# The empirical Christoffel function with applications in data analysis

## TL;DR

This paper explores the empirical Christoffel function's applications in machine learning, providing methods for support estimation, consistency results, and demonstrating its effectiveness in density estimation, outlier detection, and affine matching.

## Contribution

It introduces a thresholding scheme for support approximation, establishes a consistency result between empirical and population Christoffel functions, and demonstrates practical applications in data analysis.

## Key findings

- Effective support estimation from finite samples.
- Consistency between empirical and population Christoffel functions.
- Successful applications in density estimation, outlier detection, and affine matching.

## 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 to approximate 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.