The density of expected persistence diagrams and its kernel based estimation

Frédéric Chazal; Vincent Divol

Communication Dans Un Congrès Année : 2018

The density of expected persistence diagrams and its kernel based estimation

(1) , (1, 2)

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Frédéric Chazal

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Understanding the Shape of Data

Vincent Divol

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IdHAL : vincent-divol
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Understanding the Shape of Data

Model selection in statistical learning

Résumé

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 Čech 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 & al., Persistence images: a stable vector representation of persistent homology] 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.

Mots clés

subanalytic geometry persistence diagrams topological data analysis

Domaines

Géométrie algorithmique [cs.CG] Théorie [stat.TH]

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Soumis le : vendredi 23 février 2018-14:03:35

Dernière modification le : jeudi 21 novembre 2024-14:36:02

Archivage à long terme le : vendredi 25 mai 2018-15:04:21

Dates et versions

hal-01716181 , version 1 (23-02-2018)

hal-01716181 , version 2 (22-03-2019)

hal-01716181 , version 3 (25-04-2019)

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HAL Id : hal-01716181 , version 1

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Frédéric Chazal, Vincent Divol. The density of expected persistence diagrams and its kernel based estimation. Symposium of Computational Geometry (SoCG 2018), Jun 2018, Budapest, Hungary. ⟨hal-01716181v1⟩

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The density of expected persistence diagrams and its kernel based estimation

Résumé

Mots clés

Domaines

Dates et versions

Identifiants

Citer

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