Ten Steps of EM Suffice for Mixtures of Two Gaussians

We provide global convergence guarantees for the expectation-maximization (EM) algorithm applied to mixtures of two Gaussians with known covariance matrices. We show that EM converges geometrically to the correct mean vectors, and provide simple, closed-form expressions for the convergence rate. As a simple illustration, we show that in one dimension ten steps of the EM algorithm initialized at $+ \infty$ result in less than 1% error estimation of the means.