# Universality of persistence diagrams and the bottleneck and Wasserstein distances

arXiv: https://arxiv.org/abs/1912.02563 · HTML: https://mlsanity.com/paper/1912.02563/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Dec 2019.

## Built on by (most cited first, 2 of 2)

- [A Class of Topological Pseudodistances for Fast Comparison of Persistence Diagrams](https://mlsanity.com/paper/2402.14489/) (2024)
- [Operational Abstractions of Neural Network Concepts via Topological Representations](https://mlsanity.com/paper/2610.05017/) (2026)

## Abstract (arXiv)

We prove that persistence diagrams with the p-Wasserstein distance form the universal p-subadditive commutative monoid on an underlying metric space with a distinguished subset. This result applies to persistence diagrams, barcodes, and to multiparameter persistence modules. In addition, the 1-Wasserstein distance satisfies Kantorovich-Rubinstein duality.