  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.
