Vector quantile regression and optimal transport, from theory to numerics

TL;DR

This paper explores the theoretical foundations and numerical methods for vector quantile regression using optimal transport, including regularization and gradient descent algorithms, with applications to univariate and bivariate data.

Contribution

It introduces an entropic regularization approach and a gradient descent method for multivariate vector quantile regression based on optimal transport theory.

Findings

Feasibility demonstrated on univariate and bivariate examples.

Regularization improves computational efficiency.

Links between quantile regression and optimal transport are clarified.

Abstract

In this paper, we first revisit the Koenker and Bassett variational approach to (univariate) quantile regression, emphasizing its link with latent factor representations and correlation maximization problems. We then review the multivariate extension due to Carlier et al. (2016, 2017) which relates vector quantile regression to an optimal transport problem with mean independence constraints. We introduce an entropic regularization of this problem, implement a gradient descent numerical method and illustrate its feasibility on univariate and bivariate examples.