Hybrid Wasserstein Distance and Fast Distribution Clustering

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.