  We study the convergence behavior of the Expectation Maximization (EM)
algorithm on Gaussian mixture models with an arbitrary number of mixture
components and mixing weights. We show that as long as the means of the
components are separated by at least $\Omega(\sqrt{\min\{M,d\}})$, where $M$ is
the number of components and $d$ is the dimension, the EM algorithm converges
locally to the global optimum of the log-likelihood. Further, we show that the
convergence rate is linear and characterize the size of the basin of attraction
to the global optimum.
