Intrinsic persistent homology via density-based metric learning

BORGHINI, EUGENIO; MINDLIN, GABRIEL; FERNANDEZ, XIMENA; GROISMAN, PABLO

ArXiv

Cornell University

Lugar: Cornell; Año: 2020

2331-8422

We address the problem of estimating intrinsic distances in a manifold from afinite sample. We prove that the metric space defined by the sample endowedwith a computable metric known as sample Fermat distance converges a.s. in thesense of Gromov-Hausdorff. The limiting object is the manifold itself endowedwith the population Fermat distance, an intrinsic metric that accounts for boththe geometry of the manifold and the density that produces the sample. Thisresult is applied to obtain sample persistence diagrams that converge towardsan intrinsic persistence diagram. We show that this method outperforms morestandard approaches based on Euclidean norm with theoretical results andcomputational experiments.