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.

持久同源性的 Čech 和 Rips 构造对于输入数据的扰动是稳定的。然而，两者都对异常值不鲁棒，并且两者都对数据的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：我们从数据中证明了几种相关的密度敏感结构的双滤满足稳定性属性，以适应异常值的添加和删除。具体来说，我们考虑了多覆盖双滤、Sheehy 的细分双滤和度数双滤。对于多覆盖和细分双滤，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。我们对度双滤的结果较弱，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滤的大数定律。