Sample-Based Optimal Transport and Barycenter Problems

Max Kuang, Esteban Tabak

Research output: Contribution to journalArticle

Abstract

A methodology is developed for the numerical solution to the sample-based optimal transport and Wasserstein barycenter problems. The procedure is based on a characterization of the barycenter and of the McCann interpolants that permits the decomposition of the global problem under consideration into various local problems where the distance among successive distributions is small. These local problems can be formulated in terms of feature functions and shown to have a unique minimizer that solves a nonlinear system of equations. Both the theoretical underpinnings of the methodology and its practical implementation are developed, and illustrated with synthetic and real data sets.

Original languageEnglish (US)
Pages (from-to)1581-1630
Number of pages50
JournalCommunications on Pure and Applied Mathematics
Volume72
Issue number8
DOIs
StatePublished - Aug 1 2019

Fingerprint

Optimal Transport
Barycentre
Nonlinear systems
Decomposition
Nonlinear Systems of Equations
Methodology
Interpolants
Minimizer
Numerical Solution
Decompose

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Sample-Based Optimal Transport and Barycenter Problems. / Kuang, Max; Tabak, Esteban.

In: Communications on Pure and Applied Mathematics, Vol. 72, No. 8, 01.08.2019, p. 1581-1630.

Research output: Contribution to journalArticle

@article{e3bccb14f28f4f32a856283d03ccef67,
title = "Sample-Based Optimal Transport and Barycenter Problems",
abstract = "A methodology is developed for the numerical solution to the sample-based optimal transport and Wasserstein barycenter problems. The procedure is based on a characterization of the barycenter and of the McCann interpolants that permits the decomposition of the global problem under consideration into various local problems where the distance among successive distributions is small. These local problems can be formulated in terms of feature functions and shown to have a unique minimizer that solves a nonlinear system of equations. Both the theoretical underpinnings of the methodology and its practical implementation are developed, and illustrated with synthetic and real data sets.",
author = "Max Kuang and Esteban Tabak",
year = "2019",
month = "8",
day = "1",
doi = "10.1002/cpa.21848",
language = "English (US)",
volume = "72",
pages = "1581--1630",
journal = "Communications on Pure and Applied Mathematics",
issn = "0010-3640",
publisher = "Wiley-Liss Inc.",
number = "8",

}

TY - JOUR

T1 - Sample-Based Optimal Transport and Barycenter Problems

AU - Kuang, Max

AU - Tabak, Esteban

PY - 2019/8/1

Y1 - 2019/8/1

N2 - A methodology is developed for the numerical solution to the sample-based optimal transport and Wasserstein barycenter problems. The procedure is based on a characterization of the barycenter and of the McCann interpolants that permits the decomposition of the global problem under consideration into various local problems where the distance among successive distributions is small. These local problems can be formulated in terms of feature functions and shown to have a unique minimizer that solves a nonlinear system of equations. Both the theoretical underpinnings of the methodology and its practical implementation are developed, and illustrated with synthetic and real data sets.

AB - A methodology is developed for the numerical solution to the sample-based optimal transport and Wasserstein barycenter problems. The procedure is based on a characterization of the barycenter and of the McCann interpolants that permits the decomposition of the global problem under consideration into various local problems where the distance among successive distributions is small. These local problems can be formulated in terms of feature functions and shown to have a unique minimizer that solves a nonlinear system of equations. Both the theoretical underpinnings of the methodology and its practical implementation are developed, and illustrated with synthetic and real data sets.

UR - http://www.scopus.com/inward/record.url?scp=85067505378&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85067505378&partnerID=8YFLogxK

U2 - 10.1002/cpa.21848

DO - 10.1002/cpa.21848

M3 - Article

VL - 72

SP - 1581

EP - 1630

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 8

ER -