ABSTRACT
Extrinsic Mean 192
5.6 Exercises 199
5.1 Introduction
Recall that a Fre´chet mean of a probability measure Q on a complete metric
space (M,ρ) is the unique minimizer of the function F (x) = ∫ ρ2(x,y)Q(dy) (see Fre´chet (1948) [121]). A Fre´chet mean associated with a geodesic dis-
tance ρg determined by the Riemannian structure g on a complete manifold M is said to be an intrinsic mean, µg(Q). It is known that if Q is sufficiently concentrated then µg(Q) exists (see Theorem 5.4.2 (a) below). Also recall that the extrinsic mean µE (Q) = µ j,E (Q) of a probability measure Q on a complete manifoldM relative to an embedding j :M→ Rk is the Fre´chet mean associated with the restriction to j(M) of the Euclidian distance in Rk. In Chapter 4, it was shown that such an extrinsic mean of Q exists if the ordinary mean of
j(Q) is a nonfocal point of j(M), i.e., if there is a unique point x0 on j(M) having the smallest distance from the mean of j(Q); in this case µ j,E (Q) = j
DEFINITION 5.1.1. A probability distribution Q has small flat support on a
Riemannian manifoldM if supp(Q) is included in a flat geodesic ball. In general is is easier to compute an extrinsic mean, except for the case
when Q has small flat support. It may be pointed out that if Q is highly con-
centrated, as may be the case with medical imaging data (see the example in
Bhattacharya and Patrangenaru (2003) [42]), the Cartan and extrinsic means
are virtually indistinguishable.
