Degrees of the Wasserstein Distance to Small Toric Models

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.