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Estimating the Reach of a Manifold. (arXiv:1705.04565v2 [math.ST] UPDATED)
来源于:arXiv
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 $\hat{\tau}$ of $\tau\_{M}$ is
proposed in a framework where tangent spaces are known, and bounds assessing
its efficiency are derived. In the case of i.i.d. random point cloud
$\mathbb{X}\_{n}$, $\hat{\tau}(\mathbb{X}\_{n})$ is showed to achieve uniform
expected loss bounds over a $\mathcal{C}^3$-like model. Finally, we obtain
upper and lower bounds on the minimax rate for estimating the reach. 查看全文>>