  Computing optimal transport (OT) between measures in high dimensions is
doomed by the curse of dimensionality. A popular approach to avoid this curse
is to project input measures on lower-dimensional subspaces (1D lines in the
case of sliced Wasserstein distances), solve the OT problem between these
reduced measures, and settle for the Wasserstein distance between these
reductions, rather than that between the original measures. This approach is
however difficult to extend to the case in which one wants to compute an OT map
(a Monge map) between the original measures. Since computations are carried out
on lower-dimensional projections, classical map estimation techniques can only
produce maps operating in these reduced dimensions. We propose in this work two
methods to extrapolate, from an transport map that is optimal on a subspace,
one that is nearly optimal in the entire space. We prove that the best optimal
transport plan that takes such "subspace detours" is a generalization of the
Knothe-Rosenblatt transport. We show that these plans can be explicitly
formulated when comparing Gaussian measures (between which the Wasserstein
distance is commonly referred to as the Bures or Fr\'echet distance). We
provide an algorithm to select optimal subspaces given pairs of Gaussian
measures, and study scenarios in which that mediating subspace can be selected
using prior information. We consider applications to semantic mediation between
elliptic word embeddings and domain adaptation with Gaussian mixture models.
