# Metrics for generalized persistence modules

arXiv: https://arxiv.org/abs/1312.3829 · HTML: https://mlsanity.com/paper/1312.3829/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Dec 2013.

## Built on by (most cited first, 3 of 3)

- [Stability for Inference with Persistent Homology Rank Functions](https://mlsanity.com/paper/2307.02904/) (2023)
- [The Universal ℓp-Metric on Merge Trees](https://mlsanity.com/paper/2112.12165/) (2021)
- [Learning Transfers: Kan Extensions of Topological Invariants of Learned Representations](https://mlsanity.com/paper/2606.07627/) (2026)

## Abstract (arXiv)

We consider the question of defining interleaving metrics on generalized persistence modules over arbitrary preordered sets. Our constructions are functorial, which implies a form of stability for these metrics. We describe a large class of examples, inverse-image persistence modules, which occur whenever a topological space is mapped to a metric space. Several standard theories of persistence and their stability can be described in this framework. This includes the classical case of sublevelset persistent homology. We introduce a distinction between `soft' and `hard' stability theorems. While our treatment is direct and elementary, the approach can be explained abstractly in terms of monoidal functors.