  We propose a unified data-driven framework based on inverse optimal transport
that can learn adaptive, nonlinear interaction cost function from noisy and
incomplete empirical matching matrix and predict new matching in various
matching contexts. We emphasize that the discrete optimal transport plays the
role of a variational principle which gives rise to an optimization-based
framework for modeling the observed empirical matching data. Our formulation
leads to a non-convex optimization problem which can be solved efficiently by
an alternating optimization method. A key novel aspect of our formulation is
the incorporation of marginal relaxation via regularized Wasserstein distance,
significantly improving the robustness of the method in the face of noisy or
missing empirical matching data. Our model falls into the category of
prescriptive models, which not only predict potential future matching, but is
also able to explain what leads to empirical matching and quantifies the impact
of changes in matching factors. The proposed approach has wide applicability
including predicting matching in online dating, labor market, college
application and crowdsourcing. We back up our claims with numerical experiments
on both synthetic data and real world data sets.
