Computing the Multicover Bifiltration

TL;DR

This paper introduces two efficient combinatorial models for computing the homology of the multicover bifiltration, a 2-parameter family of spaces based on proximity and coverage, improving computational feasibility.

Contribution

The authors develop polyhedral and simplicial bifiltrations equivalent to the multicover bifiltration, reducing complexity compared to previous Čech-based models, and provide algorithms and experimental results.

Findings

Polyhedral bifiltration is based on the rhomboid tiling and can be computed efficiently.

Simplicial bifiltration aids in understanding and validating the polyhedral model.

Experimental results demonstrate the approach's effectiveness in dimensions 2 and 3.

Abstract

Given a finite set $A\subset\mathbb{R}^d$, let Cov$_{r,k}$ denote the set of all points within distance $r$ to at least $k$ points of $A$. Allowing $r$ and $k$ to vary, we obtain a 2-parameter family of spaces that grow larger when $r$ increases or $k$ decreases, called the \emph{multicover bifiltration}. Motivated by the problem of computing the homology of this bifiltration, we introduce two closely related combinatorial bifiltrations, one polyhedral and the other simplicial, which are both topologically equivalent to the multicover bifiltration and far smaller than a \v Cech-based model considered in prior work of Sheehy. Our polyhedral construction is a bifiltration of the rhomboid tiling of Edelsbrunner and Osang, and can be efficiently computed using a variant of an algorithm given by these authors. Using an implementation for dimension 2 and 3, we provide experimental results. Our simplicial construction is useful for understanding the polyhedral construction and proving its correctness.