On the measure of the cut locus of a Fréchet mean

Abstract

One of the fundamental differences between the Central Limit Theorem for empirical Fréchet means obtained in [Kendall and Le, 'Limit theorems for empirical Fréchet means of independent and non-identically distributed manifold-valued random variables ', Braz. J. Probab. Stat. 25 (2011) 323-352] and that for empirical Euclidean means lies on the assumption that the probability measure of the cut locus of the true Fréchet mean is zero. In [Hotz and Huckemann, 'Intrinsic means on the circle: uniqueness, locus and asymptotics', Preprint, 2011, arXiv:1108.2141v1], the authors show that, in the case of a circle, this assumption holds automatically. This paper shows that this holds for any complete and connected Riemannian manifold, assuming that there are at least two minimal geodesics between the Fréchet mean and any point in its cut locus and that, in fact, it can also be generalized to local minima of the p-energy function of a finite measure.

Regression Models on Riemannian Symmetric Spaces

“…In this case, Log μ ( x i , θ ) ( y i ) is not uniquely defined, whereas their geodesic distance is unique. However, according to the results in Le and Barden (2014), Log μ ( x i , q , β ) ( y i ) is well defined almost surely.…”

“…However, according to a result in Le and Barden (2014), Log μ ( x i , q , β ) ( y i ) is well defined under some very mild conditions, which require that ∫ ℳ dist ℳ (p, y i ) 2 dp ( y i ) be finite and achieve a local minimum at μ ( x i , q, β ), where p ( y i ) is any finite measure of y i on ℳ. Thus, Log μ ( x i , q , β ) ( y i ) makes it a good candidate to play the role of a ‘residual’.…”

A diffusion process associated with Fréchet means

“…with the initial condition that V ε 0 has the same distribution asξ ε 0 and where V t satisfies the stochastic differential equation (9). This implies (cf.…”

“…In contrast to Euclidean means, there is generally no closed form for Fréchet means. On the other hand, the result of [9] implies that the Euclidean random variable exp −1 µ (X) is almost surely defined, where µ is a Fréchet mean of the random variable X on M. Then, since exp −1 x (y) = − 1 2 grad 1 (ρ(x, y) 2 ), where grad 1 denotes the gradient operator acting on the first argument of a function on M × M and since…”

“…However, we do not require the support of the probability measure of X 1 to be contained in any 4 H. LE geodesic ball. Note that the result of [9] ensures that P(X 1 ∈ C µ ) = 0 under some mild condition on M. We further assume that…”