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Efficient and Robust Persistent Homology for Measures

Mickaël Buchet 1 Frédéric Chazal 1 Steve Yann Oudot 1 Donald R. Sheehy 2
1 GEOMETRICA - Geometric computing
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
Abstract : A new paradigm for point cloud data analysis has emerged recently, where point clouds are no longer treated as mere compact sets but rather as empirical measures. A notion of distance to such measures has been defined and shown to be stable with respect to perturbations of the measure. This distance can eas-ily be computed pointwise in the case of a point cloud, but its sublevel-sets, which carry the geometric infor-mation about the measure, remain hard to compute or approximate. This makes it challenging to adapt many powerful techniques based on the Euclidean distance to a point cloud to the more general setting of the distance to a measure on a metric space. We propose an efficient and reliable scheme to approximate the topological structure of the family of sublevel-sets of the distance to a measure. We obtain an algorithm for approximating the persistent homology of the distance to an empirical measure that works in arbitrary metric spaces. Precise quality and complexity guarantees are given with a discussion on the behavior of our approach in practice.
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https://hal.inria.fr/hal-01074566
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Submitted on : Tuesday, October 14, 2014 - 6:08:29 PM
Last modification on : Tuesday, June 18, 2019 - 2:36:02 PM
Long-term archiving on: : Thursday, January 15, 2015 - 10:41:01 AM

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Mickaël Buchet, Frédéric Chazal, Steve Yann Oudot, Donald R. Sheehy. Efficient and Robust Persistent Homology for Measures. ACM-SIAM Symposium on Discrete Algorithms, Jan 2015, San Diego, United States. ⟨hal-01074566⟩

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