2026-08-16T23:17:17Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:8739 2024-03-06T10:42:32Z ddc:004 open_access

The Density of Expected Persistence Diagrams and its Kernel Based Estimation Chazal, Frédéric Divol, Vincent topological data analysis persistence diagrams subanalytic geometry Persistence diagrams play a fundamental role in Topological Data Analysis where they are used as topological descriptors of filtrations built on top of data. They consist in discrete multisets of points in the plane R^2 that can equivalently be seen as discrete measures in R^2. When the data come as a random point cloud, these discrete measures become random measures whose expectation is studied in this paper. First, we show that for a wide class of filtrations, including the Cech and Rips-Vietoris filtrations, the expected persistence diagram, that is a deterministic measure on R^2, has a density with respect to the Lebesgue measure. Second, building on the previous result we show that the persistence surface recently introduced in [Adams et al., 2017] can be seen as a kernel estimator of this density. We propose a cross-validation scheme for selecting an optimal bandwidth, which is proven to be a consistent procedure to estimate the density. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Frédéric Chazal and Vincent Divol 2018 Is Part Of LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2018.26 urn:nbn:de:0030-drops-87395 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.26 eng https://creativecommons.org/licenses/by/3.0/legalcode