Stability of 2-Parameter Persistent Homology

The ?ech and Rips constructions of persistent homology are stable with respect to perturbations the input data. However, neither is robust outliers, both can be insensitive topological structure high-density regions A natural solution consider 2-parameter persistence. This paper studies stability homology: we show that several related density-sensitive bifiltrations from data satisfy properties accommodating addition removal outliers. Specifically, multicover bifiltration, Sheehy’s subdivision bifiltrations, degree bifiltrations. For get 1-Lipschitz results closely analogous standard for 1-parameter homology. Our weaker, but they tight, in a sense. As an application our theory, prove law large numbers random