Topological Stability and Latschev-type Reconstruction Theorems for CAT () Spaces

We consider the problem of homotopy-type reconstruction of compact shapes XN that are CAT () 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 CAT () spaces. Our techniques provide novel sampling conditions alternative to the existing and commonly used techniques using weak feature size and --reach. In particular, we introduce a new parameter, called the restricted distortion, which is a generalization of the well-known global distortion of embedding. We show examples of Euclidean subspaces, for which the known parameters such as the reach, --reach and weak features size vanish, whereas the restricted distortion is finite, making our reconstruction results applicable for such spaces.