Core Bifiltration

TL;DR

This paper introduces the core bifiltration, a new method inspired by HDBSCAN for recognizing shapes from noisy point clouds, with stability guarantees and efficient computations using Delaunay and Voronoi structures.

Contribution

It proposes the core bifiltration and Delaunay core bifiltration, providing a geometric and computational framework with stability analysis and experimental validation.

Findings

Core bifiltration effectively captures shape features from noisy data.

Delaunay core bifiltration admits a good cover, enabling homotopy equivalence.

Experiments demonstrate the method's ability to compute persistent homology and multipersistence modules.

Abstract

The motivation of this paper is to recognize a geometric shape from a noisy sample in the form of a point cloud. Inspired by the HDBSCAN clustering algorithm, we introduce the core dissimilarity, from which we construct the core bifiltration. We also consider the Delaunay core bifiltration by intersecting with Voronoi cells, giving us a filtered simplicial complex of smaller size. A major advantage of the (Delaunay) core bifiltration is that, for each filtration value, it admits a good cover of balls. By the persistent nerve theorem, the nerve of this cover is homotopy equivalent to the (Delaunay) core bifiltration. We show that the multicover-, core- and Delaunay core bifiltrations are all interleaved, and that they enjoy similar stability properties with respect to the Prohorov distance. We have performed experiments with the Delaunay core bifiltration. In the experiments, we calculated persistent homology along lines in the two-dimensional persistence parameter space, as well as multipersistence module approximations and Hilbert functions for the full Delaunay core bifiltration.