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Monotone Measure-Preserving Maps in Hilbert Spaces: Existence, Uniqueness, and Stability

Author

Listed:
  • Alberto González-Sanz
  • Marc Hallin
  • Bodhisattva Sen

Abstract

The contribution of this work is twofold. The first part deals with a Hilbert-space version of McCann’s celebrated result on the existence and uniqueness of monotone measure-preserving maps: given two probability measures P and Q on a separable Hilbert space H where P doesnot give mass to “small sets” (namely, Lipschitz hypersurfaces), we show, without imposing any moment assumptions, that there exists a gradient of convex function ∇ψ pushing P forward to Q. In case H is infinite-dimensional, P-a.s. uniqueness is not guaranteed, though. If, however, Q is boundedly supported (a natural assumption in several statistical applications), then this gradient is P-a.s. unique. In the second part of the paper, we establish stability results for the same transport maps in the sense of uniform convergence over compact “regularity sets”. As a consequence, we obtain a central limit theorem for the fluctuations of the optimal quadratic transport cost in a separable Hilbert space.

Suggested Citation

  • Alberto González-Sanz & Marc Hallin & Bodhisattva Sen, 2023. "Monotone Measure-Preserving Maps in Hilbert Spaces: Existence, Uniqueness, and Stability," Working Papers ECARES 2023-10, ULB -- Universite Libre de Bruxelles.
    • Handle: RePEc:eca:wpaper:2013/358740
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    References listed on IDEAS

    as
    1. Eustasio del Barrio & Alberto González-Sanz & Marc Hallin, 2022. "Nonparametric Multiple-Output Center-Outward Quantile Regression," Working Papers ECARES 2022-10, ULB -- Universite Libre de Bruxelles.
    2. repec:hal:spmain:info:hdl:2441/64itsev5509q8aa5mrbhi0g0b6 is not listed on IDEAS
    3. Victor Chernozhukov & Alfred Galichon & Marc Hallin & Marc Henry, 2014. "Monge-Kantorovich Depth, Quantiles, Ranks, and Signs," Papers 1412.8434, arXiv.org, revised Sep 2015.
    4. Victor Chernozhukov & Alfred Galichon & Marc Hallin & Marc Henry, 2014. "Monge-Kantorovich Depth, Quantiles, Ranks, and Signs," Papers 1412.8434, arXiv.org, revised Sep 2015.
    5. del Barrio, Eustasio & González-Sanz, Alberto & Hallin, Marc, 2020. "A note on the regularity of optimal-transport-based center-outward distribution and quantile functions," Journal of Multivariate Analysis, Elsevier, vol. 180(C).
    6. Nabarun Deb & Bodhisattva Sen, 2023. "Multivariate Rank-Based Distribution-Free Nonparametric Testing Using Measure Transportation," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 118(541), pages 192-207, January.
    Full references (including those not matched with items on IDEAS)

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