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DOI: 10.4230/LIPIcs.SoCG.2018.26

The Density of Expected Persistence Diagrams and its Kernel Based Estimation

Authors: Frédéric Chazal and Vincent Divol

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

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.

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Frédéric Chazal and Vincent Divol. The Density of Expected Persistence Diagrams and its Kernel Based Estimation. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chazal_et_al:LIPIcs.SoCG.2018.26,
  author =	{Chazal, Fr\'{e}d\'{e}ric and Divol, Vincent},
  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 =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.26},
  URN =		{urn:nbn:de:0030-drops-87395},
  doi =		{10.4230/LIPIcs.SoCG.2018.26},
  annote =	{Keywords: topological data analysis, persistence diagrams, subanalytic geometry}
}

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The Density of Expected Persistence Diagrams and its Kernel Based Estimation

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