Data analysis from empirical moments and the Christoffel function

TL;DR

This paper explores how spectral features of the empirical moment matrix can reveal key properties of data distributions, integrating advanced mathematical tools to enhance data analysis in machine learning.

Contribution

It introduces novel theoretical insights and practical methods combining algebraic geometry and approximation theory to analyze data supported on complex sets.

Findings

Empirical moment matrix encodes subtle data attributes.

The approach effectively uncovers density and support structures.

Numerical experiments validate the method's applicability to real data.

Abstract

Spectral features of the empirical moment matrix constitute a resourceful tool for unveiling properties of a cloud of points, among which, density, support and latent structures. It is already well known that the empirical moment matrix encodes a great deal of subtle attributes of the underlying measure. Starting from this object as base of observations we combine ideas from statistics, real algebraic geometry, orthogonal polynomials and approximation theory for opening new insights relevant for Machine Learning (ML) problems with data supported on singular sets. Refined concepts and results from real algebraic geometry and approximation theory are empowering a simple tool (the empirical moment matrix) for the task of solving non-trivial questions in data analysis. We provide (1) theoretical support, (2) numerical experiments and, (3) connections to real world data as a validation of the stamina of the empirical moment matrix approach.