Moment Varieties of Gaussian Mixtures

  • Carlos Amendola TU Berlin
  • Jean-Charles Faugere Sorbonne Universites, UPMC Univ Paris 06, CNRS, INRIA, Laboratoire d’Informatique de Paris 6 (LIP6), Equipe PolSys, 4 place Jussieu 75005 Paris
  • Bernd Sturmfels UC Berkeley
Keywords: Algebraic statistics, method of moments, mixture model, normal distribution, secant varieties


The points of a moment variety are the vectors of all moments up to some order, for a given
family of probability distributions. We study the moment varieties for mixtures of multivariate Gaussians.
Following up on Pearson's classical work from 1894, we apply current tools from computational algebra
to recover the parameters from the moments. Our moment varieties extend objects familiar to algebraic
geometers. For instance, the secant varieties of Veronese varieties are the loci obtained by setting all
covariance matrices to zero. We compute the ideals of the 5-dimensional moment varieties representing
mixtures of two univariate Gaussians, and we oer a comparison to the maximum likelihood approach.

AS2015 Special Issue articles