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arXiv · 2303.06666

Sparse Higher Order \v{C}ech Filtrations

Mickaël Buchet·Bianca B. Dornelas·Michael Kerber
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Abstract

For a finite set of balls of radius $r$, the $k$-fold cover is the space covered by at least $k$ balls. Fixing the ball centers and varying the radius, we obtain a nested sequence of spaces that is called the $k$-fold filtration of the centers. For $k=1$, the construction is the union-of-balls filtration that is popular in topological data analysis. For larger $k$, it yields a cleaner shape reconstruction in the presence of outliers. We contribute a sparsification algorithm to approximate the topology of the $k$-fold filtration. Our method is a combination and adaptation of several techniques from the well-studied case $k=1$, resulting in a sparsification of linear size that can be computed in expected near-linear time with respect to the number of input points. Our method also extends to the multicover bifiltration, composed of the $k$-fold filtrations for several values of $k$, with the same size and complexity bounds.

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Mickaël Buchet, Bianca B. Dornelas, Michael Kerber. 2023-03-12. Sparse Higher Order \v{C}ech Filtrations. https://arxiv.org/abs/2303.06666

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