Stability of 2-Parameter Persistent Homology
Stability of 2-Parameter Persistent Homology
复制标题
DOI：
10.1007/s10208-022-09576-6
复制
发表时间：
2022-10-17
影响因子：
3
通讯作者：
Lesnick, Michael
中科院分区：
文献类型：
--
作者：
Blumberg, Andrew J.;Lesnick, Michael
持久同源性的 Cech 和 Rips 结构对于输入数据的扰动是稳定的。然而，两者对于异常值都不是鲁棒的，并且都对数据高密度区域的拓扑结构不敏感。一个自然的解决方案是考虑 2 参数持久性。本文研究了 2 参数持久同源性的稳定性：我们证明了数据中几种相关的密度敏感的双过滤结构满足稳定性特性，可容纳异常值的添加和删除。具体来说，我们考虑多层双滤、Sheehy 细分双滤和分级双滤。对于多重覆盖和细分双滤，我们得到的 1-Lipschitz 稳定性结果与 1 参数持久同源性的标准稳定性结果非常相似。我们的双滤度结果较弱，但从某种意义上说它们是紧密的。作为我们理论的应用，我们证明了随机数据细分双过滤的大数定律。
The Cech and Rips constructions of persistent homology are stable with respect to perturbations of the input data. However, neither is robust to outliers, and both can be insensitive to topological structure of high-density regions of the data. A natural solution is to consider 2-parameter persistence. This paper studies the stability of 2-parameter persistent homology: we show that several related density-sensitive constructions of bifiltrations from data satisfy stability properties accommodating the addition and removal of outliers. Specifically, we consider the multicover bifiltration, Sheehy's subdivision bifiltrations, and the degree bifiltrations. For the multicover and subdivision bifiltrations, we get 1-Lipschitz stability results closely analogous to the standard stability results for 1-parameter persistent homology. Our results for the degree bifiltrations are weaker, but they are tight, in a sense. As an application of our theory, we prove a law of large numbers for subdivision bifiltrations of random data.