# Persistent Homology as Stopping-Criterion for Voronoi Interpolation

arXiv: https://arxiv.org/abs/1911.02922 · HTML: https://mlsanity.com/paper/1911.02922/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Nov 2019.

## Built on by (most cited first, 3 of 3)

- [Topology-Aware 3D Gaussian Splatting: Leveraging Persistent Homology for Optimized Structural Integrity](https://mlsanity.com/paper/2412.16619/) (2024)
- [Homological Time Series Analysis of Sensor Signals from Power Plants](https://mlsanity.com/paper/2106.02493/) (2021)
- [Learning Transfers: Kan Extensions of Topological Invariants of Learned Representations](https://mlsanity.com/paper/2606.07627/) (2026)

## Abstract (arXiv)

In this study the Voronoi interpolation is used to interpolate a set of points drawn from a topological space with higher homology groups on its filtration. The technique is based on Voronoi tessellation, which induces a natural dual map to the Delaunay triangulation. Advantage is taken from this fact calculating the persistent homology on it after each iteration to capture the changing topology of the data. The boundary points are identified as critical. The Bottleneck and Wasserstein distance serve as a measure of quality between the original point set and the interpolation. If the norm of two distances exceeds a heuristically determined threshold, the algorithm terminates. We give the theoretical basis for this approach and justify its validity with numerical experiments.