Parallel transport

Parallel transport moves a tangent vector (or a whole tangent-space model) from the tangent space at one reference point to that at another while respecting the manifold's connection, so that geometric relationships are preserved along the transporting geodesic. On the SPD manifold with the affine-invariant metric it has a closed form: transporting from reference P to Q uses E = (Q P^{-1})^{1/2}, mapping a tangent vector S to E S E^T.

In BCI it is the principled way to carry a classifier trained in one session's or subject's tangent space (centered at that domain's geometric mean) to another domain's tangent space (centered at its own mean), rather than assuming the two tangent spaces are the same. It underpins several transfer-learning methods that combine recentering with transport of features or decision boundaries, letting a model trained on source subjects apply to a target with little or no target calibration.