Moment Varieties of Gaussian Mixtures

Abstract

The points of a moment variety are the vectors of all moments up to some order of a family of probability distributions. We study this variety for mixtures of 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 offer a comparison to the maximum likelihood approach.

Moment varieties of measures on polytopes

“…Since mixture models correspond to secant varieties in P r , we can phrase our question as follows: study the secant varieties of the surfaces M {{r}} (1, 2) in Example 3.5. Pearson's hypersurface of degree 39 in [1,Theorem 1] suggests that this will not be easy.…”

Varieties of Signature Tensors

“…We focus on Brownian motion and mixtures of Brownian motion. This leads to a non-abelian refinement of the moment varieties in [2,3]. The varieties of expected signatures are computed in several cases.…”

“…Real Signature Matrices. We fix positive integers m ≤ d. Let S [d,m] axis denote the d×d matrix whose upper left m×m block is the upper triangular matrix σ (2)…”

Tapas of Algebraic Statistics

“…Recent progress with this approach has been made by Améndola et al [2,3], with partial answers. For example, it was shown [3] that considering all the moments up to order 3 − 1 will yield generically a finite number of Gaussian mixture densities with the same matching moments.…”

“…For = 2 this is Pearson's number 9. For = 3 it was found [2] that the corresponding degree is 225. In contrast, perhaps shockingly, the system of 20 polynomial equations in 20 unknowns corresponding to the moments up to order three of mixtures of two Gaussians in 3-dimensional space ℝ 3 will have generically infinitely many solutions.…”