The amplitude of an abelian category: Measures in persistence theory

Speaker: Barbara Giunti (Technische Universit├Ąt Graz)

Occasion: Topological Data Analysis

Date: April 30, 2021

Abstract: The use of persistent homology in applications is justified by the validity of certain stability results. At the core of such results is a notion of distance between the invariants that one associates to data sets. While such distances are well-understood in the one-parameter case, the situation for multiparameter persistence modules is more challenging, since there is no generalisation of the barcode.

Here we introduce a general framework to study stability questions in multiparameter persistence. We first introduce the (outer) amplitude, a functional on abelian categories that mimics the properties of an outer measure in measure theory, then study different ways to associate distances to such functionals. Our framework is very comprehensive, as many different invariants that have been introduced in the literature are examples of outer amplitudes, and similarly, we show that many known distances for multiparameter persistence are distances from outer amplitudes.

Finally, we provide new stability results using our framework.