  Optimal transport (OT) formalizes the problem of finding an optimal coupling
between probability measures given a cost matrix. The inverse problem of
inferring the cost given a coupling is Inverse Optimal Transport (IOT). IOT is
less well understood than OT. We formalize and systematically analyze the
properties of IOT using tools from the study of entropy-regularized OT.
Theoretical contributions include characterization of the manifold of
cross-ratio equivalent costs, the implications of model priors, and derivation
of an MCMC sampler. Empirical contributions include visualizations of
cross-ratio equivalent effect on basic examples and simulations validating
theoretical results.
