# Stability of 2-Parameter Persistent Homology

arXiv: https://arxiv.org/abs/2010.09628 · HTML: https://mlsanity.com/paper/2010.09628/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Oct 2020.

## Built on by (most cited first, 5 of 5)

- [A Framework for Fast and Stable Representations of Multiparameter Persistent Homology Decompositions](https://mlsanity.com/paper/2306.11170/) (2023)
- [Detection of Small Holes by the Scale-Invariant Robust Density-Aware Distance (RDAD) Filtration](https://mlsanity.com/paper/2204.07821/) (2022)
- [Topological data analysis using persistent discrete homology](https://mlsanity.com/paper/2506.15020/) (2025)
- [G-LoG Bi-filtration for Medical Image Classification](https://mlsanity.com/paper/2602.18329/) (2026)
- [The sublevel Flood bifiltration: towards scalable 2-parameter persistent homology](https://mlsanity.com/paper/2610.05441/) (2026)

## Abstract (arXiv)

The vCech 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.