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Conference Papers Year : 2015

Efficient and Robuste persistent homology for measures

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Frédéric Chazal
Steve Oudot
  • Function : Author
  • PersonId : 845393
Donald Sheehy
  • Function : Author
  • PersonId : 950510

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.

Dates and versions

hal-01368613 , version 1 (19-09-2016)

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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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