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

## TL;DR

This paper develops a multivariate theory of Christoffel-Darboux kernels to detect outliers in large data sets by analyzing perturbations of the kernels and leveraging statistical measures like leverage scores.

## Contribution

It introduces a new multivariate framework for analyzing Christoffel-Darboux kernels under perturbations, enhancing outlier detection methods.

## Key findings

- Perturbation formulas for Christoffel functions and level sets.
- Leverage scores as a criterion for outlier detection.
- Numerical simulations demonstrating the theory in 2D cases.

## 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.