Verschmierte Grenzwertsätze
Zusammenfassung der Projektergebnisse
In spite of limited funding and the pandemic hitting, with the new general CLT for Fréchet means on singular spaces (formulation and proof require approx. 100 pages or, equivalently, four papers), major progress has been achieved. It is the nature of this CLT that it has direct application for stickiness and this has spurred a separate investigation of various flavors (types) of stickiness and their relationship with respect to geometric properties of underlying spaces. In particular it led to the insight that, e.g. Hadamard manifolds cannot feature stickiness. Although not addressed in the project, we conjecture, that they can feature finite sample stickiness, however, making quantile based tests inappropriate. In fact, using the variance modulation’s sticky flavor allows for exact definition of finite sample stickiness and according statistical methods. Further investigation (e.g. discriminating different types as is the case with FSS) which may have direct impact on statistics of phylogenetic trees is left for further research, in particular exploiting stickiness as a blessing to identify hybridisation effects in a nascent collaboration with the Deutsche Primatenzentrum in Göttingen. Also on the side of smeariness – the original avenue of the project – major progress could be achieved. On the one hand, the variance modulation allowed not only for a precise definition of FSS – not available at the start of the project – but also for its statistical assessment. The connection between smeariness (begetting FSS) and compact manifolds is now much clearer, and so is smeariness as a phase transition between uniqueness and nonuniqueness, but, by no means, fully understood yet: For instance we have not been able to produce smeariness of arbitrary polynomial degree in spheres, let alone on general compact manifolds – although we conjecture even log-smeariness can be found there. Notably, log-smeariness has been recently established by Kroshnin (2021) on Wasserstein spaces. Also a precise formulation of an asymptotic measure on population Fréchet mean sets induced by sampling, and its consequences, e.g. for statistical testing, has not yet been addressed in the project and is left for future research. One major obstacle we frequently encountered is the very restrictive condition for the support of a probability distribution within a geodesic half ball in CAT(κ) spaces for κ > 0 to ensure uniqueness, cf. Afsari (2011). We expect that combining the many results of this project with a weakening of this restriction, e.g. in the spirit of Arnaudon and Miclo (2014) (the sample mean from a distribution with density w.r.t. Riemannian volume is a.s. unique) will lead to much stronger results; for instance that no Riemannian manifold may feature stickiness – thus far we could only show it for distributions restricted to geodesic half balls. Besides addressing the specific research goals, a number of overview articles could be published. Linking geometry and statistics with direct impact on applications in non-Euclidean statistics, we were able to answer a number of fundamental questions, a considerable number of questions remain open and many more new questions arose.
Projektbezogene Publikationen (Auswahl)
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Finite Sample Smeariness on Spheres. Lecture Notes in Computer Science, 12-19. Springer International Publishing.
Eltzner, Benjamin; Hundrieser, Shayan & Huckemann, Stephan F.
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Smeariness Begets Finite Sample Smeariness. Lecture Notes in Computer Science, 29-36. Springer International Publishing.
Tran Van, Do ; Eltzner, Benjamin & Huckemann, Stephan
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Stability of the cut locus and a central limit theorem for Fréchet means of Riemannian manifolds. Proceedings of the American Mathematical Society, 149(9), 3947-3963.
Eltzner, Benjamin; Galaz-García, Fernando; Huckemann, Stephan & Tuschmann, Wilderich
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Generalized asymptotic theory and uniqueness of M-estimators on manifolds. Habilitation thesis, Univ. Göttingen.
Eltzner, Benjamin
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Geometrical smeariness – a new phenomenon of Fréchet means. Bernoulli, 28(1).
Eltzner, Benjamin
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A central limit theorem for random tangent fields on stratified spaces.
Mattingly, J. C.; Miller, E. & Tran Van, Do
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Central limit theorems for Fréchet means on stratified spaces.
Mattingly, J. C.; Miller, E. & Tran Van, Do
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Diffusion means in geometric spaces. Bernoulli, 29(4).
Eltzner, Benjamin; Hansen, Pernille E.H.; Huckemann, Stephan F. & Sommer, Stefan
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Exploring Uniform Finite Sample Stickiness. Lecture Notes in Computer Science, 349-356. Springer Nature Switzerland.
Ulmer, Susanne; Tran Van, Do & Huckemann, Stephan F.
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Geometry of measures on smoothly stratified metric spaces.
Mattingly, J. C.; Miller, E. & Tran Van, Do
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Shadow geometry at singular points of CAT(κ) spaces.
Mattingly, J. C.; Miller, E. & Tran Van, Do
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Sticky flavors.
Lammers, Lars; Tran Van, Do & Huckemann, Stephan F.
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Types of Stickiness in BHV Phylogenetic Tree Spaces and Their Degree. Lecture Notes in Computer Science, 357-365. Springer Nature Switzerland.
Lammers, Lars; Tran Van, Do; Nye, Tom M. W. & Huckemann, Stephan F.
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A lower bound for estimating Fréchet means.
Hundrieser, Shayan.; Eltzner, Benjamin & Huckemann, Stephan F.
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Finite sample smeariness of Fréchet means with application to climate. Electronic Journal of Statistics, 18(2).
Hundrieser, Shayan; Eltzner, Benjamin & Huckemann, Stephan F.
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The long diffusion time limit of diffusion means on spheres and real projective spaces. In T. Shioya (Ed.), Proceedings of the HeKKSaGOn conference at RIMS and Tohoku 2023. Springer Tohoku Series in Mathematics.
Düsberg, T. & Eltzner, Benjamin
