↪︎ Core Bifiltration
The core bifiltration is a parameter-free density-based bifiltration interleaved with the well-known multicover bifiltration. This makes it suitable for extracting topological features from noisy point cloud data. In particulat, both bifiltrations enjoy stability with respect to the Prohorov distance between the point measures. For point clouds in $\mathbb{R}^n$, a smaller and more efficient version based on the Delaunay complex also exists: The Delaunay Core bifiltration is implemented in the multipers library making it accessible to both researchers and practitioners.
For a quick start: check out the multipers documentation for a tutorial on how to use the Delaunay Core bifiltration in Python.
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, and computed multipersistence module approximations.
