Metric thickenings of Euclidean submanifolds
Metric thickenings of Euclidean submanifolds
复制标题
欧几里得子流形的公制增厚
DOI：
10.1016/j.topol.2018.12.014
复制
发表时间：
2017
影响因子：
0.6
通讯作者：
Joshua Mirth
中科院分区：
文献类型：
--
作者：
Henry Adams;Joshua Mirth
给定一个样本Y，它来自一个嵌入在欧氏空间中的未知流形X，通过在顶点集Y上建立一个Vietoris-Rips或Čech单纯复形，可以恢复X的同调群。然而，这些单纯复形不需要继承流形的度量结构，特别是当Y是无限的。事实上，如果单纯复形不是局部有限的，它甚至不是可度量化的。相反，我们考虑度量加厚，称为Vietoris-RipsandČech加厚，它配备了1-Wasserstein度量代替单纯复拓扑。我们证明了对于具有正到达的欧氏子集X，其增厚满足豪斯曼定理和神经引理的度量类似物（X的度量Vietoris-Rips和Čech增厚对于小于到达的尺度参数同伦等价于X）.据我们所知，这是第一个版本的豪斯曼的定理Vietoris-Rips建设整个欧几里德子流形（而不是黎曼流形），我们的结果也延伸到非流形形状（因为不是所有集的积极达到流形）。与豪斯曼的原证明相反，我们的同伦等价是一个变形收缩，是由两个方向上的正则映射实现的，而且可以通过简单的线性同伦证明从映射组合到相应的恒等映射是同伦等价的.
Given a sampleYfrom an unknown manifoldXembedded in Euclidean space, it is possible to recover the homology groups ofXby building a Vietoris–Rips or Čech simplicial complex on top of the vertex setY. However, these simplicial complexes need not inherit the metric structure of the manifold, in particular whenYis infinite. Indeed, a simplicial complex is not even metrizable if it is not locally finite. We instead consider metric thickenings, called theVietoris–RipsandČech thickenings, which are equipped with the 1-Wasserstein metric in place of the simplicial complex topology. We show that for Euclidean subsetsXwith positive reach, the thickenings satisfy metric analogues of Hausmann's theorem and the nerve lemma (the metric Vietoris–Rips and Čech thickenings ofXare homotopy equivalent toXfor scale parameters less than the reach). To our knowledge this is the first version of Hausmann's theorem for Vietoris–Rips constructions on entire Euclidean submanifolds (as opposed to Riemannian manifolds), and our result also extends to non-manifold shapes (as not all sets of positive reach are manifolds). In contrast to Hausmann's original proof, our homotopy equivalence is a deformation retraction, is realized by canonical maps in both directions, and furthermore can be proven to be a homotopy equivalence via simple linear homotopies from the map compositions to the corresponding identity maps.
影响因子：
1.1
作者：
Aamari, Eddie;Kim, Jisu;Wasserman, Larry
通讯作者：
Wasserman, Larry