Fréchet (Karcher) mean

The Fréchet mean generalizes averaging to a manifold: it is the point minimizing the sum of squared geodesic distances to the data, argmin_G sum_i delta(G, P_i)^2. On the SPD manifold with the affine-invariant metric this geometric mean has no closed form for more than two matrices and is computed by an iterative gradient or fixed-point algorithm (Karcher flow), which converges because the squared-distance function is geodesically convex; the two-matrix case does have the closed form P^{1/2}(P^{-1/2}QP^{-1/2})^{1/2}P^{1/2}.

The geometric mean is the reference point for tangent-space projection, the class prototype in MDM, and the domain center in Riemannian alignment, so it is the single most-used computation in Riemannian BCI. Different metrics give different means: the log-Euclidean mean has the closed form exp of the mean of the logs, which is much faster but lacks full affine invariance.