Euler Characteristics and Homotopy Types of Definable Sublevel Sets, with Applications to Topological Data Analysis
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866908913522180096 |
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| author | Ji, Mattie Meng, Kun |
| author_facet | Ji, Mattie Meng, Kun |
| contents | Given a definable function $f: S \to \mathbb{R}$ on a definable set $S$, we study sublevel sets of the form $S^f_t \coloneqq \{x \in S: f(x) \leq t\}$ for all $t \in \mathbb{R}$. Using o-minimal structures, we prove that the Euler characteristic of $S^f_t$ is right-continuous with respect to $t$. Furthermore, when $S$ is compact, we show that $S^f_{t+δ}$ deformation retracts to $S^f_t$ for all sufficiently small $δ> 0$. Applying these results, we also characterize the connections between the following concepts in topological data analysis: the Euler characteristic transform (ECT), smooth ECT, Euler-Radon transform (ERT), and smooth ERT. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_03142 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Euler Characteristics and Homotopy Types of Definable Sublevel Sets, with Applications to Topological Data Analysis Ji, Mattie Meng, Kun Algebraic Topology Statistics Theory Primary: 03C64, 46M20. Secondary: 55N31 Given a definable function $f: S \to \mathbb{R}$ on a definable set $S$, we study sublevel sets of the form $S^f_t \coloneqq \{x \in S: f(x) \leq t\}$ for all $t \in \mathbb{R}$. Using o-minimal structures, we prove that the Euler characteristic of $S^f_t$ is right-continuous with respect to $t$. Furthermore, when $S$ is compact, we show that $S^f_{t+δ}$ deformation retracts to $S^f_t$ for all sufficiently small $δ> 0$. Applying these results, we also characterize the connections between the following concepts in topological data analysis: the Euler characteristic transform (ECT), smooth ECT, Euler-Radon transform (ERT), and smooth ERT. |
| title | Euler Characteristics and Homotopy Types of Definable Sublevel Sets, with Applications to Topological Data Analysis |
| topic | Algebraic Topology Statistics Theory Primary: 03C64, 46M20. Secondary: 55N31 |
| url | https://arxiv.org/abs/2309.03142 |