Demystifying Latschev's Theorem: Manifold Reconstruction from Noisy Data
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866929285525143552 |
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| author | Majhi, Sushovan |
| author_facet | Majhi, Sushovan |
| contents | For a closed Riemannian manifold $\mathcal{M}$ and a metric space $S$ with a small Gromov$\unicode{x2013}$Hausdorff distance to it, Latschev's theorem guarantees the existence of a sufficiently small scale $β>0$ at which the Vietoris$\unicode{x2013}$Rips complex of $S$ is homotopy equivalent to $\mathcal{M}$. Despite being regarded as a stepping stone to the topological reconstruction of Riemannian manifolds from a noisy data, the result is only a qualitative guarantee. Until now, it had been elusive how to quantitatively choose such a proximity scale $β$ in order to provide sampling conditions for $S$ to be homotopy equivalent to $\mathcal{M}$. In this paper, we prove a stronger and pragmatic version of Latschev's theorem, facilitating a simple description of $β$ using the sectional curvatures and convexity radius of $\mathcal{M}$ as the sampling parameters. Our study also delves into the topological recovery of a closed Euclidean submanifold from the Vietoris$\unicode{x2013}$Rips complexes of a Hausdorff close Euclidean subset. As already known for Čech complexes, we show that Vietoris$\unicode{x2013}$Rips complexes also provide topologically faithful reconstruction guarantees for submanifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_17288 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Demystifying Latschev's Theorem: Manifold Reconstruction from Noisy Data Majhi, Sushovan Algebraic Topology Metric Geometry For a closed Riemannian manifold $\mathcal{M}$ and a metric space $S$ with a small Gromov$\unicode{x2013}$Hausdorff distance to it, Latschev's theorem guarantees the existence of a sufficiently small scale $β>0$ at which the Vietoris$\unicode{x2013}$Rips complex of $S$ is homotopy equivalent to $\mathcal{M}$. Despite being regarded as a stepping stone to the topological reconstruction of Riemannian manifolds from a noisy data, the result is only a qualitative guarantee. Until now, it had been elusive how to quantitatively choose such a proximity scale $β$ in order to provide sampling conditions for $S$ to be homotopy equivalent to $\mathcal{M}$. In this paper, we prove a stronger and pragmatic version of Latschev's theorem, facilitating a simple description of $β$ using the sectional curvatures and convexity radius of $\mathcal{M}$ as the sampling parameters. Our study also delves into the topological recovery of a closed Euclidean submanifold from the Vietoris$\unicode{x2013}$Rips complexes of a Hausdorff close Euclidean subset. As already known for Čech complexes, we show that Vietoris$\unicode{x2013}$Rips complexes also provide topologically faithful reconstruction guarantees for submanifolds. |
| title | Demystifying Latschev's Theorem: Manifold Reconstruction from Noisy Data |
| topic | Algebraic Topology Metric Geometry |
| url | https://arxiv.org/abs/2305.17288 |