Demystifying Latschev’s Theorem: Manifold Reconstruction from Noisy Data

Abstract

For a closed Riemannian manifold \(\mathcal {M}\) and a metric space S with a small Gromov–Hausdorff distance to it, Latschev’s theorem guarantees the existence of a sufficiently small scale \(\beta >0\) at which the Vietoris–Rips complex of S is homotopy equivalent to \(\mathcal {M}\). Despite being regarded as a stepping stone to the topological reconstruction of Riemannian manifolds from noisy data, the result is only a qualitative guarantee. Until now, it had been elusive how to quantitatively choose such a proximity scale \(\beta \) in order to provide sampling conditions for S to be homotopy equivalent to \(\mathcal {M}\). In this paper, we prove a stronger and pragmatic version of Latschev’s theorem, facilitating a simple description of \(\beta \) using the sectional curvatures and convexity radius of \(\mathcal {M}\) as the sampling parameters. Our study also delves into the topological recovery of a closed Euclidean submanifold from the Vietoris–Rips complexes of a Hausdorff close Euclidean subset. As already known for Čech complexes, we show that Vietoris–Rips complexes also provide topologically faithful reconstruction guarantees for submanifolds.