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Efficient and Robuste persistent homology for measures

Mickael Buchet 1 Frédéric Chazal 2 Steve Oudot 1 Donald Sheehy 1
1 GEOMETRICA - Geometric computing
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
2 DATASHAPE - Understanding the Shape of Data
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 de ned 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 ecient 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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Submitted on : Monday, September 19, 2016 - 5:28:36 PM
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Mickael Buchet, Frédéric Chazal, Steve Oudot, Donald Sheehy. Efficient and Robuste persistent homology for measures. Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms , Jan 2015, San Diego, United States. ⟨10.1137/1.9781611973730.13⟩. ⟨hal-01368613⟩



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