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The k-PDTM : a coreset for robust geometric inference

Abstract : Analyzing the sub-level sets of the distance to a compact sub-manifold of R d is a common method in TDA to understand its topology. The distance to measure (DTM) was introduced by Chazal, Cohen-Steiner and Mérigot in [7] to face the non-robustness of the distance to a compact set to noise and outliers. This function makes possible the inference of the topology of a compact subset of R d from a noisy cloud of n points lying nearby in the Wasserstein sense. In practice, these sub-level sets may be computed using approximations of the DTM such as the q-witnessed distance [10] or other power distance [6]. These approaches lead eventually to compute the homology of unions of n growing balls, that might become intractable whenever n is large. To simultaneously face the two problems of large number of points and noise, we introduce the k-power distance to measure (k-PDTM). This new approximation of the distance to measure may be thought of as a k-coreset based approximation of the DTM. Its sublevel sets consist in union of k-balls, k << n, and this distance is also proved robust to noise. We assess the quality of this approximation for k possibly dramatically smaller than n, for instance k = n 1 3 is proved to be optimal for 2-dimensional shapes. We also provide an algorithm to compute this k-PDTM.
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https://hal.archives-ouvertes.fr/hal-01694542
Contributor : Claire Brécheteau <>
Submitted on : Saturday, January 27, 2018 - 6:44:41 PM
Last modification on : Friday, April 10, 2020 - 5:26:45 PM
Document(s) archivé(s) le : Friday, May 25, 2018 - 11:05:56 AM

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  • HAL Id : hal-01694542, version 1
  • ARXIV : 1801.10346

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Claire Brécheteau, Clément Levrard. The k-PDTM : a coreset for robust geometric inference. 2018. ⟨hal-01694542⟩

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