$$A_\infty $$ Persistent Homology Estimates Detailed Topology from Pointcloud Datasets
$$A_\infty $$ Persistent Homology Estimates Detailed Topology from Pointcloud Datasets
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$$A_infty $$ 持久同源估计点云数据集的详细拓扑
DOI：
10.1007/s00454-021-00319-y
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发表时间：
2021
影响因子：
0.8
通讯作者：
Belchí F
中科院分区：
文献类型：
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作者：
Belchí F
设X是度量空间M的闭子空间。众所周知，在温和的假设下，人们可以从近似X的有限点集估计X的Betti数。在本文中，我们证明了人们也可以使用P来估计X的更详细的拓扑性质。我们实现这一点，通过证明的稳定性持久的同源性。在最一般的情况下，这种稳定性意味着给定拓扑空间Y上的连续函数，函数中的小扰动最多只能在-条形码族中有小扰动。这一工作可以看作是杯积和广义Massey积持久性稳定性的证明。本文的技术关键在于找到一个使-持久化成为函子的设置。
LetXbe a closed subspace of a metric spaceM. It is well known that, under mild hypotheses, one can estimate the Betti numbers ofXfrom a finite setof points approximatingX. In this paper, we show that one can also usePto estimate much more detailed topological properties ofX. We achieve this by proving the stability of-persistent homology. In its most general case, this stability means that given a continuous functionon a topological spaceY, small perturbations in the functionfimply at most small perturbations in the family of-barcodes. This work can be viewed as a proof of the stability of cup-product and generalized-Massey-products persistence. The technical key of this paper consists of figuring out a setting which makes-persistence functorial.