论文标题
歧管推理中的最小自适应估计
Minimax adaptive estimation in manifold inference
论文作者
论文摘要
我们专注于多种估计的问题：给定一组观测值接近一些未知的Submanifold $ m $，一个人想恢复有关$ M $的几何信息的信息。到目前为止提出的最小估计器都至关重要地取决于对某些参数的先验知识，这些参数量化了产生样品的基本分布（例如其密度的界限），而这些数量在实践中是未知的。 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和在Hausdorff损失下进行单数反卷积]。其次，我们为参数$ t $提出了一个完全数据驱动的选择过程，从而导致了此类模型的最小自适应歧管估计器。此选择过程实际上使我们能够恢复一组观测值和$ M $之间的Hausdorff距离，因此可以用作其他设置的比例参数，例如切线空间估计。
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.
