Wasserstein Stability for Persistence Diagrams

Abstract: The stability of persistence diagrams is among the most important results in applied and computational topology. Most results in the literature phrase stability in terms of the bottleneck distance between diagrams and the infinity-norm of perturbations. This has two main implications: it makes the space of persistence diagrams rather pathological and it is often provides very pessimistic bounds with respect to outliers. In this talk I will discuss new stability results with respect to the p-Wasserstein distance between persistence diagrams. I will give an elementary proof for the setting of functions on sufficiently finite spaces in terms of the p-norm of the perturbations. I will also apply the results to a wide range of applications in topological data analysis (TDA) including topological summaries, persistence transforms and the special but important case of Vietoris-Rips complexes. This is joint work with Primoz Skraba (see arxiv.org/abs/2006.16824).

The assumed knowledge for the talk:

The persistent homology and persistence diagram for the sub-level set filtration of a real-valued function on a finite simplicial complex.

The Vietoris-Rips complex of a set of points in Euclidean space.

computational geometryalgebraic topologycombinatoricsgeometric topologyprobability

Audience: advanced learners

Asia Pacific Seminar on Applied Topology and Geometry