Geometric Bounds for Persistence
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866908672162004992 |
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| author | Balitskiy, Alexey Coskunuzer, Baris Mémoli, Facundo |
| author_facet | Balitskiy, Alexey Coskunuzer, Baris Mémoli, Facundo |
| contents | In this paper, we offer a new perspective on persistent homology by integrating key concepts from metric geometry. For a given compact subset $\mathcal{X}$ of a Banach space $\mathbf{Y}$, we analyze the topological features arising in the family $\mathcal{N}_\bullet(\mathcal{X} \subset \mathbf{Y})$ of nested neighborhoods of $\mathcal{X}$ in $\mathbf{Y}$ and provide several geometric bounds on their persistence (lifespans).
We begin by examining the lifespans of these homology classes in terms of their filling radii in $\mathbf{Y}$, establishing connections between these lifespans and fundamental invariants in metric geometry, such as the Urysohn width. We then derive bounds on these lifespans by considering the $\ell^\infty$-principal components of $\mathcal{X}$, also known as Kolmogorov widths.
Additionally, we introduce and investigate the concept of extinction time of a metric space $\mathcal{X}$: the critical threshold beyond which no homological features persist in any degree. We propose methods for estimating the Čech and Vietoris-Rips extinction times of $\mathcal{X}$ by relating $\mathcal{X}$ to its convex hull and to its tight span, respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_13980 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Geometric Bounds for Persistence Balitskiy, Alexey Coskunuzer, Baris Mémoli, Facundo Algebraic Topology Metric Geometry In this paper, we offer a new perspective on persistent homology by integrating key concepts from metric geometry. For a given compact subset $\mathcal{X}$ of a Banach space $\mathbf{Y}$, we analyze the topological features arising in the family $\mathcal{N}_\bullet(\mathcal{X} \subset \mathbf{Y})$ of nested neighborhoods of $\mathcal{X}$ in $\mathbf{Y}$ and provide several geometric bounds on their persistence (lifespans). We begin by examining the lifespans of these homology classes in terms of their filling radii in $\mathbf{Y}$, establishing connections between these lifespans and fundamental invariants in metric geometry, such as the Urysohn width. We then derive bounds on these lifespans by considering the $\ell^\infty$-principal components of $\mathcal{X}$, also known as Kolmogorov widths. Additionally, we introduce and investigate the concept of extinction time of a metric space $\mathcal{X}$: the critical threshold beyond which no homological features persist in any degree. We propose methods for estimating the Čech and Vietoris-Rips extinction times of $\mathcal{X}$ by relating $\mathcal{X}$ to its convex hull and to its tight span, respectively. |
| title | Geometric Bounds for Persistence |
| topic | Algebraic Topology Metric Geometry |
| url | https://arxiv.org/abs/2403.13980 |