The Seminar in Geometry and Statistics takes place monthly at the Department of Mathematics and Data Science of VUB (Building G, Sixth Floor, Room 6.60). It is flexible in terms of schedule and topics, though topics revolve around geometry and statistics.
DATE: Thursday 12 December 2024 at 14:00
SPEAKER: Susovan Pal (VUB)
TITLE: Optimal lift, a tool for statistical testing on Singular Shape Spaces
ABSTRACT: Shape spaces arise naturally in biomedical imaging and computer vision to study objects like hippocampi, often associated with disease progression. However, these spaces typically lack smooth Riemannian structures in certain regions ('singularities'), complicating statistical analysis. To address this, we lift shape-valued data to Riemannian manifolds ('preshapes'), which are more amenable to statistical methods.
In this talk, I will introduce a test for the equality of means for Euclidean and Riemannian manifold-valued data, building on concepts like Fréchet means. I will also discuss Kendall's shape spaces as examples, and optimal lifts from shapes to preshapes, sharing theoretical results on the uniqueness and regularity of these lifts. The talk concludes with statistical tests on shapes using optimal lifts and experimental findings. This work is part of an ongoing project with Stephan Huckemann, Benjamin Eltzner, Do Tran (affiliated with University of Göttingen, Max Planck Institute at Göttingen and Deutsche Bank respectively).
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DATE: Thursday 23 January 2025
SPEAKER: Eddie Aamari (École Normale Supérieure de Paris)
TITLE: A theory of stratification learning: clustering-by-dimensionality with reconstruction
ABSTRACT: Given i.i.d. random variables X_1, ..., X_n in R^D drawn from a stratified mixture \cup_k M_k of immersed C2-manifolds of different dimensions d_k with k at most K, we study the minimax estimation of the family M_k and the associated unsupervised clustering problem. We provide a constructive algorithm allowing to estimate each mixture component M_k at its optimal dimension-specific rate (log n /n)^{2/d_k} adaptively. The method is based on an ascending hierarchical co-detection of points belonging to different layers which also identifies the number of layers K, the dimensions d_k, assign each point X_i to a layer accurately, and estimate tangent spaces optimally. The results hold regardless of any reach assumption on the M_k's nor on intersection configurations M_k \cap M_{k'}. They open the way to a broad clustering framework, where each mixture component (or stratum) M_k models a cluster, emanating from a specific nonlinear correlation phenomenon leaving only d_k local degrees of freedom.
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DATE: Somewhere between 11 and 13 February 2025
SPEAKER: Christian Rose (Potsdam Universität)
TITLE: TBA
ABSTRACT: TBA
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DATE: Somewhere between 17 and 21 March 2024
SPEAKER: Mauricio Che (University of Vienna)
TITLE: TBA
ABSTRACT: TBA
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DATE: Wednesday 13 November 2024 at 14:00
SPEAKER: Claire Brécheteau (Nantes Université)
TITLE: Statistical tests for uniformity and iidness on homogeneous spaces
DATE: Wednesday 11 September 2024 at 11:00
SPEAKER: Pierre-Antoine Absil (UCLouvain)
TITLE: Feasible and Infeasible Optimization on Manifolds