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The empirical Christoffel function with applications in data analysis

Authors
  • Lasserre, Jean B.1
  • Pauwels, Edouard2
  • 1 University of Toulouse, LAAS-CNRS and Institute of Mathematics, LAAS, 7 avenue du Colonel Roche, Toulouse, 31077, France , Toulouse (France)
  • 2 Université Toulouse 3 Paul Sabatier, IRIT, 118 route de Narbonne, Toulouse, 31062, France , Toulouse (France)
Type
Published Article
Journal
Advances in Computational Mathematics
Publisher
Springer US
Publication Date
Mar 07, 2019
Volume
45
Issue
3
Pages
1439–1468
Identifiers
DOI: 10.1007/s10444-019-09673-1
Source
Springer Nature
Keywords
License
Yellow

Abstract

We illustrate the potential applications in machine learning of the Christoffel function, or, more precisely, its empirical counterpart associated with a counting measure uniformly supported on a finite set of points. Firstly, we provide a thresholding scheme which allows approximating the support of a measure from a finite subset of its moments with strong asymptotic guaranties. Secondly, we provide a consistency result which relates the empirical Christoffel function and its population counterpart in the limit of large samples. Finally, we illustrate the relevance of our results on simulated and real-world datasets for several applications in statistics and machine learning: (a) density and support estimation from finite samples, (b) outlier and novelty detection, and (c) affine matching.

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