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Journal Articles Discrete and Computational Geometry Year : 2013

Linear-size approximations to the vietoris-rips filtration

Approximations de la filtration de Vietoris-Rips de taille linéaire

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Donald Sheehy
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Abstract

The Vietoris–Rips filtration is a versatile tool in topological data analysis. It is a sequence of simplicial complexes built on a metric space to add topological structure to an otherwise disconnected set of points. It is widely used because it encodes useful information about the topology of the underlying metric space. This information is of-ten extracted from its so-called persistence diagram. Unfortunately, this filtration is often too large to construct in full. We show how to construct an O(n)-size filtered simplicial complex on an n-point metric space such that its persistence diagram is a good approximation to that of the Vietoris–Rips filtration. This new filtration can be constructed in O(n log n) time. The constant factors in both the size and the run-ning time depend only on the doubling dimension of the metric space and the desired tightness of the approximation. For the first time, this makes it computationally tractable to approximate the persistence di-agram of the Vietoris–Rips filtration across all scales for large data sets. We describe two different sparse filtrations. The first is a zigzag filtration that removes points as the scale increases. The second is a (non-zigzag) filtration that yields the same persistence diagram. Both methods are based on a hierarchical net-tree and yield the same guar-antees.
La filtration de Vietoris-Rips est un outil très versatile en analyse topologique des données. C'est une séquence de complexes simpliciaux construits sur une métrique pour ajouter de la structure topologique à un nuage de points. Malheureusement, cette filtration est souvent trop large pour tenir entièerement en mémoire. Nous montrons comment construire un complexe simplicial filtré de taille O(n) à partir d'un espace métrique fini composé de n points, de manièere à ce que le diagramme de persistance de ce complexe filtré soit une bonne approximation de celui de la filtration de Vietoris-Rips.
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Dates and versions

hal-01111878 , version 1 (31-01-2015)

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Donald Sheehy. Linear-size approximations to the vietoris-rips filtration. Discrete and Computational Geometry, 2013, pp.778-796. ⟨10.1145/2261250.2261286⟩. ⟨hal-01111878⟩

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