Wasserstein Stability of Persistence Landscapes and Barcodes
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912717042876416 |
|---|---|
| author | Zhao, Wanchen Bubenik, Peter |
| author_facet | Zhao, Wanchen Bubenik, Peter |
| contents | Barcodes form a complete set of invariants for interval decomposable persistence modules and are an important summary in topological data analysis. The set of barcodes is equipped with a canonical one-parameter family of metrics, the $p-$Wasserstein distances. However, the $p-$Wasserstein distances depend on a choice of a metric on the set of interval modules, and there is no canonical choice. One convention is to use the length of the symmetric difference between 2 intervals, which equals to the 1-norm of the difference between their Hilbert functions. We propose a new metric for interval modules based on the rank invariant instead of the dimension invariant. Our metric is topologically equivalent to the metrics induced by the $p-$norms on $\R^2$. We establish stability results from filtered CW complexes to barcodes, as well as from barcodes to persistence landscapes. In particular, we show that vectorization via persistence landscapes is 1-Lipschitz with a sharp bound, with respect to the 1-Wasserstein distance on barcodes and the 1-norm on persistence landscapes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_20921 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Wasserstein Stability of Persistence Landscapes and Barcodes Zhao, Wanchen Bubenik, Peter Algebraic Topology Barcodes form a complete set of invariants for interval decomposable persistence modules and are an important summary in topological data analysis. The set of barcodes is equipped with a canonical one-parameter family of metrics, the $p-$Wasserstein distances. However, the $p-$Wasserstein distances depend on a choice of a metric on the set of interval modules, and there is no canonical choice. One convention is to use the length of the symmetric difference between 2 intervals, which equals to the 1-norm of the difference between their Hilbert functions. We propose a new metric for interval modules based on the rank invariant instead of the dimension invariant. Our metric is topologically equivalent to the metrics induced by the $p-$norms on $\R^2$. We establish stability results from filtered CW complexes to barcodes, as well as from barcodes to persistence landscapes. In particular, we show that vectorization via persistence landscapes is 1-Lipschitz with a sharp bound, with respect to the 1-Wasserstein distance on barcodes and the 1-norm on persistence landscapes. |
| title | Wasserstein Stability of Persistence Landscapes and Barcodes |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2509.20921 |