Algebraic Identifiability of Gaussian Mixtures
Carlos Amendola, Kristian Ranestad, and Bernd Sturmfels
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Submission date: 04. Dec. 2016
published in: International mathematics research notices, 2018 (2018) 21, p. 6556-6580
DOI number (of the published article): 10.1093/imrn/rnx090
Keywords and phrases: mixture models, secant variety, Gaussian
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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.