# Center-outward quantiles and the measurement of multivariate risk

arXiv: https://arxiv.org/abs/1912.04924 · HTML: https://mlsanity.com/paper/1912.04924/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Dec 2019.

## Built on by (most cited first, 4 of 4)

- [Transport-based Counterfactual Models](https://mlsanity.com/paper/2108.13025/) (2021)
- [GAN Estimation of Lipschitz Optimal Transport Maps](https://mlsanity.com/paper/2202.07965/) (2022)
- [Stochastic optimal transport in Banach Spaces for regularized estimation of multivariate quantiles](https://mlsanity.com/paper/2302.00982/) (2023)
- [What is a good matching of probability measures? A counterfactual lens on transport maps](https://mlsanity.com/paper/2509.16027/) (2025)

## Abstract (arXiv)

All multivariate extensions of the univariate theory of risk measurement run into the same fundamental problem of the absence, in dimension d > 1, of a canonical ordering of Rd. Based on measure transportation ideas, several attempts have been made recently in the statistical literature to overcome that conceptual difficulty. In Hallin (2017), the concepts of center-outward distribution and quantile functions are developed as generalisations of the classical univariate concepts of distribution and quantile functions, along with their empirical versions. We propose a class of smooth approximations as an alternative to the interpolation developed in del Barrio et al. (2018). This approximation allows for the computation of some new empirical risk measures, based either on the convex potential associated with the proposed transports, or on the volumes of the resulting empirical quantile regions. We also discuss the role of such transports in the evaluation of the risk associated with multivariate regularly varying distributions. Some simulations and applications to case studies illustrate the value of the approach.