A Comparison Framework for Interleaved Persistence Modules

TL;DR

This paper generalizes key theorems in persistence module theory to provide more flexible and rigorous error bounds for persistence diagrams, enhancing stability analysis in topological data analysis.

Contribution

It introduces a generalized algebraic stability theorem for $ eal$-indexed persistence modules, extending previous results and enabling more precise error bounds in persistence diagram comparisons.

Findings

Generalized induced matching theorem

Extended algebraic stability theorem

Improved error bounds in persistence diagrams

Abstract

We present a generalization of the induced matching theorem and use it to prove a generalization of the algebraic stability theorem for $\mathbb{R}$-indexed pointwise finite-dimensional persistence modules. Via numerous examples, we show how the generalized algebraic stability theorem enables the computation of rigorous error bounds in the space of persistence diagrams that go beyond the typical formulation in terms of bottleneck (or log bottleneck) distance.