  We consider the problem of estimating the parameters a Gaussian Mixture Model
with K components of known weights, all with an identity covariance matrix. We
make two contributions. First, at the population level, we present a sharper
analysis of the local convergence of EM and gradient EM, compared to previous
works. Assuming a separation of $\Omega(\sqrt{\log K})$, we prove convergence
of both methods to the global optima from an initialization region larger than
those of previous works. Specifically, the initial guess of each component can
be as far as (almost) half its distance to the nearest Gaussian. This is
essentially the largest possible contraction region. Our second contribution
are improved sample size requirements for accurate estimation by EM and
gradient EM. In previous works, the required number of samples had a quadratic
dependence on the maximal separation between the K components, and the
resulting error estimate increased linearly with this maximal separation. In
this manuscript we show that both quantities depend only logarithmically on the
maximal separation.
