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We provide two fundamental results on the population (infinite-sample) likelihood function of Gaussian mixture models with $M \geq 3$ components.
Identifiability of finite mixtures
Henry Teicher · 1963
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Maximum likelihood from incomplete data via the EM algorithm
Arthur P Dempster, Nan M Laird, and Donald B Rubin · 1977
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Statistical Analysis of Finite Mixture Distributions
D Michael Titterington · 1985
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The “automatic” robustness of minimum distance functionals
David L Donoho and Richard C Liu · 1988
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Optimal rate of convergence for finite mixture models
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Rates of convergence for the Gaussian mixture sieve
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Entropies and rates of convergence for maximum likelihood and Bayes estimation for mixtures of normal densities
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Razvan Pascanu, Yann N Dauphin, Surya Ganguli, and Yoshua Bengio · 2014
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Optimal computational and statistical rates of convergence for sparse nonconvex learning problems
Zhaoran Wang, Han Liu, and Tong Zhang · 2014
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Maximum likelihood estimates for Gaussian mixtures are transcendental
Carlos Améndola, Mathias Drton, and Bernd Sturmfels · 2015
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Statistical guarantees for the EM algorithm: From population to sample-based analysis
Sivaraman Balakrishnan, Martin J Wainwright, and Bin Yu · 2015
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Escaping from saddle points—online stochastic gradient for tensor decomposition
Rong Ge, Furong Huang, Chi Jin, and Yang Yuan · 2015
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Learning mixtures of spherical Gaussians: Moment methods and spectral decompositions
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Regularized M-estimators with nonconvexity: Statistical and algorithmic theory for local optima
Po-Ling Loh and Martin J Wainwright · 2013
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Identifiability and optimal rates of convergence for parameters of multiple types in finite mixtures
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Gradient descent converges to minimizers
Jason D Lee, Max Simchowitz, Michael I Jordan, and Benjamin Recht · 2016
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