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

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. In particular, we leverage the concept of the {\em large scale distortion}, and show examples of Euclidean subspaces, for which the known parameters such as the reach, $\mu$--reach and weak features size vanish, whereas the large scale distortion is finite, making our reconstruction results applicable for such cases.