Data-Driven Optimal Transport

The problem of optimal transport between two distributions ρ(x) and μ(y) is extended to situations where the distributions are only known through a finite number of samples (x_i) and (y_j). A weak formulation is proposed, based on the dual of the Kantorovich formulation, with two main modifications: replacing the expected values in the objective function by their empirical means over the (x_i) and (y_j), and restricting the dual variables u(x) and v(y) to a suitable set of test functions adapted to the local availability of sample points. A procedure is proposed and tested for the numerical solution of this problem, based on a fluidlike flow in phase space, where the sample points play the role of active Lagrangian markers.