Mini-Workshop: Asymptotic Statistics on Stratified Spaces

Abstract

Statistical analysis of non-Euclidean data such as data on manifolds is an active and established topic of research, for instance, in the statistical analysis of shape. However, many types of data naturally reside in metric spaces which are not smooth manifolds as a whole, rather they are unions of manifold strata of varying dimensions. These spaces form a key general family of geometric spaces for data analysis. Statistics in stratified spaces has recently found great interest in applications and mathematical theory building. While the fundamental theory is still in its beginnings, as a centerpiece the derivation and investigation of statistics and their asymptotics has materialized. Only a few basic results are known, but it is clear that the geometric constraints imposed by stratified spaces lead to unexpected asymptotic behavior of standard statistical properties, such as “stickiness” of means, see [4]. It is the scope of the proposed workshop to better understand fundamental relations between asymptotic behavior of statistical descriptors and global as well as local geometric and topological structures. This investigation calls for an intense collaboration of the fields involved: statistics & stochastics; geometry & topology; combinatorics, algorithms & numerics. This workshop sought to bring together world-leading scientists and high-potential early career researchers working in this field to collaborate on a focused set of fundamental questions.