Moment Varieties of Gaussian Mixtures

Abstract

The points of a moment variety are the vectors of all moments up to some order, for a givenfamily 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 algebrato recover the parameters from the moments. Our moment varieties extend objects familiar to algebraicgeometers. For instance, the secant varieties of Veronese varieties are the loci obtained by setting allcovariance matrices to zero. We compute the ideals of the 5-dimensional moment varieties representingmixtures of two univariate Gaussians, and we oer a comparison to the maximum likelihood approach.