NDLI: Geometric Inference for Probability Measures

Content Provider	Springer Nature Link
Author	Cohen Steiner, David Mérigot, Quentin Chazal, Frédéric
Copyright Year	2011
Abstract	Data often comes in the form of a point cloud sampled from an unknown compact subset of Euclidean space. The general goal of geometric inference is then to recover geometric and topological features (e.g., Betti numbers, normals) of this subset from the approximating point cloud data. It appears that the study of distance functions allows one to address many of these questions successfully. However, one of the main limitations of this framework is that it does not cope well with outliers or with background noise. In this paper, we show how to extend the framework of distance functions to overcome this problem. Replacing compact subsets by measures, we introduce a notion of distance function to a probability distribution in ℝ d . These functions share many properties with classical distance functions, which make them suitable for inference purposes. In particular, by considering appropriate level sets of these distance functions, we show that it is possible to reconstruct offsets of sampled shapes with topological guarantees even in the presence of outliers. Moreover, in settings where empirical measures are considered, these functions can be easily evaluated, making them of particular practical interest.
Ending Page	751
Page Count	19
Starting Page	733
File Format	PDF
ISSN	16153375
e-ISSN	16153383
Journal	Foundations of Computational Mathematics
Issue Number	6
Volume Number	11
Language	English
Publisher	Springer-Verlag
Publisher Date	2011-07-28
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Data analysis Economics general Estimation Linear and Multilinear Algebras, Matrix Theory Spaces of measures, convergence of measures Computational topology Geometric inference Optimal transportation Surface reconstruction Numerical Analysis Computer Science Nearest neighbor Applications of Mathematics Math Applications in Computer Science
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis Computational Theory and Mathematics Computational Mathematics

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