These are some notes on the Karcher mean. I will be updating this post hopefully in the coming weeks.
In a Euclidean space, for a set of k points, x_1, x_2 ... x_k, the sample mean is:
In a nonlinear manifold, a simple summation is no longer possible. We can, however, make an extrinsic computation by embedding the manifold in a vector space, computing the Euclidean mean and projecting the result back into the manifold. A disadvantage of this approach is that the mean computed depends on the choice of embedding.
A second possibility is an intrinsic computation, i.e., one where we use intrinsic manifold computations to compute the mean.
To compute an intrinsic mean within a manifold, M, we use the concept of the mean as the centroid of a density. This idea was put forward by Fréchet to calculate means in a Riemanniann manifold. The computation involved a minimization but the existence and uniqueness of the resulting mean could not be guaranteed (see Pennec's 1999 NSIP paper for details). Karcher's proposal that a local instead of a global mean be used (see Karcher's 1977 paper), led to a practical implementation. We shall henceforth refer to this local mean as the Karcher Mean.
(To be updated ...)
Karcher Mean references I found helpful
Ricardo Ferreira et al. have a paper entitled Newton Method for Riemannian centroid computation in naturally reductive homogeneous spaces which has implementation details such as the intrinsic manifold computations for well known manifolds such as the sphere, the special orthogonal group, SO(n), and the space of positive definite matrices.
1) M. Fréchet, "Les elements aléatoires de nature quelconque dans un espace
distancié," Annales de l'Institut Henri Poincaré, Vol. 10, (1948) pp. 215-310.
2) X. Pennec, “Probabilities and statistics on Riemannian manifolds: Basic tools for geometric measurements,” in Proc. NSIP'99, Vol. 1, (1999), pp. 194–198.
3) H. Karcher, Riemannian center of mass and mollifier smoothing. Commun. Pure and Appl. Math. 30 (1977), pp. 509–541.