The empirical Christoffel function with applications in data analysis
This work addresses foundational challenges in data analysis for statisticians and machine learning practitioners, but it appears incremental as it builds on existing Christoffel function theory.
The paper tackles the problem of approximating measure supports and relating empirical to population Christoffel functions, providing strong asymptotic guarantees and consistency results, with applications demonstrated on simulated and real datasets for tasks like density estimation and outlier detection.
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.