  We present an optimal transport framework for performing regression when both
the covariate and the response are probability distributions on a compact
Euclidean subset $\Omega\subset\mathbb{R}^d$, where $d>1$. Extending beyond
compactly supported distributions, this method also applies when both the
predictor and responses are Gaussian distributions on $\mathbb{R}^d$. Our
approach generalizes an existing transportation-based regression model to
higher dimensions. This model postulates that the conditional Fr\'echet mean of
the response distribution is linked to the covariate distribution via an
optimal transport map. We establish an upper bound for the rate of convergence
of a plug-in estimator.
  We propose an iterative algorithm for computing the estimator, which is based
on DC (Difference of Convex Functions) Programming. In the Gaussian case, the
estimator achieves a parametric rate of convergence, and the computation of the
estimator simplifies to a finite-dimensional optimization over positive
definite matrices, allowing for an efficient solution. The performance of the
estimator is demonstrated in a simulation study.
