A smeary central limit theorem for manifolds with application to high-dimensional spheres
By Benjamin Eltzner, Stephan F. Huckemann. 2018
58 citations recorded by the source corpus.
The (CLT) central limit theorems for generalized Frechet means (data descriptors assuming values in stratified spaces, such as intrinsic means, geodesics, etc.) on manifolds from the literature are only valid if a certain empirical process of Hessians of the Frechet function converges suitably, as in the proof of the prototypical BP-CLT (Bhattacharya and Patrangenaru (2005)). This is not valid in many realistic scenarios and we provide for a new very general CLT. In particular this includes scenarios where, in a suitable chart, the sample mean fluctuates asymptotically at a scale $n^{\alpha}$ with exponents ${\alpha} < 1/2$ with a non-normal distribution. As the BP-CLT yields only fluctuations that are, rescaled with $n^{1/2}$ , asymptotically normal, just as the classical CLT for random vectors, these lower rates, somewhat loosely called smeariness, had to date been observed only on the circle (Hotz and Huckemann (2015)). We make the concept of smeariness on manifolds precise, give an example for two-smeariness on spheres of arbitrary dimension, and show that smeariness, although "almost never" occurring, may have serious statistical implications on a continuum of sample scenarios nearby. In fact, this effect increases with dimension, striking in particular in high dimension low sample size scenarios.
Subfields: Applied Mathematics, Artificial Intelligence, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/00e4dbedb5d19f41e3bd972ce61d25aa27ff0297
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In Applied Mathematics
In Artificial Intelligence
In Geometry and Topology
Citations in the corpus
- Convergence rates for the generalized Fréchet mean via the quadruple inequality (2018, background): “enter. In the context of Riemannian manifolds, it has been studied – among others – by [BP03]. An asymptotic normality result for generalized Fr´echet means on finite dimensional manifolds is shown in [EH18]. For complete metric spaces of nonpositive curvature, called Hadamard spaces, [Stu03] shows h”