  We study the optimization landscape of the log-likelihood function and the
convergence of the Expectation-Maximization (EM) algorithm in latent Gaussian
tree models, i.e. tree-structured Gaussian graphical models whose leaf nodes
are observable and non-leaf nodes are unobservable. We show that the unique
non-trivial stationary point of the population log-likelihood is its global
maximum, and establish that the expectation-maximization algorithm is
guaranteed to converge to it in the single latent variable case. Our results
for the landscape of the log-likelihood function in general latent tree models
provide support for the extensive practical use of maximum likelihood
based-methods in this setting. Our results for the EM algorithm extend an
emerging line of work on obtaining global convergence guarantees for this
celebrated algorithm. We show our results for the non-trivial stationary points
of the log-likelihood by arguing that a certain system of polynomial equations
obtained from the EM updates has a unique non-trivial solution. The global
convergence of the EM algorithm follows by arguing that all trivial fixed
points are higher-order saddle points.
