The \v{C}ech 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.
翻译：\ v{C}C}C}和裂纹的持久性同族体构造在输入数据的扰动方面是稳定的。 但是, 两者都对异常点没有强力, 两者都可能对数据密度高的区域的地形结构不敏感。 一个自然的解决方案是考虑 2 参数的持久性。 本文研究了 2 参数 持久性同族体的稳定性 : 我们显示, 从数据中分离的多个相关密度敏感构造满足了稳定特性, 适应了外部线的增减。 具体地说, 我们考虑的是多覆盖的双过滤、 Sheehy的子分割和程度的双过滤。 对于多覆盖和次分割, 我们得到的1 参数的稳定性结果与 1 参数 标准稳定性结果非常相似。 我们关于分解程度的结果较弱, 但从某种意义上说, 它们是紧凑紧的。 作为我们理论的应用, 我们证明有大量法则用于随机数据子分割的多数值法则。
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