  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.
