  Classical regression models do not cover non-Euclidean data that reside in a
general metric space, while the current literature on non-Euclidean regression
by and large has focused on scenarios where either predictors or responses are
random objects, i.e., non-Euclidean, but not both. In this paper we propose
geodesic optimal transport regression models for the case where both predictors
and responses lie in a common geodesic metric space and predictors may include
not only one but also several random objects. This provides an extension of
classical multiple regression to the case where both predictors and responses
reside in non-Euclidean metric spaces, a scenario that has not been considered
before. It is based on the concept of optimal geodesic transports, which we
define as an extension of the notion of optimal transports in distribution
spaces to more general geodesic metric spaces, where we characterize optimal
transports as transports along geodesics. The proposed regression models cover
the relation between non-Euclidean responses and vectors of non-Euclidean
predictors in many spaces of practical statistical interest. These include
one-dimensional distributions viewed as elements of the 2-Wasserstein space and
multidimensional distributions with the Fisher-Rao metric that are represented
as data on the Hilbert sphere. Also included are data on finite-dimensional
Riemannian manifolds, with an emphasis on spheres, covering directional and
compositional data, as well as data that consist of symmetric positive definite
matrices. We illustrate the utility of geodesic optimal transport regression
with data on summer temperature distributions and human mortality.
