A Smeary Central Limit Theorem for Manifolds with Application to High Dimensional Spheres

TL;DR

This paper introduces a new, more general central limit theorem for manifold data, capturing non-normal fluctuations at rates below the classical 1/2 scale, especially relevant for high-dimensional spheres.

Contribution

It establishes a broad CLT for Frechet means on manifolds that includes non-normal, lower-rate fluctuations, extending beyond classical results.

Findings

Includes scenarios with non-normal fluctuations at rates less than n^{1/2}

Provides an example of two-smeariness on spheres of arbitrary dimension

Shows smeariness effects increase with dimension and impact high-dimensional, low-sample-size data

Abstract

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.