A k-points-based distance 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 topological data analysis, to understand its topology. Therefore, topological inference procedures usually rely on a distance estimate based on n sample points [41]. In the case where sample points are corrupted by noise, the distance-to-measure function (DTM, [16]) is a surrogate for the distance-to-compact-set function. In practice, computing the homology of its sub-level sets requires to compute the homology of unions of n balls ([28, 14]), that might become intractable whenever n is large. To simultaneously face the two problems of a large number of points and noise, we introduce the k-power-distance-to-measure function (k-PDTM). This new approximation of the distance-to-measure may be thought of as a k-pointbased approximation of the DTM. Its sublevel sets consist in unions of k balls, and this distance is also proved robust to noise. We assess the quality of this approximation for k possibly drastically smaller than n, and provide an algorithm to compute this k-PDTM from a sample. Numerical experiments illustrate the good behavior of this k-points approximation in a noisy topological inference framework.
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Submitted on : Wednesday, August 14, 2019 - 12:03:16 PM
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Claire Brécheteau, Clément Levrard. A k-points-based distance for robust geometric inference. 2019. ⟨hal-02266408⟩

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