Topological Stability and Latschev-type Reconstruction Theorems for Spaces of Curvature Bounded Above

TL;DR

This paper develops a new method for homotopy-type reconstruction of compact spaces with curvature bounded above, using a path-based metric and Vietoris--Rips complexes, providing guarantees even for spaces with vanishing reach.

Contribution

It introduces a novel sampling condition and reconstruction technique for spaces with curvature bounds, extending guarantees to cases with zero reach.

Findings

Reconstruction guarantees for spaces with vanishing reach.

New sampling conditions based on curvature and topology.

Analysis of Gromov--Hausdorff stability for such spaces.

Abstract

We consider the problem of homotopy-type reconstruction of compact subsets $X\subset\R^N$ that have the Alexandrov curvature bounded above ($\leq$ $\kappa$) in the intrinsic length metric. The reconstructed spaces are in the form of Vietoris--Rips complexes computed from a compact sample $S$, Hausdorff--close to the unknown shape $X$. Instead of the Euclidean metric on the sample, our reconstruction technique leverages a path-based metric to compute these complexes. As naturally emerging in the framework of reconstruction, we also study the Gromov--Hausdorff topological stability and finiteness problem for general compact for subspaces of curvature bounded above. Our techniques provide novel sampling conditions as an alternative to the existing and commonly used techniques using weak feature size and $\mu$--reach. To the best of our knowledge, this is the first work that establishes homotopy-type reconstruction guarantees for spaces with vanishing reach and $\mu$--reach, a regime not covered by existing sampling conditions.