Monoidal Rips: Stable Multiparameter Filtrations of Directed Networks
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866908615817822208 |
|---|---|
| author | Blaser, Nello Brun, Morten Gardaa, Odin Hoff Salbu, Lars M. |
| author_facet | Blaser, Nello Brun, Morten Gardaa, Odin Hoff Salbu, Lars M. |
| contents | We introduce the monoidal Rips filtration, a filtered simplicial set for weighted directed graphs and other lattice-valued networks. Our construction generalizes the Vietoris-Rips filtration for metric spaces by replacing the maximum operator, determining the filtration values, with a more general monoidal product. We establish interleaving guarantees for the monoidal Rips persistent homology, capturing existing stability results for real-valued networks. When the lattice is a product of totally ordered sets, we are in the setting of multiparameter persistence. Here, the interleaving distance is bounded in terms of a generalized network distance. We use this to prove a novel stability result for the sublevel Rips bifiltration. Our experimental results show that our method performs better than Flagser in a graph regression task, and that combining different monoidal products in point cloud classification can improve performance. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_14019 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Monoidal Rips: Stable Multiparameter Filtrations of Directed Networks Blaser, Nello Brun, Morten Gardaa, Odin Hoff Salbu, Lars M. Algebraic Topology Computational Geometry 55N31, 62R40, 55U10 We introduce the monoidal Rips filtration, a filtered simplicial set for weighted directed graphs and other lattice-valued networks. Our construction generalizes the Vietoris-Rips filtration for metric spaces by replacing the maximum operator, determining the filtration values, with a more general monoidal product. We establish interleaving guarantees for the monoidal Rips persistent homology, capturing existing stability results for real-valued networks. When the lattice is a product of totally ordered sets, we are in the setting of multiparameter persistence. Here, the interleaving distance is bounded in terms of a generalized network distance. We use this to prove a novel stability result for the sublevel Rips bifiltration. Our experimental results show that our method performs better than Flagser in a graph regression task, and that combining different monoidal products in point cloud classification can improve performance. |
| title | Monoidal Rips: Stable Multiparameter Filtrations of Directed Networks |
| topic | Algebraic Topology Computational Geometry 55N31, 62R40, 55U10 |
| url | https://arxiv.org/abs/2503.14019 |