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From Mahalanobis to Bregman via Monge and Kantorovich

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  • Marc Hallin

    (Université Libre de Bruxelles)

Abstract

In his celebrated 1936 paper on “the generalized distance in statistics,” P.C. Mahalanobis pioneered the idea that, when defined over a space equipped with some probability measure P, a meaningful distance should be P-specific, with data-driven empirical counterpart. The so-called Mahalanobis distance and the related Mahalanobis outlyingness achieve this objective in the case of a Gaussian P by mapping P to the spherical standard Gaussian, via a transformation based on second-order moments which appears to be an optimal transport in the Monge-Kantorovich sense. In a non-Gaussian context, though, one may feel that second-order moments are not informative enough, or inappropriate; moreover, they might not exist. We therefore propose a distance that fully takes the underlying P into account—not just its second-order features—by considering the potential that characterizes the optimal transport mapping P to the uniform over the unit ball, along with a symmetrized version of the corresponding Bregman divergence.

Suggested Citation

  • Marc Hallin, 2018. "From Mahalanobis to Bregman via Monge and Kantorovich," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 80(1), pages 135-146, December.
    • Handle: RePEc:spr:sankhb:v:80:y:2018:i:1:d:10.1007_s13571-018-0163-4
    DOI: 10.1007/s13571-018-0163-4
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    References listed on IDEAS

    as
    1. Victor Chernozhukov & Alfred Galichon & Marc Hallin & Marc Henry, 2014. "Monge-Kantorovich Depth, Quantiles, Ranks, and Signs," Papers 1412.8434, arXiv.org, revised Sep 2015.
    2. Alfred Galichon, 2016. "Optimal Transport Methods in Economics," Economics Books, Princeton University Press, edition 1, number 10870.
    3. Martin Burger, 2016. "Bregman Distances in Inverse Problems and Partial Differential Equations," Springer Optimization and Its Applications, in: Jean-Baptiste Hiriart-Urruty & Adam Korytowski & Helmut Maurer & Maciej Szymkat (ed.), Advances in Mathematical Modeling, Optimization and Optimal Control, pages 3-33, Springer.
    4. Alfred Galichon, 2016. "Optimal transport methods in economics," Post-Print hal-03256830, HAL.
    5. repec:hal:spmain:info:hdl:2441/64itsev5509q8aa5mrbhi0g0b6 is not listed on IDEAS
    6. repec:hal:spmain:info:hdl:2441/4c5431jp6o888pdrcs0fuirl40 is not listed on IDEAS
    7. Eustasio Del Barrio & Juan Cuesta Albertos & Marc Hallin & Carlos Matran, 2018. "Smooth Cyclically Monotone Interpolation and Empirical Center-Outward Distribution Functions," Working Papers ECARES 2018-15, ULB -- Universite Libre de Bruxelles.
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