Randomly initialized EM estimates the true center of a symmetric two-component Gaussian mixture to within logarithmic factors of the minimax rate in O(√n) iterations, given n=Ω(d log^3 d) samples.
Optimal rates for finite mixture estimation
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abstract
We study the rates of estimation of finite mixing distributions, that is, the parameters of the mixture. We prove that under some regularity and strong identifiability conditions, around a given mixing distribution with $m_0$ components, the optimal local minimax rate of estimation of a mixing distribution with $m$ components is $n^{-1/(4(m-m_0) + 2)}$. This corrects a previous paper by Chen (1995) in The Annals of Statistics. By contrast, it turns out that there are estimators with a (non-uniform) pointwise rate of estimation of $n^{-1/2}$ for all mixing distributions with a finite number of components.
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Randomly initialized EM algorithm for two-component Gaussian mixture achieves near optimality in $O(\sqrt{n})$ iterations
Randomly initialized EM estimates the true center of a symmetric two-component Gaussian mixture to within logarithmic factors of the minimax rate in O(√n) iterations, given n=Ω(d log^3 d) samples.