Data-Driven Optimal Transport

The problem of optimal transport between two distributions rho(x) and mu(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. (C) 2016 Wiley Periodicals, Inc.