Intrinsic Persistent Homology via density-based metric learning

Speaker: Ximena Fernández (Swansea University)

Occasion: Topological Data Analysis

Date: April 28, 2021

Abstract: 

Typically, persistence diagrams computed from a sample depend strongly on the distance associated to the data. When the point cloud is a sample of a Riemannian manifold embedded in a Euclidean space, an estimator of the intrinsic distance is relevant to obtain persistence diagrams from data that capture its intrinsic geometry.

In this talk, we consider a computable estimator of a Riemannian metric known as Fermat distance, that accounts for both the geometry of the manifold and the density that produces the sample. We prove that the metric space defined by the sample endowed with this estimator (known as sample Fermat distance) converges a.s. in the sense of Gromov-Hausdorff to the manifold itself endowed with the (population) Fermat distance. This result is applied to obtain sample persistence diagrams that converge towards an intrinsic persistence diagram. We show that this approach outperforms more standard methods based on Euclidean norm, with theoretical results and computational experiments [1].

[1] E. Borghini, X. Fernández, P. Groisman, G. Mindlin. ‘Intrinsic persistent homology via density-based metric learning’. arXiv:2012.07621 (2020)