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:我们为Hausdorf 估算的美元值得出了非非表面的迷你界限。 美元值为M = subset\ mathbb{R<unk> D$, 与( 可能) 非空边界为$ 美元。 模型集合并扩展了最常用的 $mathcal{C<unk> 2 型估算模型: 没有边界的多元值, 和全维域。 我们既考虑对元值本身的估计, 也考虑对美元值为美元( 如果非空的话) 的边界值的估计 。 根据美元样本, 如果 $\\ bigl( log n/n)\\\\\\ d<unk> bigr$( =\ spregyset $) 和 $O\ bigl( log n/n) 2/ (d+1) <unk> bigr$。 我们开发了一个基于Voronioioi \ $\\\\\\\\\ mag_\\ 美元( rig_) 美元/ m_\\\ big1) 足够重点( 美元) 。</s>