  We study the gradient Expectation-Maximization (EM) algorithm for Gaussian
Mixture Models (GMM) in the over-parameterized setting, where a general GMM
with $n>1$ components learns from data that are generated by a single ground
truth Gaussian distribution. While results for the special case of 2-Gaussian
mixtures are well-known, a general global convergence analysis for arbitrary
$n$ remains unresolved and faces several new technical barriers since the
convergence becomes sub-linear and non-monotonic. To address these challenges,
we construct a novel likelihood-based convergence analysis framework and
rigorously prove that gradient EM converges globally with a sublinear rate
$O(1/\sqrt{t})$. This is the first global convergence result for Gaussian
mixtures with more than $2$ components. The sublinear convergence rate is due
to the algorithmic nature of learning over-parameterized GMM with gradient EM.
We also identify a new emerging technical challenge for learning general
over-parameterized GMM: the existence of bad local regions that can trap
gradient EM for an exponential number of steps.
