphical models whose underlying graphs are star trees. Also, the polar degrees of the graphical models with four binary random variables where the graphs are a path on four vertices and the four-cycle, as well as for small, no-three-way interaction models, were computed. We investigate the algebraic degree of computing the Wasserstein distance to a small subset of these models. It was observed that this algebraic degree is typically smaller than the corresponding polar degree.

翻译：在给定Wasserstein度量下，概率单纯形中统计模型与给定分布之间的最近点（一个或多个）的研究引出了一个多面体范数距离优化问题。确定数据点到模型的Wasserstein距离的复杂度包含两个组成部分：一是由所使用有限度量Lipschitz多面体的组合结构决定的组合复杂度；二是由模型Zariski闭包的极度数决定的代数复杂度。我们给出了有理正规旋转曲面和基图为星树的图模型极度数的公式。此外，计算了四个二元随机变量图模型（图分别为四顶点路径和四环）以及小型无三向交互作用模型的极度数。我们研究了计算距这些模型子集Wasserstein距离的代数度，发现该代数度通常小于对应的极度数。