eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-08
26:1
26:15
10.4230/LIPIcs.SoCG.2018.26
article
The Density of Expected Persistence Diagrams and its Kernel Based Estimation
Chazal, Frédéric
Divol, Vincent
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol099-socg2018/LIPIcs.SoCG.2018.26/LIPIcs.SoCG.2018.26.pdf
topological data analysis
persistence diagrams
subanalytic geometry