Multidimensional distribution-to-distribution regression via optimal transport maps

The distribution-to-distribution regression models have received considerable attention over the past years. The existing studies on distribution-to-distribution regression models mainly focus on univariate distributions, and it is quite challenging to extend the existing methods to multivariate settings due to no analytical solution for the optimal transport problem and complicated theories involved. We propose a novel method to make inference on multidimensional distribution-to-distribution regression with both covariates and response variables following multidimensional probability distributions based on the theory of optimal transportation. The considered model links the conditional Fréchet mean of response variable to the covariates via the optimal transport map, which is taken as the gradient of the input convex neural network (ICNN) trained with the minimax optimization. We develop the Fréchet-least-square (FLS) estimator of the regression map by learning the optimal Kantorovich potential, and investigate the identifiability, consistency and convergence rate of the FLS estimator based on the theory of empirical process. We also construct a case-deletion diagnostic measure to identify influential observations. Simulation studies and a real example are used to illustrate the proposed methodologies.