Persistent Magnitude

TL;DR

This paper introduces the persistent magnitude, a new numerical invariant for graded persistence modules, which generalizes magnitude concepts and connects to existing persistent homology theories, offering new computational strategies.

Contribution

It defines persistent magnitude with formal properties, links it to blurred magnitude homology, and proposes a method to derive new invariants from existing persistent homology theories.

Findings

Persistent magnitude is additive and compatible with tensor products.

It equals the magnitude of a finite metric space via blurred magnitude homology.

The approach enables transforming existing persistent homology into new numerical invariants.

Abstract

In this paper we introduce the persistent magnitude, a new numerical invariant of (sufficiently nice) graded persistence modules. It is a weighted and signed count of the bars of the persistence module, in which a bar of the form $[a,b)$ in degree $d$ is counted with weight $(e^{-a}-e^{-b})$ and sign $(-1)^d$. Persistent magnitude has good formal properties, such as additivity with respect to exact sequences and compatibility with tensor products, and has interpretations in terms of both the associated graded functor, and the Laplace transform. Our definition is inspired by Otter's notion of blurred magnitude homology: we show that the magnitude of a finite metric space is precisely the persistent magnitude of its blurred magnitude homology. Turning this result on its head, we obtain a strategy for turning existing persistent homology theories into new numerical invariants by applying the persistent magnitude. We explore this strategy in detail in the case of persistent homology of Morse functions, and in the case of Rips homology.