English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

In the past decades, the central limit theorem (CLT) has been generalized tonon-Euclidean data spaces. Some years ago, it was found that for some randomvariables on the circle, the sample Fr\'echet mean fluctuates around thepopulation mean asymptotically at a scale $n^{-\tau}$ with exponent $\tau <1/2$ with a non-normal distribution if the probability density at the antipodalpoint of the mean is $\frac{1}{2\pi}$. The author and his collaborator recentlydiscovered that $\tau = 1/6$ for some random variables on higher dimensionalspheres. In this article we show that, even more surprisingly, the phenomenonon spheres of higher dimension is qualitatively different from that on thecircle, as it depends purely on geometrical properties of the space, namely itscurvature, and not on the density at the antipodal point. This gives rise tothe new concept of geometrical smeariness. In consequence, the sphere can bedeformed, say, by removing a neighborhood of the antipodal point of the meanand gluing a flat space there, with a smooth transition piece. This yieldssmeariness on a manifold, which is diffeomorphic to Euclidean space. We give anexample family of random variables with 2-smeary mean, i.e. with $\tau = 1/6$,whose range has a hole containing the cut locus of the mean. The hole sizeexhibits a curse of dimensionality as it can increase with dimension,converging to the whole hemisphere opposite a local Fr\'echet mean. We observesmeariness in simulated landmark shapes on Kendall pre-shape space and in realdata of geomagnetic north pole positions on the two-dimensional sphere.

Geometrical Smeariness -- A new Phenomenon of Fr\'echet Means