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Mathematics > Complex Variables

arXiv:1812.06560 (math)
[Submitted on 16 Dec 2018 (v1), last revised 27 Apr 2019 (this version, v2)]

Title:Perturbations of Christoffel-Darboux kernels. I: detection of outliers

Authors:Bernhard Beckermann, Mihai Putinar, Edward B. Saff, Nikos Stylianopoulos
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Abstract:Two central objects in constructive approximation, the Christoffel-Darboux kernel and the Christoffel function, are encoding ample information about the associated moment data and ultimately about the possible generating measures. We develop a multivariate theory of the Christoffel-Darboux kernel in C^d, with emphasis on the perturbation of Christoffel functions and their level sets with respect to perturbations of small norm or low rank. The statistical notion of leverage score provides a quantitative criterion for the detection of outliers in large data. Using the refined theory of Bergman orthogonal polynomials, we illustrate the main results, including some numerical simulations, in the case of finite atomic perturbations of area measure of a 2D region. Methods of function theory of a complex variable and (pluri)potential theory are widely used in the derivation of our perturbation formulas.
Comments: second version, 53 pages
Subjects: Complex Variables (math.CV)
MSC classes: 34E10, 42C05, 46E22, 30H20, 30E05
Cite as: arXiv:1812.06560 [math.CV]
  (or arXiv:1812.06560v2 [math.CV] for this version)
  https://doi.org/10.48550/arXiv.1812.06560

Submission history

From: Mihai Putinar [view email]
[v1] Sun, 16 Dec 2018 23:54:57 UTC (664 KB)
[v2] Sat, 27 Apr 2019 22:37:11 UTC (668 KB)
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