Stability and Approximations for Decorated Reeb Spaces
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2023
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866913274221559808 |
|---|---|
| author | Curry, Justin Mio, Washington Needham, Tom Okutan, Osman Berat Russold, Florian |
| author_facet | Curry, Justin Mio, Washington Needham, Tom Okutan, Osman Berat Russold, Florian |
| contents | Given a map $f:X \to M$ from a topological space $X$ to a metric space $M$, a decorated Reeb space consists of the Reeb space, together with an attribution function whose values recover geometric information lost during the construction of the Reeb space. For example, when $M=\mathbb{R}$ is the real line, the Reeb space is the well-known Reeb graph, and the attributions may consist of persistence diagrams summarizing the level set topology of $f$. In this paper, we introduce decorated Reeb spaces in various flavors and prove that our constructions are Gromov-Hausdorff stable. We also provide results on approximating decorated Reeb spaces from finite samples and leverage these to develop a computational framework for applying these constructions to point cloud data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_01982 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Stability and Approximations for Decorated Reeb Spaces Curry, Justin Mio, Washington Needham, Tom Okutan, Osman Berat Russold, Florian Metric Geometry Computational Geometry Algebraic Topology Given a map $f:X \to M$ from a topological space $X$ to a metric space $M$, a decorated Reeb space consists of the Reeb space, together with an attribution function whose values recover geometric information lost during the construction of the Reeb space. For example, when $M=\mathbb{R}$ is the real line, the Reeb space is the well-known Reeb graph, and the attributions may consist of persistence diagrams summarizing the level set topology of $f$. In this paper, we introduce decorated Reeb spaces in various flavors and prove that our constructions are Gromov-Hausdorff stable. We also provide results on approximating decorated Reeb spaces from finite samples and leverage these to develop a computational framework for applying these constructions to point cloud data. |
| title | Stability and Approximations for Decorated Reeb Spaces |
| topic | Metric Geometry Computational Geometry Algebraic Topology |
| url | https://arxiv.org/abs/2312.01982 |