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

Data analysis from empirical moments and the Christoffel function