Efficient and Robuste persistent homology for measures
Résumé
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.