  Inverse optimal transport (OT) refers to the problem of learning the cost
function for OT from observed transport plan or its samples. In this paper, we
derive an unconstrained convex optimization formulation of the inverse OT
problem, which can be further augmented by any customizable regularization. We
provide a comprehensive characterization of the properties of inverse OT,
including uniqueness of solutions. We also develop two numerical algorithms,
one is a fast matrix scaling method based on the Sinkhorn-Knopp algorithm for
discrete OT, and the other one is a learning based algorithm that parameterizes
the cost function as a deep neural network for continuous OT. The novel
framework proposed in the work avoids repeatedly solving a forward OT in each
iteration which has been a thorny computational bottleneck for the bi-level
optimization in existing inverse OT approaches. Numerical results demonstrate
promising efficiency and accuracy advantages of the proposed algorithms over
existing state-of-the-art methods.
