论文标题
2参数持续同源性的稳定性
Stability of 2-Parameter Persistent Homology
论文作者
论文摘要
对于输入数据的扰动，持续同源性的čeCh和RIPS结构是稳定的。但是，对离群值也不强大，并且两者都不对数据的高密度区域的拓扑结构不敏感。自然解决方案是考虑2参数持久性。本文研究了2参数持续的同源性的稳定性：我们表明，来自数据的几种相关密度敏感构建体满足了适应异常值的添加和去除的稳定性。具体而言，我们考虑了多重分叉，Sheehy的细分分叉和程度分叉。对于多跨性别和细分分叉，我们获得了1- lipschitz稳定性结果与1参数持续同源性的标准稳定性结果非常相似。从某种意义上说，我们的分叉结果较弱，但它们很紧。作为我们理论的应用，我们证明了一项大量定律，用于随机数据的细分分叉。
The Čech 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.
