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The inverse optimal transport problem is to find the underlying cost functionfrom the knowledge of optimal transport plans. While this amounts to solving alinear inverse problem, in this work we will be concerned with the nonlinearinverse problem to identify the cost function when only a set of marginals andits corresponding optimal values are given. We focus on absolutely continuousprobability distributions with respect to the $d$-dimensional Lebesgue measureand classes of concave and convex cost functions. Our main result implies thatthe cost function is uniquely determined from the union of the ranges of thegradients of the optimal potentials. Since, in general, the optimal potentialsmay not be observed, we derive sufficient conditions for their identifiability- if an open set of marginals is observed, the optimal potentials are thenidentified via the value of the optimal costs. We conclude with a more in-depthstudy of this problem in the univariate case, where an explicit representationof the transport plan is available. Here, we link the notion of identifiabilityof the cost function with that of statistical completeness.

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