Inverse Optimal Transport: Learning Cost Functions for Optimal Transport, Xiaojing Ye@Georgia SU

We consider the inverse problem of optimal transport (OT) to recover the ground cost function from observed transport plan or its samples. The inverse OT problem has been cast as a bi-level optimization problem in the literature, where each iteration requires solving a forward OT problem and causes substantial computational cost overall. In this work, we derive an equivalent but much simpler unconstrained and convex optimization formulation of the inverse OT problem, which can be further augmented by customizable regularization and solved easily. We provide a complete characterization of the new optimization problem and its solution space. To validate the effectiveness of this framework, we develop two numerical algorithms, one is a fast matrix scaling method based on the Sinkhorn-Knopp algorithm for the discrete case, and the other one is a deep learning based method that trains the cost function as a deep neural network using the samples of transport plan for the continuous case. Numerical results demonstrate promising efficiency and accuracy advantages of the proposed methods.