The empirical Christoffel function with applications in data analysis

• Published in: Advances in computational mathematics Vol. 45; no. 3; pp. 1439 - 1468
• Main Authors: Lasserre, Jean B., Pauwels, Edouard
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2019; Springer Nature B.V; Springer Verlag
• Subjects: 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; Density estimation; Support inference; 68T05; Consistency; 62-07; 62H99; Christoffel function; Statistics
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-019-09673-1

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.