A Sparse Multicover Bifiltration of Linear Size

Ángel Javier Alonso

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arXiv · 2411.06986

A Sparse Multicover Bifiltration of Linear Size

Ángel Javier Alonso

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Abstract

The $k$-cover of a point cloud $X$ in $\mathbb{R}^{d}$ at radius $r$ is the set of all points within distance $r$ of at least $k$ points of $X$. By varying $r$ and $k$ we obtain a two-parameter filtration known as the multicover bifiltration. This bifiltration has received attention recently due to being choice-free and robust to outliers. However, it is hard to compute: the smallest known equivalent simplicial bifiltration has $O(|X|^{d+1})$ simplices. We introduce a $(1+\epsilon)$-approximation of the multicover bifiltration of linear size $O(|X|)$, for fixed $d$ and $\epsilon$. The methods also apply to the subdivision Rips bifiltration on metric spaces of bounded doubling dimension, yielding analogous results.

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Ángel Javier Alonso. 2024-11-11. A Sparse Multicover Bifiltration of Linear Size. https://arxiv.org/abs/2411.06986

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