Amplitudes in persistence theory

TL;DR

This paper introduces a new framework for comparing invariants in multiparameter persistence, defining amplitudes and related invariants to enhance stability and discrimination in topological data analysis.

Contribution

It develops a general framework for comparing distances and invariants in multiparameter persistence, including the definition and classification of amplitudes and their relationships.

Findings

Introduces amplitudes, monotone, and subadditive invariants in persistence theory.

Provides methods to associate and classify distances based on these invariants.

Analyzes the discriminative power of amplitude-based distances in TDA scenarios.

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 with data sets. Here we introduce a general framework to compare distances and invariants in multiparameter persistence, where there is no natural choice of invariants and distances between them. We define amplitudes, monotone, and subadditive invariants that arise from assigning a non-negative real number to objects of an abelian category. We then present different ways to associate distances to such invariants, and we provide a classification of classes of amplitudes relevant to topological data analysis. In addition, we study the the relationships as well as the discriminitative power of such amplitude distances arising in topological data analysis scenarios.