The EM Algorithm is Adaptively-Optimal for Unbalanced Symmetric Gaussian Mixtures

This paper studies the problem of estimating the means $\pm\theta_{*}\in\mathbb{R}^{d}$ of a symmetric two-component Gaussian mixture $\delta_{*}\cdot N(\theta_{*},I)+(1-\delta_{*})\cdot N(-\theta_{*},I)$, where the weights $\delta_{*}$ and $1-\delta_{*}$ are unequal. Assuming that $\delta_{*}$ is known, we show that the population version of the EM algorithm globally converges if the initial estimate has non-negative inner product with the mean of the larger weight component. This can be achieved by the trivial initialization $\theta_{0}=0$. For the empirical iteration based on $n$ samples, we show that …

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