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

Jisu Kim (1) , Jaehyeok Shin (2) , Frédéric Chazal (1) , Alessandro Rinaldo (2) , Larry Wasserman (2, 3)

1 DATASHAPE - Understanding the Shape of Data
2 Statistics Department, Carnegie Mellon University
3 Machine Learning Department [Carnegie Mellon Univ.]

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

Mots clés: Computational topology; Homotopy reconstruction; Homotopy Equivalence; Vietoris-Rips complex; $\check{C}$ech complex; Reach; $\mu$-reach; Nerve Theorem; Offset; Double offset; Consistency

Domaines: Topologie algébrique [math.AT]; Géométrie algorithmique [cs.CG]

Jisu Kim, Jaehyeok Shin, Frédéric Chazal, Alessandro Rinaldo, Larry Wasserman. Homotopy Reconstruction via the Cech Complex and the Vietoris-Rips Complex. SoCG 2020 - 36th International Symposium on Computational Geometry, Jun 2020, Zurich, Switzerland. ⟨hal-02425686v2⟩