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Empirical optimal transport under estimated costs: Distributional limits and statistical applications

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  • Hundrieser, Shayan
  • Mordant, Gilles
  • Weitkamp, Christoph A.
  • Munk, Axel

Abstract

Optimal transport (OT) based data analysis is often faced with the issue that the underlying cost function is (partially) unknown. This is addressed in this paper with the derivation of distributional limits for the empirical OT value when the cost function and the measures are estimated from data. For statistical inference purposes, but also from the viewpoint of a stability analysis, understanding the fluctuation of such quantities is paramount. Our results find direct application in the problem of goodness-of-fit testing for group families, in machine learning applications where invariant transport costs arise, in the problem of estimating the distance between mixtures of distributions, and for the analysis of empirical sliced OT quantities.

Suggested Citation

  • Hundrieser, Shayan & Mordant, Gilles & Weitkamp, Christoph A. & Munk, Axel, 2024. "Empirical optimal transport under estimated costs: Distributional limits and statistical applications," Stochastic Processes and their Applications, Elsevier, vol. 178(C).
    • Handle: RePEc:eee:spapps:v:178:y:2024:i:c:s0304414924001686
    DOI: 10.1016/j.spa.2024.104462
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    References listed on IDEAS

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    1. Marc Hallin & Gilles Mordant & Johan Segers, 2020. "Multivariate Goodness-of-Fit Tests Based on Wasserstein Distance," Working Papers ECARES 2020-06, ULB -- Universite Libre de Bruxelles.
    2. Mordant, Gilles & Segers, Johan, 2022. "Measuring dependence between random vectors via optimal transport," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
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    9. Christoph Alexander Weitkamp & Katharina Proksch & Carla Tameling & Axel Munk, 2024. "Distribution of Distances based Object Matching: Asymptotic Inference," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 119(545), pages 538-551, January.
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