The Stability of Persistence Diagrams Under Non-Uniform Scaling
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866916494382727168 |
|---|---|
| author | Le, Vu-Anh Dik, Mehmet |
| author_facet | Le, Vu-Anh Dik, Mehmet |
| contents | We investigate the stability of persistence diagrams \( D \) under non-uniform scaling transformations \( S \) in \( \mathbb{R}^n \). Given a finite metric space \( X \subset \mathbb{R}^n \) with Euclidean distance \( d_X \), and scaling factors \( s_1, s_2, \ldots, s_n > 0 \) applied to each coordinate, we derive explicit bounds on the bottleneck distance \( d_B(D, D_S) \) between the persistence diagrams of \( X \) and its scaled version \( S(X) \). Specifically, we show that \[ d_B(D, D_S) \leq \frac{1}{2} (s_{\max} - s_{\min}) \cdot \operatorname{diam}(X), \] where \( s_{\min} \) and \( s_{\max} \) are the smallest and largest scaling factors, respectively, and \( \operatorname{diam}(X) \) is the diameter of \( X \). We extend this analysis to higher-dimensional homological features, alternative metrics such as the Wasserstein distance, and iterative or probabilistic scaling scenarios. Our results provide a framework for quantifying the effects of non-uniform scaling on persistence diagrams. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_16126 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Stability of Persistence Diagrams Under Non-Uniform Scaling Le, Vu-Anh Dik, Mehmet Algebraic Topology Geometric Topology Metric Geometry We investigate the stability of persistence diagrams \( D \) under non-uniform scaling transformations \( S \) in \( \mathbb{R}^n \). Given a finite metric space \( X \subset \mathbb{R}^n \) with Euclidean distance \( d_X \), and scaling factors \( s_1, s_2, \ldots, s_n > 0 \) applied to each coordinate, we derive explicit bounds on the bottleneck distance \( d_B(D, D_S) \) between the persistence diagrams of \( X \) and its scaled version \( S(X) \). Specifically, we show that \[ d_B(D, D_S) \leq \frac{1}{2} (s_{\max} - s_{\min}) \cdot \operatorname{diam}(X), \] where \( s_{\min} \) and \( s_{\max} \) are the smallest and largest scaling factors, respectively, and \( \operatorname{diam}(X) \) is the diameter of \( X \). We extend this analysis to higher-dimensional homological features, alternative metrics such as the Wasserstein distance, and iterative or probabilistic scaling scenarios. Our results provide a framework for quantifying the effects of non-uniform scaling on persistence diagrams. |
| title | The Stability of Persistence Diagrams Under Non-Uniform Scaling |
| topic | Algebraic Topology Geometric Topology Metric Geometry |
| url | https://arxiv.org/abs/2411.16126 |