# Homotopy Reconstruction via the Cech Complex and the Vietoris-Rips Complex

arXiv: https://arxiv.org/abs/1903.06955 · HTML: https://mlsanity.com/paper/1903.06955/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Mar 2019.

## Built on by (most cited first, 4 of 4)

- [On the effectiveness of persistent homology](https://mlsanity.com/paper/2206.10551/) (2022)
- [The Impact of Changes in Resolution on the Persistent Homology of Images](https://mlsanity.com/paper/2111.05663/) (2021)
- [Cover Learning for Large-Scale Topology Representation](https://mlsanity.com/paper/2503.09767/) (2025)
- [Topology-Informed Machine Learning for Efficient Prediction of Solid Oxide Fuel Cell Electrode Polarization](https://mlsanity.com/paper/2410.05307/) (2024)

## Abstract (arXiv)

We derive conditions under which the reconstruction of a target space is topologically correct via the vCech complex or the Vietoris-Rips complex obtained from possibly noisy point cloud data. We provide two novel theoretical results. First, we describe sufficient conditions under which any non-empty intersection of finitely many Euclidean balls intersected with a positive reach set is contractible, so that the Nerve theorem applies for the restricted vCech complex. Second, we demonstrate the homotopy equivalence of a positive μ-reach set and its offsets. Applying these results to the restricted vCech complex and using the interleaving relations with the vCech complex (or the Vietoris-Rips complex), we formulate conditions guaranteeing that the target space is homotopy equivalent to the vCech complex (or the Vietoris-Rips complex), in terms of the μ-reach. Our results sharpen existing results.