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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.23460 |
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Table of Contents:
- Latschev's theorem provides sufficient conditions on a metric space $M$ and $δ> 0$ for the homotopy type of $M$ to agree with that of the Vietoris-Rips complex $\mathcal{R}^δ(N)$ of any nearby space $N$ in the Gromov-Hausdorff distance. We prove a persistent version of this theorem, providing sufficient conditions on a pair $(M, f \colon M \to \mathbb{R}^N)$ and $δ> 0$ for the persistent homotopy type of the sublevel set filtration of $(M,f)$ to be interleaved with that of the function-Rips complex $\mathcal{R}^δ(N^{\bullet})$ of any nearby pair $(N,g)$. In particular, our result answers a longstanding question on the related topic of estimating sublevel set persistent homology from finite point samples.