Core Bifiltration
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915488857063424 |
|---|---|
| author | Blaser, Nello Brun, Morten Gardaa, Odin Hoff Salbu, Lars M. |
| author_facet | Blaser, Nello Brun, Morten Gardaa, Odin Hoff Salbu, Lars M. |
| contents | The motivation of this paper is to recognize a geometric shape from a noisy sample in the form of a point cloud. Inspired by the HDBSCAN clustering algorithm, we introduce the core dissimilarity, from which we construct the core bifiltration. We also consider the Delaunay core bifiltration by intersecting with Voronoi cells, giving us a filtered simplicial complex of smaller size. A major advantage of the (Delaunay) core bifiltration is that, for each filtration value, it admits a good cover of balls. By the persistent nerve theorem, the nerve of this cover is homotopy equivalent to the (Delaunay) core bifiltration. We show that the multicover-, core- and Delaunay core bifiltrations are all interleaved, and that they enjoy similar stability properties with respect to the Prohorov distance. We have performed experiments with the Delaunay core bifiltration. In the experiments, we calculated persistent homology along lines in the two-dimensional persistence parameter space, as well as multipersistence module approximations and Hilbert functions for the full Delaunay core bifiltration. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_01214 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Core Bifiltration Blaser, Nello Brun, Morten Gardaa, Odin Hoff Salbu, Lars M. Algebraic Topology Computational Geometry 55N31, 62R40 The motivation of this paper is to recognize a geometric shape from a noisy sample in the form of a point cloud. Inspired by the HDBSCAN clustering algorithm, we introduce the core dissimilarity, from which we construct the core bifiltration. We also consider the Delaunay core bifiltration by intersecting with Voronoi cells, giving us a filtered simplicial complex of smaller size. A major advantage of the (Delaunay) core bifiltration is that, for each filtration value, it admits a good cover of balls. By the persistent nerve theorem, the nerve of this cover is homotopy equivalent to the (Delaunay) core bifiltration. We show that the multicover-, core- and Delaunay core bifiltrations are all interleaved, and that they enjoy similar stability properties with respect to the Prohorov distance. We have performed experiments with the Delaunay core bifiltration. In the experiments, we calculated persistent homology along lines in the two-dimensional persistence parameter space, as well as multipersistence module approximations and Hilbert functions for the full Delaunay core bifiltration. |
| title | Core Bifiltration |
| topic | Algebraic Topology Computational Geometry 55N31, 62R40 |
| url | https://arxiv.org/abs/2405.01214 |