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2 results on '"62G05, 62G30, 62-07"'

1. Rates of convergence for robust geometric inference

Author

Chazal, Frédéric, Massart, Pascal, and Michel, Bertrand

Subjects
Mathematics - Statistics Theory, Computer Science - Computational Geometry, 62G05, 62G30, 62-07
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. (2011) 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 rates of convergence of the DTEM directly depends on the regularity at zero of a particular quantile fonction 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.

Published
2015

2. Rates of convergence for robust geometric inference

Author

Bertrand Michel, Frédéric Chazal, Pascal Massart, Geometric computing (GEOMETRICA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria), Model selection in statistical learning (SELECT), Laboratoire de Mathématiques d'Orsay (LMO), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire de Mathématiques d'Orsay (LM-Orsay), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Statistique Théorique et Appliquée (LSTA), Université Pierre et Marie Curie - Paris 6 (UPMC), Understanding the Shape of Data (DATASHAPE), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de Mathématiques d'Orsay (LMO), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), Université Pierre et Marie Curie - Paris 6 (UPMC)-Centre National de la Recherche Scientifique (CNRS), ANR-13-BS01-0008,TopData,Analyse Topologique des Données : Méthodes Statistiques et Estimation(2013), European Project: 339025,EC:FP7:ERC,ERC-2013-ADG,GUDHI(2014), INRIA, Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11), Michel, Bertrand, Chazal, Frédéric, Analyse Topologique des Données : Méthodes Statistiques et Estimation - - TopData2013 - ANR-13-BS01-0008 - Blanc 2013 - VALID, and Algorithmic Foundations of Geometry Understanding in Higher Dimensions - GUDHI - - EC:FP7:ERC2014-02-01 - 2019-01-31 - 339025 - VALID

Subjects
Statistics and Probability, Computational Geometry (cs.CG), FOS: Computer and information sciences, 62G30, Mathematical optimization, Geometric inference, rates of convergence, Context (language use), Mathematics - Statistics Theory, 02 engineering and technology, Statistics Theory (math.ST), [INFO.INFO-CG]Computer Science [cs]/Computational Geometry [cs.CG], 01 natural sciences, 010104 statistics & probability, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Convergence (routing), 0202 electrical engineering, electronic engineering, information engineering, FOS: Mathematics, Applied mathematics, 62G05, 0101 mathematics, [MATH.MATH-ST] Mathematics [math]/Statistics [math.ST], Mathematics, Probability measure, distance to measure, [STAT.TH] Statistics [stat]/Statistics Theory [stat.TH], [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], Quantile function, Empirical measure, 68U05, 62-07, Compact space, Rate of convergence, [INFO.INFO-CG] Computer Science [cs]/Computational Geometry [cs.CG], Computer Science - Computational Geometry, 020201 artificial intelligence & image processing, Topological data analysis, 28A33, Statistics, Probability and Uncertainty, 62G05, 62G30, 62-07
Abstract

International audience; 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.

Published
2015

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