  In this paper, we address the problem of Domain Adaptation (DA) using Optimal
Transport (OT) on Riemannian manifolds. We model the difference between two
domains by a diffeomorphism and use the polar factorization theorem to claim
that OT is indeed optimal for DA in a well-defined sense, up to a volume
preserving map. We then focus on the manifold of Symmetric and
Positive-Definite (SPD) matrices, whose structure provided a useful context in
recent applications. We demonstrate the polar factorization theorem on this
manifold. Due to the uniqueness of the weighted Riemannian mean, and by
exploiting existing regularized OT algorithms, we formulate a simple algorithm
that maps the source domain to the target domain. We test our algorithm on two
Brain-Computer Interface (BCI) data sets and observe state of the art
performance.
