# Metrics and stabilization in one parameter persistence

## TL;DR

This paper introduces a new perspective on one-parameter persistence by emphasizing the importance of metric choices, leading to stabilized invariants and practical data analysis applications.

## Contribution

It proposes a metric-based framework for one-parameter persistence, focusing on stabilization of invariants rather than decomposition theorems, with theoretical development and empirical evidence.

## Key findings

- Stabilization of discrete invariants via pseudometrics
- Development of stable rank invariant theory
- Evidence of practical usefulness in data analysis

## Abstract

We propose a new way of thinking about one parameter persistence. We believe topological persistence is fundamentally not about decomposition theorems but a central role is played by a choice of metrics. Choosing a pseudometric between persistent vector spaces leads to stabilization of discrete invariants. We develop theory behind this stabilization and stable rank invariant. We give evidence of the usefulness of this approach in concrete data analysis.