This talk will try to tour the geometric inference field by motivating and presenting the optimal rates for the Hausdorff estimation of 𝑑-dimensional manifolds 𝑀 in high ambient dimension. We will focus on the influence of the possible presence of a boundary 𝜕𝑀, on noise, on different mixed dimensions, and on computational constraints.
The studied geometric class of target manifolds will unite and extend the most prevalent C²-type models: manifolds without boundary, and full-dimensional (convex) domains. A Voronoi-based procedure that allows to identify points close to 𝜕𝑀 will be presented. When data exhibits mixed dimensions, we will adress the problem of clustering the different components. If time permits, the noise and computational constraints will be addressed jointly through the statistical query (SQ) framework, which consists in replacing the usual access to samples from a distribution by the access to adversarially perturbed expected values of functions interactively chosen by the learner.
