  We define a modified Wasserstein distance for distribution clustering which
inherits many of the properties of the Wasserstein distance but which can be
estimated easily and computed quickly. The modified distance is the sum of two
terms. The first term --- which has a closed form --- measures the
location-scale differences between the distributions. The second term is an
approximation that measures the remaining distance after accounting for
location-scale differences. We consider several forms of approximation with our
main emphasis being a tangent space approximation that can be estimated using
nonparametric regression. We evaluate the strengths and weaknesses of this
approach on simulated and real examples.
