  We consider the problem of spherical Gaussian Mixture models with $k \geq 3$
components when the components are well separated. A fundamental previous
result established that separation of $\Omega(\sqrt{\log k})$ is necessary and
sufficient for identifiability of the parameters with polynomial sample
complexity (Regev and Vijayaraghavan, 2017). In the same context, we show that
$\tilde{O} (kd/\epsilon^2)$ samples suffice for any $\epsilon \lesssim 1/k$,
closing the gap from polynomial to linear, and thus giving the first optimal
sample upper bound for the parameter estimation of well-separated Gaussian
mixtures. We accomplish this by proving a new result for the
Expectation-Maximization (EM) algorithm: we show that EM converges locally,
under separation $\Omega(\sqrt{\log k})$. The previous best-known guarantee
required $\Omega(\sqrt{k})$ separation (Yan, et al., 2017). Unlike prior work,
our results do not assume or use prior knowledge of the (potentially different)
mixing weights or variances of the Gaussian components. Furthermore, our
results show that the finite-sample error of EM does not depend on
non-universal quantities such as pairwise distances between means of Gaussian
components.
