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

TL;DR

This paper establishes new theoretical conditions under which the topological reconstruction of a space from noisy point cloud data using cech and Vietoris-Rips complexes is guaranteed to be accurate, improving upon previous results.

Contribution

It introduces two novel theorems on contractibility and homotopy equivalence related to cech and Vietoris-Rips complexes, sharpening existing topological data analysis results.

Findings

Conditions for contractibility of intersections of Euclidean balls with positive reach sets.

Homotopy equivalence between positive reach sets and their offsets.

Sharpened guarantees for topological reconstruction from point cloud data.

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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