License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2018.26
URN: urn:nbn:de:0030-drops-87395
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8739/
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Chazal, Frédéric ; Divol, Vincent

The Density of Expected Persistence Diagrams and its Kernel Based Estimation

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LIPIcs-SoCG-2018-26.pdf (0.8 MB)

Abstract

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.

BibTeX - Entry

@InProceedings{chazal_et_al:LIPIcs:2018:8739,
  author =	{Fr{\'e}d{\'e}ric Chazal and Vincent Divol},
  title =	{{The Density of Expected Persistence Diagrams and its Kernel Based Estimation}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{26:1--26:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Bettina Speckmann and Csaba D. T{\'o}th},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8739},
  URN =		{urn:nbn:de:0030-drops-87395},
  doi =		{10.4230/LIPIcs.SoCG.2018.26},
  annote =	{Keywords: topological data analysis, persistence diagrams, subanalytic geometry}
}

Keywords: topological data analysis, persistence diagrams, subanalytic geometry
Collection: 34th International Symposium on Computational Geometry (SoCG 2018)
Issue Date: 2018
Date of publication: 08.06.2018

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