DATE: Wednesday 13 November 2024 at 14:00

SPEAKER: Claire Brécheteau (Nantes Université)

TITLE: Statistical tests for uniformity and iidness on homogeneous spaces

ABSTRACT: I will introduce two families of statistical tests aiming at testing uniformity of samples of data points on homogeneous compact Polish spaces. Such tests are based on the computation of nearest neighbours distances, as in [1]. Such tests are consistent and come with separation rates. I will show numerical results on the flat torus, the circle, the sphere, and a subset of the Poincaré disk. In particular, the tests will be compared to classical tests on the sphere and the circle [2].

[1] Brécheteau, A statistical test of isomorphism between metric-measure spaces using the distance-to-a-measure signature, 2019

[2] Garciá-Portugués Verdebout, An overview of uniformity tests on the hypersphere, 2018

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DATE: December

SPEAKER: TBA

TITLE: TBA

ABSTRACT: TBA

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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.

DATE: Wednesday 11 September 2024 at 11:00

SPEAKER: Pierre-Antoine Absil (UCLouvain)

TITLE: Feasible and Infeasible Optimization on Manifolds