  Quantile regression (QR) is a statistical tool for distribution-free
estimation of conditional quantiles of a target variable given explanatory
features. QR is limited by the assumption that the target distribution is
univariate and defined on an Euclidean domain. Although the notion of quantiles
was recently extended to multi-variate distributions, QR for multi-variate
distributions on manifolds remains underexplored, even though many important
applications inherently involve data distributed on, e.g., spheres (climate and
geological phenomena), and tori (dihedral angles in proteins). By leveraging
optimal transport theory and c-concave functions, we meaningfully define
conditional vector quantile functions of high-dimensional variables on
manifolds (M-CVQFs). Our approach allows for quantile estimation, regression,
and computation of conditional confidence sets and likelihoods. We demonstrate
the approach's efficacy and provide insights regarding the meaning of
non-Euclidean quantiles through synthetic and real data experiments.
