Persistence Modules of Metric Space Approximations: Stability under Perturbations

Date

Aug 4, 2025

Time

1:00 PM - 2:00 PM

Speaker

Philip Sandt

Affiliation

ETH Zürich

Language

en

Main Topic

Biologie

Host

Heather Harrington

Description

We suppose that we are given a point cloud, which we think of as a finite sample of points from some geometric object. This situation can arise in data analysis, where high-dimensional data has to be analyzed, but it could also be a sample from a submanifold of Rn. The mathematical formalism that studies such point clouds is called persistence. It works by assigning a persistence module to a point cloud. A persistence module is a purely algebraic object but is able to capture a lot of topological information about the data set. We can look at the data set from the point of view of di!erent scales, and for each scale the persistence module gives us feedback on the number of connected components, the number of holes and their higher dimensional analogues at each scale. At di!erent scales, di!erent geometric shapes become apparent in the data set. The question which we investigate is the following. How does the persistence module associated to a point cloud change when the point cloud is replaced by a perturbed version of itself, for example if we sample with noise? Investigating such a question will require introducing distance functions on the level of both the persistence modules and the perturbation between sets. The main result is that the amount by which the persistence modules di!er is bounded by the amount of perturbation between the data sets. This is called the stability inequality.

Last modified: Aug 3, 2025, 7:37:17 AM