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arXiv · 1903.06955

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

Jisu Kim·Jaehyeok Shin·Frédéric Chazal·Alessandro Rinaldo·Larry Wasserman
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Abstract

We derive conditions under which the reconstruction of a target space is topologically correct via the \v{C}ech 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 \v{C}ech complex. Second, we demonstrate the homotopy equivalence of a positive $\mu$-reach set and its offsets. Applying these results to the restricted \v{C}ech complex and using the interleaving relations with the \v{C}ech complex (or the Vietoris-Rips complex), we formulate conditions guaranteeing that the target space is homotopy equivalent to the \v{C}ech complex (or the Vietoris-Rips complex), in terms of the $\mu$-reach. Our results sharpen existing results.

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Jisu Kim, Jaehyeok Shin, Frédéric Chazal, Alessandro Rinaldo, Larry Wasserman. 2019-03-16. Homotopy Reconstruction via the Cech Complex and the Vietoris-Rips Complex. https://arxiv.org/abs/1903.06955

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