Estimating the reach of a manifold
Estimating the reach of a manifold
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DOI：
10.1214/19-ejs1551
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发表时间：
2019-01-01
影响因子：
1.1
通讯作者：
Wasserman, Larry
中科院分区：
文献类型：
--
作者：
Aamari, Eddie;Kim, Jisu;Wasserman, Larry
流形估计中的各种问题都使用了一个称为到达的量，用tau（M）表示，它是流形正则性的度量。本文是首次探讨如何估计可达性的问题。首先，我们通过近似的角度来研究到达的几何形状。我们得到了关于无边界子流形可达性的新的几何结果。在已知切空间的oracle框架下，提出了一种τ（M）的帽上的估计量（τ），并给出了估计量的有效界.在i.i.d.的情况下。随机点云X-n，（τ）在cap-（X-n）上实现了类C-3模型的均匀期望损失界。最后，我们得到的上界和下界的极大极小率估计的范围。
Various problems in manifold estimation make use of a quantity called the reach, denoted by tau(M), which is a measure of the regularity of the manifold. This paper is the first investigation into the problem of how to estimate the reach. First, we study the geometry of the reach through an approximation perspective. We derive new geometric results on the reach for submanifolds without boundary. An estimator (tau) over cap of tau(M) is proposed in an oracle framework where tangent spaces are known, and bounds assessing its efficiency are derived. In the case of i.i.d. random point cloud X-n, (tau) over cap-(X-n) is showed to achieve uniform expected loss bounds over a C-3-like model. Finally, we obtain upper and lower bounds on the minimax rate for estimating the reach.