Minimax boundary estimation and estimation with boundary

Abstract

We derive non-asymptotic minimax bounds for the Hausdorff estimation of d-dimensional submanifolds $M\subset {\mathbb{R}}^D$ with (possibly) non-empty boundary $\partial M$. The model reunites and extends the most prevalent ${\mathcal{C}}^2$-type set estimation models: manifolds without boundary, and full-dimensional domains. We consider both the estimation of the manifold M itself and that of its boundary $\partial M$ if non-empty. Given n samples, the minimax rates are of order $O\left({(logn∕n)}^{2∕d}\right)$ if $\partial M=\varnothing$ and $O\left({(logn∕n)}^{2∕(d+1)}\right)$ if $\partial M\ne \varnothing$, up to logarithmic factors. In the process, we develop a Voronoi-based procedure that allows to identify enough points $O\left({(logn∕n)}^{2∕(d+1)}\right)$-close to $\partial M$ for reconstructing it. Explicit constant derivations are given, showing that these rates do not depend on the ambient dimension $D\gg d$.