375 research outputs found
Moment ideals of local Dirac mixtures
In this paper we study ideals arising from moments of local Dirac measures
and their mixtures. We provide generators for the case of first order local
Diracs and explain how to obtain the moment ideal of the Pareto distribution
from them. We then use elimination theory and Prony's method for parameter
estimation of finite mixtures. Our results are showcased with applications in
signal processing and statistics. We highlight the natural connections to
algebraic statistics, combinatorics and applications in analysis throughout the
paper.Comment: 26 pages, 3 figure
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
Gronwall's conjecture for 3-webs with infinitesimal symmetries
We study non-flat planar 3-webs with infinitesimal symmetries. Using
multi-dimensional Schwarzian derivative we give a criterion for linearization
of such webs and present a projective classification thereof. Using this
classification we show that the Gronwall conjecture is true for 3-webs
admitting infinitesimal symmetries.Comment: 20 page
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