The empirical Christoffel function with applications in data analysis
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 o...
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| Published in | Advances in computational mathematics Vol. 45; no. 3; pp. 1439 - 1468 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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New York
Springer US
01.06.2019
Springer Nature B.V Springer Verlag |
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| 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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| AbstractList | 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. We illustrate the potential applications in machine learning of theChristoffel function, or more precisely, its empirical counterpart associatedwith a counting measure uniformly supported on a finite set of points.Firstly, we provide a thresholding scheme which allows to approximatethe support of a measure from a finite subset of its moments with strongasymptotic guaranties. Secondly, we provide a consistency result whichrelates the empirical Christoffel function and its population counterpartin the limit of large samples. Finally, we illustrate the relevance of ourresults on simulated and real world datasets for several applications instatistics and machine learning: (a) density and support estimation fromfinite samples, (b) outlier and novelty detection and (c) affine matching |
| Author | Pauwels, Edouard Lasserre, Jean B. |
| Author_xml | – sequence: 1 givenname: Jean B. surname: Lasserre fullname: Lasserre, Jean B. organization: LAAS-CNRS and Institute of Mathematics, University of Toulouse – sequence: 2 givenname: Edouard orcidid: 0000-0002-8180-075X surname: Pauwels fullname: Pauwels, Edouard email: edouard.pauwels@irit.fr organization: IRIT, Université Toulouse 3 Paul Sabatier |
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| Cites_doi | 10.1016/0021-9045(86)90016-X 10.1214/aoms/1177728190 10.2307/1971219 10.1007/s10474-012-0283-7 10.1111/j.1467-842X.2006.00421.x 10.1007/BF02788993 10.1214/13-AOS1188 10.1137/0138038 10.1016/j.aim.2009.06.010 10.1017/CBO9780511565717 10.2307/3315915 10.2307/2944352 10.1016/j.jmva.2016.01.003 10.1214/aos/1069362732 10.1214/aos/1030741073 10.1006/jmva.1995.1075 10.1214/07-AOP365 10.1214/08-AOS661 10.1214/15-AOS1366 10.3150/09-BEJ184 10.1214/aos/1176324533 10.1006/jath.1998.3312 10.1006/jath.1995.1075 10.1111/1467-9469.00100 10.1214/aoms/1177704472 10.1162/089976601750264965 10.1512/iumj.2009.58.3644 10.1214/aos/1176324626 10.1145/1143844.1143874 10.1145/1060590.1060636 10.4310/MAA.1996.v3.n2.a6 10.2139/ssrn.3239041 |
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| Keywords | Density estimation Support inference 68T05 Consistency 62-07 62H99 Christoffel function Statistics |
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| Snippet | We illustrate the potential applications in machine learning of the Christoffel function, or, more precisely, its empirical counterpart associated with a... We illustrate the potential applications in machine learning of theChristoffel function, or more precisely, its empirical counterpart associatedwith a counting... |
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| SubjectTerms | Artificial intelligence Asymptotic methods Computational mathematics Computational Mathematics and Numerical Analysis Computational Science and Engineering Computer Science Data analysis Empirical analysis Machine Learning Mathematical and Computational Biology Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Outliers (statistics) Statistical methods Visualization |
| Title | The empirical Christoffel function with applications in data analysis |
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