Data-Driven Optimal Transport

Giulio Trigila, Esteban G. Tabak

Research output: Contribution to journalArticlepeer-review


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 (xi) and (yj). 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 (xi) and (yj), 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.

Original languageEnglish (US)
Pages (from-to)613-648
Number of pages36
JournalCommunications on Pure and Applied Mathematics
Issue number4
StatePublished - Apr 1 2016

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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