Metrics on Probability Distributions Through Optimal Commuting Maps

We describe a class of metrics introduced in [9] in which two probability distributions are compared via their interaction with a fixed reference measure \(\nu \) , which, in many cases of interest, concentrates on a lower dimensional submanifold. This is a variant of the well known Wasserstein metric from optimal transport, in which we modify the optimal transport problem to only allow couplings which are compatible with optimal transport between the marginals and \(\nu \) . We describe several characterizations of this metric, as well as a theorem identifying conditions under which the optimal coupling is unique and concentrates on a graph over the first marginal. We also describe the metric explicitly for several choices of the reference measure, illustrating that different base measures can result in very different geometry. Finally, we augment a result in [9] by showing that our metric arises as the derivative of the cost in a multi-marginal optimal transport problem with respect to a parameter expressing the relative weights of the interactions between the variables.