Identifiability and Exact Reconstruction of the Optimal Transport Cost on Finite Spaces

The goal of optimal transport (OT) is to find optimal assignments or matchings between data sets which minimize the total cost for a given cost function. However, sometimes the cost function is unknown but we have access to (parts of) the solution to the OT problem, e.g.\ the OT plan or the value of the objective function. Recovering the cost from such information is called inverse OT and has become recently of certain interest triggered by novel applications, e.g.\ in social science and economics. This raises the issue under which circumstances such cost is identifiable, i.e., it can be uniquely recovered from other OT quantities. In this work we provide sufficient and necessary conditions for the identifiability of the cost function on finite ground spaces. We find that such conditions correspond to the combinatorial structure of the corresponding linear program and discuss its computational complexity and implications for cost estimation in statistical linear models.