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Topological Inference via Meshing

Benoît Hudson 1 Gary Miller 2 Steve Oudot 3, 4 Donald Sheehy 2
3 GEOMETRICA - Geometric computing
CRISAM - Inria Sophia Antipolis - Méditerranée
4 GEOMETRICA - Geometric computing
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
Abstract : We apply ideas from mesh generation to improve the time and space complexities of computing the full persistent homological information associated with a point cloud $P$ in Euclidean space $\R^d$. Classical approaches rely on the \v Cech, Rips, $\alpha$-complex, or witness complex filtrations of $P$, whose complexities scale up very badly with $d$. For instance, the $\alpha$-complex filtration incurs the $n^{\Omega(d)}$ size of the Delaunay triangulation, where $n$ is the size of $P$. The common alternative is to truncate the filtrations when the sizes of the complexes become prohibitive, possibly before discovering the most relevant topological features. In this paper we propose a new collection of filtrations, based on the Delaunay triangulation of a carefully-chosen superset of $P$, whose sizes are reduced to $2^{O(d^2)}n$. A nice property of these filtrations is to be interleaved multiplicatively with the family of offsets of $P$, so that the persistence diagram of $P$ can be approximated in $2^{O(d^2)}n^3$ time in theory, with a near-linear observed running time in practice (ignoring the constant factors depending exponentially on $d$). Thus, our approach remains tractable in medium dimensions, say 4 to 10.
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Submitted on : Tuesday, December 1, 2009 - 12:03:35 AM
Last modification on : Friday, June 11, 2021 - 5:12:08 PM
Long-term archiving on: : Tuesday, October 16, 2012 - 3:01:30 PM


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  • HAL Id : inria-00436891, version 1


Benoît Hudson, Gary Miller, Steve Oudot, Donald Sheehy. Topological Inference via Meshing. [Research Report] RR-7125, 2009. ⟨inria-00436891v1⟩



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