Robustness and Computability of 2-Parameter Persistent Homology

Abstract: The Vietoris-Rips filtration, the standard filtration on metric data in topological data analysis, is notoriously sensitive to outliers and can be insensitive to variations in density. A natural solution is to consider 2-parameter persistence, treating density and spatial scale as separate parameters. In this talk, I will present results on the stability, robustness, and computability of 2-parameter persistence. A main focus will be Sheehy's subdivision-Rips bifiltration, the only density-sensitive bifiltration on metric data known to satisfy a strong robustness property. This filtration is too large to compute directly, but we will see that it can be approximated by much smaller objects. Our results reveal an apparent tension between robustness and computability for 2-parameter persistence, which in spite of substantial progress, is not yet fully understood.

The talk will be based on three papers, the first with Andrew Blumberg and the others with KenMcCabe:

• https://link.springer.com/article/10.1007/s10208-022-09576-6
• https://arxiv.org/abs/2406.07679
• https://arxiv.org/abs/2408.16716

geometric topology

Audience: researchers in the topic

GEOTOP-A seminar

Series comments: Web-seminar series on Applications of Geometry and Topology