Minimax Boundary Estimation and Estimation with Boundary

In this talk, we study the non-asymptotic minimax rates for the Hausdorff estimation of 𝑑-dimensional manifolds 𝑀 with (possibly) non-empty boundary 𝜕𝑀. The class of target sets that we consider reunites and extends the most prevalent C²-type models: manifolds without boundary, and full-dimensional domains. We will consider both the estimation of the manifold 𝑀 itself and that of its boundary 𝜕𝑀 if non-empty. In the process, we will present a Voronoi-based procedure that allows to identify enough points close to 𝜕𝑀 for reconstructing it. Explicit constant derivations are given, showing that these rates do not depend on the ambient dimension. If time permits, we will talk about possible extensions of the estimation procedure to smoother manifolds with corners.

Joint work with Catherine Aaron & Clément Levrard