Topological inference of manifolds with boundary
Topological inference of manifolds with boundary
复制标题
有边界流形的拓扑推理
DOI：
10.1016/j.comgeo.2019.101606
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
Wang, Bei
中科院分区：
文献类型：
--
作者：
Wang, Yuan;Wang, Bei
给定从某些底层空间采样的一组数据点，在处理采样数据时，几何和拓扑数据分析存在两个重要挑战：重构——如何将离散样本组装成全局结构，以及推理——如何从高维、不完整和噪声的数据中提取几何和拓扑信息。 Niyogi 等人。 （2008）表明，通过使用合适的偏移参数构建样本的偏移，可以提供保留同伦类型的重建，因此对于无边界的欧几里德空间的密集采样的平滑子流形来说具有同源性。查扎尔等人。 （2009）和阿塔利等人。 (2013) 引入了一组参数化的采样条件，扩展了 Niyogi 等人的结果。到欧几里得空间的一大类紧子集。我们的工作解决了数据问题，填补了 Niyogi 等人的工作之间的空白。和查扎尔等人。特别是，我们给出了现有理论无法处理的边界流形采样条件的概率概念。我们还给出了更强有力的结果，将偏移和流形之间的拓扑等价关系为变形收缩。
Given a set of data points sampled from some underlying space, there are two important challenges in geometric and topological data analysis when dealing with sampled data: reconstruction – how to assemble discrete samples into global structures, and inference – how to extract geometric and topological information from data that are high-dimensional, incomplete and noisy. Niyogi et al. (2008) have shown that by constructing an offset of the samples using a suitable offset parameter could provide reconstructions that preserve homotopy types therefore homology for densely sampled smooth submanifolds of Euclidean space without boundary. Chazal et al. (2009) and Attali et al. (2013) have introduced a parameterized set of sampling conditions that extend the results of Niyogi et al. to a large class of compact subsets of Euclidean space. Our work tackles data problems that fill a gap between the work of Niyogi et al. and Chazal et al. In particular, we give a probabilistic notion of sampling conditions for manifolds with boundary that could not be handled by existing theories. We also give stronger results that relate topological equivalence between the offset and the manifold as a deformation retract.