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 regularity of $M$ (such as its reach), 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 $(\hat{M}_t)_{t\geq 0}$, and show that for some choice of $t$ (depending on the regularity parameters), 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 to recover the sample rate of the set of observations, and can therefore be used as an hyperparameter in other settings, such as tangent space estimation.

中文翻译：

---

流形推理中的极小极大自适应估计

我们专注于流形估计的问题：给定一组靠近某个未知子流形 $M$ 采样的观测值，我们希望恢复有关 $M$ 几何形状的信息。迄今为止提出的极大极小估计量都主要依赖于量化 $M$ 的规律性（例如其范围）的某些参数的先验知识，而这些数量在实践中是未知的。我们对这个问题的贡献是双重的：首先，我们引入了一个单参数族流形估计量 $(\hat{M}_t)_{t\geq 0}$，并表明对于 $t$ 的某些选择（取决于在正则性参数上），相应的估计量是在 [Genovese 等人，流形估计和 Hausdorff 损失下的奇异反卷积] 中介绍的 $C^2$ 流形模型类上的极小极大值。第二，我们为参数 $t$ 提出了一个完全数据驱动的选择程序，从而在此类模型上产生一个极小极大自适应流形估计器。这个选择过程实际上允许恢复观察集的采样率，因此可以用作其他设置中的超参数，例如切线空间估计。