Exploring Stickiness in CAT(κ) Spaces

by Lars Lammers
Doctoral thesis
Date of Examination:2024-09-16
Date of issue:2024-10-10
Advisor:Prof. Dr. Stephan F. Huckemann
Referee:Prof. Dr. Stephan F. Huckemann
Referee:Prof. Dr. Russell Luke
Persistent Address: http://dx.doi.org/10.53846/goediss-10795

 

 

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Abstract

English

The Fréchet mean is a natural generalization of the expectation to probability distributions on arbitrary metric spaces. In recent years, several seemingly related phenomena were observed for certain distributions on singular CAT(κ) spaces and are often collectively referred to as stickiness. Most notably, the sample Fréchet means of sticky distributions is confined to lower-dimensional subsets of the space. In addition, these distributions seem to be extraordinarily robust: the Fréchet mean remains in the lower-dimensional subset it sticks to when the distribution is perturbed by a point mass or when another distribution is considered which is sufficiently close in Wasserstein distance. Due to the restricted asymptotic variation of sample Fréchet means, stickiness limits the usefulness of sample Fréchet means for the discrimination between two distributions sticking to the same subset. Among the spaces affected by the phenomenon of stickiness are Billera-Holmes- Vogtmann (BHV) spaces of phylogenetic trees, in which stickiness manifests as the tendency of Fréchet means sticking to unresolved tree topologies. In that context, stickiness also causes computational issues. Conventional iterative methods for the computation of sample Fréchet means often fail in finding the correct tree topology when the topology is not binary. In this thesis, we introduce multiple sticky flavors capturing different phenomena commonly associated with the umbrella term of stickiness, and investigate their relationships in different scenarios, with a focus on CAT(κ) spaces and BHV-spaces, in particular. Furthermore, we develop hypothesis tests for sticky distributions on BHV spaces, and propose new methods for determining edges in Fréchet means of distributions on BHV spaces.
Keywords: Fréchet mean; CAT(κ) spaces; Stickiness; Wasserstein distance; Phylogenetic trees; Statisical discrimination
 
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