Computing the Distance between unbalanced Distributions -- The flat
  Metric
Henri Schmidt Christian Düll
We provide an implementation to compute the flat metric in any dimension. The
flat metric, also called dual bounded Lipschitz distance, generalizes the
well-known Wasserstein distance W1 to the case that the distributions are of
unequal total mass. This is of particular interest for unbalanced optimal
transport tasks and for the analysis of data distributions where the sample
size is important or normalization is not possible. The core of the method is
based on a neural network to determine on optimal test function realizing the
distance between two given measures. Special focus was put on achieving
comparability of pairwise computed distances from independently trained
networks. We tested the quality of the output in several experiments where
ground truth was available as well as with simulated data.