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 $O\bigl((\log n/n)^{2/d}\bigr)$ if $\partial M = \emptyset$ and $O\bigl((\log n/n)^{2/(d+1)}\bigr)$ if $\partial M \neq \emptyset$, up to logarithmic factors. In the process, we develop a Voronoi-based procedure that allows to identify enough points $O\bigl((\log n/n)^{2/(d+1)}\bigr)$-close to $\partial M$ for reconstructing it.

翻译：我们推导了针对（可能）具有非空边界 ∂M 的 d 维子流形 M ⊂ ℝ^D 的豪斯多夫估计的非渐近极小极大界。该模型统一并扩展了最普遍的 𝒞² 型集估计模型：无边界的流形和全维区域。我们同时考虑流形 M 本身及其边界 ∂M（若非空）的估计。给定 n 个样本，极小极大率在 ∂M = ∅ 时为 O((log n/n)^{2/d})（对数因子不计），在 ∂M ≠ ∅ 时为 O((log n/n)^{2/(d+1)})。在此过程中，我们开发了一种基于 Voronoi 的程序，能够识别足够多与 ∂M 距离为 O((log n/n)^{2/(d+1)}) 的点，以重构边界。