Computing extended persistent homology of radial distance filtrations of Euclidean shapes

What the math gives to ML

The paper provides a concrete radial-filtration construction for extracting extended topological structure from Euclidean shapes, together with a reduction from a manifold to its boundary and a certified discretization error bound. The transferable asset is not persistent homology by itself, but the ability to replace an expensive volumetric topology computation with signed interval computations on a lower-dimensional boundary and to relate pixelwise approximations to a known interleaving distance. This suggests topology-aware image losses and shape representations that focus on radial connectivity, holes, and essential classes around a chosen anatomical or object center. The most practical first use is a boundary-based radial persistence loss or feature encoder, followed by experiments testing whether it preserves shape topology at lower memory and computation than volumetric persistence.