Approximating Local Homology from Samples

Recently, multi-scale notions of local homology (a variant of persistent homology) have been used to study the local structure of spaces around a given point from a point cloud sample. Current reconstruction guarantees rely on constructing embedded complexes which become difficult in high dimensions. We show that the persistence diagrams used for estimating local homology, can be approximated using families of Vietoris-Rips complexes, whose simple constructions are robust in any dimension. To the best of our knowledge, our results, for the first time, make applications based on local homology, such as stratification learning, feasible in high dimensions.

💡 Research Summary

The paper addresses a fundamental bottleneck in the use of local homology—a variant of persistent homology that captures the topological structure of a space in the neighborhood of a point—for high‑dimensional point‑cloud data. Traditional reconstruction guarantees rely on embedded complexes such as Čech or α‑complexes. While mathematically sound, constructing these complexes requires explicit knowledge of the underlying embedding and suffers from exponential growth in the number of simplices as the ambient dimension increases, making them impractical beyond a modest number of dimensions.

To overcome this limitation, the authors propose a novel approximation scheme based entirely on Vietoris‑Rips (VR) complexes. The key insight is that, for a given query point (x) and a radius (r), the persistence diagram of the local homology of the intersection (B(x,r)\cap X) can be approximated by a family of VR complexes built on the same subset with a distance threshold (\varepsilon). By varying both (r) and (\varepsilon) one obtains a two‑parameter filtration whose associated multi‑parameter persistence module approximates the true local homology module.

The theoretical contributions are threefold. First, the authors prove a “sampling‑density” theorem: if the point cloud (X) is a (\delta)-net of the underlying compact space, then there exist intervals (