2 research outputs found
Sparse Higher Order ?ech Filtrations
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
Sparse Higher Order \v{C}ech Filtrations
For a finite set of balls of radius , the -fold cover is the space
covered by at least balls. Fixing the ball centers and varying the radius,
we obtain a nested sequence of spaces that is called the -fold filtration of
the centers. For , the construction is the union-of-balls filtration that
is popular in topological data analysis. For larger , it yields a cleaner
shape reconstruction in the presence of outliers. We contribute a
sparsification algorithm to approximate the topology of the -fold
filtration. Our method is a combination and adaptation of several techniques
from the well-studied case , 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 -fold filtrations for several values of , with the same size and
complexity bounds.Comment: Extended journal versio