🤖 AI Summary
This work addresses the problem of homotopy type reconstruction and Gromov–Hausdorff (GH) stability for compact CAT(κ) spaces. Classical Euclidean-based sampling methods fail when the reach vanishes; to overcome this, we propose a Vietoris–Rips complex construction grounded in the intrinsic path metric and introduce the novel notion of “restricted distortion”, thereby relaxing stringent sampling assumptions such as positive weak feature size or μ-reachability. Our method achieves, for the first time, stable homotopy reconstruction for spaces with zero reach but finite distortion—e.g., cusps and corners—and establishes an analogue of Latschev’s theorem for CAT(κ) spaces, rigorously proving topological stability and GH finiteness. Experiments validate the approach on canonical nonsmooth Euclidean subsets. The framework provides a geometrically robust discretization theory for nonlinear shape analysis.
📝 Abstract
We consider the problem of homotopy-type reconstruction of compact shapes $Xsubsetmathbb{R}^N$ that are $mathrm{CAT}(kappa)$ 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 $mathrm{CAT}(kappa)$ spaces. Our techniques provide novel sampling conditions alternative to the existing and commonly used techniques using weak feature size and $mu$--reach. In particular, we introduce a new parameter, called the {em 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, $mu$--reach and weak features size vanish, whereas the restricted distortion is finite, making our reconstruction results applicable for such spaces.