Data-Driven Optimal Transport

Giulio Trigila, Esteban Tabak

Research output: Contribution to journalArticle

Abstract

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
Volume69
Issue number4
DOIs
StatePublished - Apr 1 2016

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Optimal Transport
Sample point
Data-driven
Weak Formulation
Test function
Expected Value
Phase Space
Availability
Objective function
Numerical Solution
Formulation

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Data-Driven Optimal Transport. / Trigila, Giulio; Tabak, Esteban.

In: Communications on Pure and Applied Mathematics, Vol. 69, No. 4, 01.04.2016, p. 613-648.

Research output: Contribution to journalArticle

Trigila, Giulio ; Tabak, Esteban. / Data-Driven Optimal Transport. In: Communications on Pure and Applied Mathematics. 2016 ; Vol. 69, No. 4. pp. 613-648.
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