On the Computation of Wasserstein Barycenters

18 Pages Posted: 28 Nov 2018 Last revised: 11 May 2019

See all articles by Giovanni Puccetti

Giovanni Puccetti

University of Milan - Department of Economics, Management and Quantitative Methods (DEMM)

Ludger Rüschendorf

University of Freiburg

Steven Vanduffel

Vrije Universiteit Brussel (VUB)

Date Written: May 9, 2019

Abstract

The Wasserstein barycenter is an important notion in the analysis of high dimensional data with a broad range of applications in applied probability, economics, statistics, and in particular to clustering and image processing. We state a general version of the equivalence of the Wasserstein barycenter problem to the n-coupling problem. As a consequence, the coupling to the sum principle (characterizing solutions to the n-coupling problem), also provides a new criterion for the explicit characterization of barycenters. Based on this criterion, we provide as a main contribution a simple to implement algorithm for computing barycenters. We find in several examples that our approach has a similar time complexity with respect to well established algorithms and is able to provide an accurate and fast visualization of barycenters of a large number of measures. The proposed algorithm can also be applied to morecomplex optimization problems like the k-barycenter problem.

Keywords: Wasserstein barycenter, swapping algorithm, optimal transportations, k-means clustering, image processing

Suggested Citation

Puccetti, Giovanni and Rüschendorf, Ludger and Vanduffel, Steven, On the Computation of Wasserstein Barycenters (May 9, 2019). Available at SSRN: https://ssrn.com/abstract=3276147 or http://dx.doi.org/10.2139/ssrn.3276147

Giovanni Puccetti (Contact Author)

University of Milan - Department of Economics, Management and Quantitative Methods (DEMM) ( email )

Via Conservatorio, 7
Milan, 20122
Italy

Ludger Rüschendorf

University of Freiburg ( email )

Fahnenbergplatz
Freiburg, D-79085
Germany

Steven Vanduffel

Vrije Universiteit Brussel (VUB) ( email )

Pleinlaan 2
Brussels, Brabant 1050
Belgium

HOME PAGE: http://www.stevenvanduffel.com

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