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Rates of convergence for robust geometric inference

F Chazal 1 P Massart 2, 3 B Michel 4
1 DATASHAPE - Understanding the Shape of Data
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
3 SELECT - Model selection in statistical learning
LMO - Laboratoire de Mathématiques d'Orsay, Inria Saclay - Ile de France
Abstract : Distances to compact sets are widely used in the field of Topological Data Analysis for inferring geometric and topological features from point clouds. In this context, the distance to a probability measure (DTM) has been introduced by Chazal et al. (2011b) as a robust alternative to the distance a compact set. In practice, the DTM can be estimated by its empirical counterpart, that is the distance to the empirical measure (DTEM). In this paper we give a tight control of the deviation of the DTEM. Our analysis relies on a local analysis of empirical processes. In particular, we show that the rate of convergence of the DTEM directly depends on the regularity at zero of a particular quantile function which contains some local information about the geometry of the support. This quantile function is the relevant quantity to describe precisely how difficult is a geometric inference problem. Several numerical experiments illustrate the convergence of the DTEM and also confirm that our bounds are tight.
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Contributor : Frédéric Chazal Connect in order to contact the contributor
Submitted on : Wednesday, October 19, 2016 - 9:55:23 AM
Last modification on : Tuesday, July 13, 2021 - 3:14:11 AM


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  • HAL Id : hal-01336913, version 2


F Chazal, P Massart, B Michel. Rates of convergence for robust geometric inference. Electronic Journal of Statistics , Shaker Heights, OH : Institute of Mathematical Statistics, 2016, 10 (2), pp.44. ⟨hal-01336913v2⟩



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