The structure and stability of persistence modules

TL;DR

This paper provides a simplified, self-contained framework for the theory of persistence modules, introducing weaker conditions for stability and constructing persistence diagrams using measure theory, making the theory more accessible and broadly applicable.

Contribution

It introduces a new, simplified approach to persistence modules, relaxing finiteness conditions and employing measure theory for diagram construction, enhancing clarity and applicability.

Findings

Persistence diagrams can be constructed without strict finiteness conditions.

The new framework simplifies proofs and calculations in persistence theory.

Weaker hypotheses suffice for stability and existence results.

Abstract

We give a self-contained treatment of the theory of persistence modules indexed over the real line. We give new proofs of the standard results. Persistence diagrams are constructed using measure theory. Linear algebra lemmas are simplified using a new notation for calculations on quiver representations. We show that the stringent finiteness conditions required by traditional methods are not necessary to prove the existence and stability of the persistence diagram. We introduce weaker hypotheses for taming persistence modules, which are met in practice and are strong enough for the theory still to work. The constructions and proofs enabled by our framework are, we claim, cleaner and simpler.