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
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
References: View references in EconPapers View complete reference list from CitEc
Citations: Track citations by RSS feed
Downloads: (external link)
https://dipot.ulb.ac.be/dspace/bitstream/2013/270860/3/2018-12-HALLIN-from.pdf Œuvre complète ou partie de l'œuvre (application/pdf)
This item may be available elsewhere in EconPapers: Search for items with the same title.
Export reference: BibTeX
RIS (EndNote, ProCite, RefMan)
Persistent link: https://EconPapers.repec.org/RePEc:eca:wpaper:2013/270860
Ordering information: This working paper can be ordered from
http://hdl.handle.ne ... lb.ac.be:2013/270860
Access Statistics for this paper
More papers in Working Papers ECARES from ULB -- Universite Libre de Bruxelles Contact information at EDIRC.
Bibliographic data for series maintained by Benoit Pauwels ().