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Discrete Wasserstein barycenters: optimal transport for discrete data

Author

Listed:
  • Ethan Anderes

    (University of California Davis)

  • Steffen Borgwardt

    (Technische Universität München)

  • Jacob Miller

    (University of California Davis)

Abstract

Wasserstein barycenters correspond to optimal solutions of transportation problems for several marginals, and as such have a wide range of applications ranging from economics to statistics and computer science. When the marginal probability measures are absolutely continuous (or vanish on small sets) the theory of Wasserstein barycenters is well-developed [see the seminal paper (Agueh and Carlier in SIAM J Math Anal 43(2):904–924, 2011)]. However, exact continuous computation of Wasserstein barycenters in this setting is tractable in only a small number of specialized cases. Moreover, in many applications data is given as a set of probability measures with finite support. In this paper, we develop theoretical results for Wasserstein barycenters in this discrete setting. Our results rely heavily on polyhedral theory which is possible due to the discrete structure of the marginals. The results closely mirror those in the continuous case with a few exceptions. In this discrete setting we establish that Wasserstein barycenters must also be discrete measures and there is always a barycenter which is provably sparse. Moreover, for each Wasserstein barycenter there exists a non-mass-splitting optimal transport to each of the discrete marginals. Such non-mass-splitting transports do not generally exist between two discrete measures unless special mass balance conditions hold. This makes Wasserstein barycenters in this discrete setting special in this regard. We illustrate the results of our discrete barycenter theory with a proof-of-concept computation for a hypothetical transportation problem with multiple marginals: distributing a fixed set of goods when the demand can take on different distributional shapes characterized by the discrete marginal distributions. A Wasserstein barycenter, in this case, represents an optimal distribution of inventory facilities which minimize the squared distance/transportation cost totaled over all demands.

Suggested Citation

  • Ethan Anderes & Steffen Borgwardt & Jacob Miller, 2016. "Discrete Wasserstein barycenters: optimal transport for discrete data," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 84(2), pages 389-409, October.
  • Handle: RePEc:spr:mathme:v:84:y:2016:i:2:d:10.1007_s00186-016-0549-x
    DOI: 10.1007/s00186-016-0549-x
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    References listed on IDEAS

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    1. Pierre-André Chiappori & Robert McCann & Lars Nesheim, 2010. "Hedonic price equilibria, stable matching, and optimal transport: equivalence, topology, and uniqueness," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 42(2), pages 317-354, February.
    2. Alfred Galichon & Pierre Henri-Labordère & Nizar Touzi, 2013. "A stochastic control approach to No-Arbitrage bounds given marginals, with an application to Lookback options," Sciences Po publications info:hdl:2441/5rkqqmvrn4t, Sciences Po.
    3. Mathias Beiglbock & Pierre Henry-Labord`ere & Friedrich Penkner, 2011. "Model-independent Bounds for Option Prices: A Mass Transport Approach," Papers 1106.5929, arXiv.org, revised Feb 2013.
    4. Mathias Beiglböck & Pierre Henry-Labordère & Friedrich Penkner, 2013. "Model-independent bounds for option prices—a mass transport approach," Finance and Stochastics, Springer, vol. 17(3), pages 477-501, July.
    5. G. Carlier & I. Ekeland, 2010. "Matching for teams," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 42(2), pages 397-418, February.
    6. Miles Lubin & Iain Dunning, 2015. "Computing in Operations Research Using Julia," INFORMS Journal on Computing, INFORMS, vol. 27(2), pages 238-248, May.
    7. repec:dau:papers:123456789/6728 is not listed on IDEAS
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    Cited by:

    1. Steffen Borgwardt & Felix Happach, 2019. "Good Clusterings Have Large Volume," Operations Research, INFORMS, vol. 67(1), pages 215-231, January.
    2. Puccetti, Giovanni & Rüschendorf, Ludger & Vanduffel, Steven, 2020. "On the computation of Wasserstein barycenters," Journal of Multivariate Analysis, Elsevier, vol. 176(C).
    3. Steffen Borgwardt, 2022. "An LP-based, strongly-polynomial 2-approximation algorithm for sparse Wasserstein barycenters," Operational Research, Springer, vol. 22(2), pages 1511-1551, April.
    4. Johannes von Lindheim, 2023. "Simple approximative algorithms for free-support Wasserstein barycenters," Computational Optimization and Applications, Springer, vol. 85(1), pages 213-246, May.
    5. Steffen Borgwardt & Stephan Patterson, 2021. "On the computational complexity of finding a sparse Wasserstein barycenter," Journal of Combinatorial Optimization, Springer, vol. 41(3), pages 736-761, April.

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