From Mahalanobis to Bregman via Monge and Kantorovich

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.