375 research outputs found

    Moment ideals of local Dirac mixtures

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    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

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    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

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    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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