Tangent space projection

At any reference point (usually the Fréchet mean of the training covariances) the manifold can be locally approximated by its tangent space, a Euclidean vector space of dimension C(C+1)/2. A covariance P_i is mapped to a tangent vector by the logarithmic map at reference G: S_i = log_G(P_i) = G^{1/2} log(G^{-1/2} P_i G^{-1/2}) G^{1/2}, then vectorized by half-vectorization with sqrt(2) weighting of off-diagonal entries to preserve the inner product, giving a feature vector.

This flattening is the workhorse of practical Riemannian BCI: once in the tangent space you can use any standard Euclidean classifier (LDA, logistic regression, SVM) while still having respected the manifold's geometry in the mapping. The approximation is exact only at the reference point and degrades for covariances far from it, so choosing a good reference (the geometric mean) and keeping data reasonably concentrated matters.