Optimal transport with a density-dependent cost function

A new pairwise cost function is proposed for the optimal transport barycenter problem, adopting the form of the minimal action between two points, with a Lagrangian that takes into account an underlying probability distribution. Under this notion of distance, two points can only be close if there exist paths joining them that do not traverse areas of small probability. A framework is proposed and developed for the numerical solution of the corresponding data-driven optimal transport problem. The procedure parameterizes the paths of minimal action through path dependent Chebyshev polynomials and enforces the agreement between the paths' endpoints and the given source and target distributions through an adversarial penalization. The methodology and its application to clustering and matching problems is illustrated through synthetic examples.