Advances in Computational Mathematics

, Volume 45, Issue 3, pp 1439–1468 | Cite as

The empirical Christoffel function with applications in data analysis

  • Jean B. Lasserre
  • Edouard PauwelsEmail author


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.


Christoffel function Statistics Support inference Density estimation Consistency 

Mathematics Subject Classification (2010)

62-07 62H99 68T05 


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The research of the first author was funded by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement 666981 TAMING).


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Authors and Affiliations

  1. 1.LAAS-CNRS and Institute of MathematicsUniversity of ToulouseToulouseFrance
  2. 2.IRITUniversité Toulouse 3 Paul SabatierToulouseFrance

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