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
2 UCONN - University of Connecticut

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.

Mots clés: Persistent Homology; Rips complex; Distance to a measure

Domaines: Géométrie algorithmique [cs.CG]

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⟩