Nonlinear Inverse Optimal Transport: Identifiability of the Transport Cost from its Marginals and Optimal Values

The inverse optimal transport problem is to find the underlying cost function from the knowledge of optimal transport plans. While this amounts to solving a linear inverse problem, in this work we will be concerned with the nonlinear inverse problem to identify the cost function when only a set of marginals and its corresponding optimal values are given. We focus on absolutely continuous probability distributions with respect to the $d$-dimensional Lebesgue measure and classes of concave and convex cost functions. Our main result implies that the cost function is uniquely determined from the union of the ranges of the gradients of the optimal potentials. Since, in general, the optimal potentials may not be observed, we derive sufficient conditions for their identifiability - if an open set of marginals is observed, the optimal potentials are then identified via the value of the optimal costs. We conclude with a more in-depth study of this problem in the univariate case, where an explicit representation of the transport plan is available. Here, we link the notion of identifiability of the cost function with that of statistical completeness.