We focus on the problem of manifold estimation: given a set of observations sampled close to some unknown submanifold $M$, one wants to recover information about the geometry of $M$. Minimax estimators which have been proposed so far all depend crucially on the a priori knowledge of some parameters quantifying the underlying distribution generating the sample (such as bounds on its density), whereas those quantities will be unknown in practice. Our contribution to the matter is twofold: first, we introduce a one-parameter family of manifold estimators $(\hat{M}_t)_{t\geq 0}$ based on a localized version of convex hulls, and show that for some choice of $t$, the corresponding estimator is minimax on the class of models of $C^2$ manifolds introduced in [Genovese et al., Manifold estimation and singular deconvolution under Hausdorff loss]. Second, we propose a completely data-driven selection procedure for the parameter $t$, leading to a minimax adaptive manifold estimator on this class of models. This selection procedure actually allows us to recover the Hausdorff distance between the set of observations and $M$, and can therefore be used as a scale parameter in other settings, such as tangent space estimation.

翻译：我们侧重于多重估算问题:鉴于一系列抽样观察,其抽样范围接近于一些未知的亚币美元,人们希望恢复关于美元元的几何信息。迄今为止提出的迷你马克思估计值都主要取决于某些参数的先验性知识,这些参数可以量化产生样本的基本分布(例如其密度的界限),而这些数量在实践中并不为人所知。我们对此问题的贡献是双重的:首先,我们引入一个以本地版的 convex 船体为基础的多元估计值(hat{M ⁇ t) Qt\ge Q 0.} 的单数组,从而导致一个小型的调制成数,对于某些选择,则相应的估计值都取决于某些选择的美元。相应的估计值对于在[Genovese 和(Genoves al.,Manideworp 估计值和Hausdordorf 损失下的单项混和单项调 模型的类别,这个选择程序实际上使我们能够在模型中恢复其他的距离设置。