23 research outputs found
Algebraic Identifiability of Gaussian Mixtures
We prove that all moment varieties of univariate Gaussian mixtures have the
expected dimension. Our approach rests on intersection theory and Terracini's
classification of defective surfaces. The analogous identifiability result is
shown to be false for mixtures of Gaussians in dimension three and higher.
Their moments up to third order define projective varieties that are defective.
Our geometric study suggests an extension of the Alexander-Hirschowitz Theorem
for Veronese varieties to the Gaussian setting.Comment: 18 pages, to appear in International Mathematics Research Notice
Moment Identifiability of Homoscedastic Gaussian Mixtures
We consider the problem of identifying a mixture of Gaussian distributions
with same unknown covariance matrix by their sequence of moments up to certain
order. Our approach rests on studying the moment varieties obtained by taking
special secants to the Gaussian moment varieties, defined by their natural
polynomial parametrization in terms of the model parameters. When the order of
the moments is at most three, we prove an analogue of the Alexander-Hirschowitz
theorem classifying all cases of homoscedastic Gaussian mixtures that produce
defective moment varieties. As a consequence, identifiability is determined
when the number of mixed distributions is smaller than the dimension of the
space. In the two component setting we provide a closed form solution for
parameter recovery based on moments up to order four, while in the one
dimensional case we interpret the rank estimation problem in terms of secant
varieties of rational normal curves.Comment: 27 pages, 1 table, 1 figur
Moment Varieties of Gaussian Mixtures
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.Comment: 17 pages, 2 figure
Likelihood geometry of correlation models
Correlation matrices are standardized covariance matrices. They form an affine space of symmetric matrices defined by setting the diagonal entries to one. We study the geometry of maximum likelihood estimation for this model and linear submodels that encode additional symmetries. We also consider the problem of minimizing two closely related functions of the covariance matrix: the Stein's loss and the symmetrized Stein's loss. Unlike the Gaussian log-likelihood these two functions are convex and hence admit a unique positive definite optimum. Some of our results hold for general affine covariance models
Convex Hulls of Curves: Volumes and Signatures
Taking the convex hull of a curve is a natural construction in computational
geometry. On the other hand, path signatures, central in stochastic analysis,
capture geometric properties of curves, although their exact interpretation for
levels larger than two is not well understood. In this paper, we study the use
of path signatures to compute the volume of the convex hull of a curve. We
present sufficient conditions for a curve so that the volume of its convex hull
can be computed by such formulae. The canonical example is the classical moment
curve, and our class of curves, which we call cyclic, includes other known
classes such as -order curves and curves with totally positive torsion. We
also conjecture a necessary and sufficient condition on curves for the
signature volume formula to hold. Finally, we give a concrete geometric
interpretation of the volume formula in terms of lengths and signed areas.Comment: 15 pages, 5 figures. Comments are welcome
Estimating Gaussian mixtures using sparse polynomial moment systems
The method of moments is a statistical technique for density estimation that
solves a system of moment equations to estimate the parameters of an unknown
distribution. A fundamental question critical to understanding identifiability
asks how many moment equations are needed to get finitely many solutions and
how many solutions there are. We answer this question for classes of Gaussian
mixture models using the tools of polyhedral geometry. Using these results, we
present an algorithm that performs parameter recovery, and therefore density
estimation, for high dimensional Gaussian mixture models that scales linearly
in the dimension.Comment: 30 page
Varieties of Signature Tensors
The signature of a parametric curve is a sequence of tensors whose entries
are iterated integrals. This construction is central to the theory of rough
paths in stochastic analysis. It is here examined through the lens of algebraic
geometry. We introduce varieties of signature tensors for both deterministic
paths and random paths. For the former, we focus on piecewise linear paths, on
polynomial paths, and on varieties derived from free nilpotent Lie groups. For
the latter, we focus on Brownian motion and its mixtures.Comment: 52 pages, 1 figure, 6 tables; to appear in Forum of Mathematics,
Sigm