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From Mahalanobis to Bregman via Monge and Kantorovich towards a “General Generalised Distance”

Marc Hallin ()

No 2018-12, Working Papers ECARES from ULB -- Universite 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 corresponding 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.

Keywords: Bregman divergence; gradient of convex function; Mahalanobis distance; measure transportation; McCann theorem; Monge-Kantorovich problem; multivariate distribution function; multivariate quantiles; outlyingness (search for similar items in EconPapers)
New Economics Papers: this item is included in nep-ecm
Date: 2018-05
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