A0272
Title: Wasserstein convergence of persistence diagrams on generic manifolds
Authors: Vincent Divol - Universite Paris Dauphine PSL (France) [presenting]
Abstract: Persistence diagrams (PDs) are routinely used in Topological Data Analysis to describe the topology of a sample in a multiscale fashion. They consist of a multiset of points in the upper half-plane, where each point in the PD intuitively corresponds to a topological feature of the underlying point cloud. When the sample lies on a submanifold of the Euclidean space, the PD of the sample (with respect to the Cech filtration) is known to be separated into two parts. A small number of points in the PD, which lie far away from the diagonal of the upper half-plane, correspond to the PD of the underlying manifold. On the other hand, a large collection of points lying close to the diagonal informally represents "topological noise". A complete asymptotic description of the structure of this topological noise is provided in the case where the sample lies on a generic submanifold. In particular, limit laws are offered for the total persistence of such PDs and prove convergence results with respect to Wasserstein distances. This generalizes previous results proven in another study in the case of points sampled in the cube $[0,1]^m$.
