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Perturbations of Christoffel–Darboux Kernels: Detection of Outliers
Foundations of Computational Mathematics ( IF 3 ) Pub Date : 2020-04-20 , DOI: 10.1007/s10208-020-09458-9
Bernhard Beckermann , Mihai Putinar , Edward B. Saff , Nikos Stylianopoulos

Two central objects in constructive approximation, the Christoffel–Darboux kernel and the Christoffel function, encode 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 \(\mathbb {C}^d\), 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.



中文翻译:

Christoffel–Darboux内核的扰动:异常值的检测

结构上近似的两个中心对象Christoffel–Darboux内核和Christoffel函数对有关矩数据以及最终可能的生成度量的大量信息进行编码。我们在\(\ mathbb {C} ^ d \)中开发Christoffel–Darboux内核的多元理论。,重点是Christoffel函数的扰动及其相对于小规范或低等级扰动的水平集。杠杆得分的统计概念为检测大数据中的异常值提供了定量标准。使用Bergman正交多项式的精细理论,我们举例说明了2D区域面积测量的有限原子扰动的主要结果,包括一些数值模拟。复变量的函数理论和(复数)势理论的方法被广泛用于我们的扰动公式的推导中。

更新日期:2020-04-21
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