Positivity of Multiparameter Persistence Diagrams and Bottleneck Stability

TL;DR

This paper investigates the positivity and stability of multiparameter persistence diagrams derived from multifiltrations, aiming to ensure reliable topological summaries in data analysis.

Contribution

It introduces conditions under which multiparameter persistence diagrams are positive and proves their bottleneck stability, advancing the theoretical understanding of multiparameter persistent homology.

Findings

Establishment of positivity conditions for multiparameter persistence diagrams

Proof of bottleneck stability for these diagrams

Framework for reliable topological data analysis using multiparameter persistence

Abstract

Persistent homology studies the birth and death of cycles in a parameterized family of spaces. In this paper, we study the birth and death of cycles in a multifiltration of a chain complex with the goal of producing a persistence diagram that satisfies bottleneck stability.