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.

翻译：我们研究了日志-最大值函数的优化景观,并研究了潜伏高山树模型中期待-最大值算法(EM)的趋同性,即叶节目可见且非叶节节不可观测的叶状高斯树结构图形模型。我们发现,人口日志类似值独特的非三角固定点是其全球最大值,并证明期望-最大值算法保证在单一潜伏变量中与它趋同。我们在一般潜伏树模型中对日志-最值函数景观的结果为在这一环境中广泛实际使用最大可能性的基于方法提供了支持。我们的EM算法结果扩展了为这一值得庆祝的算法获得全球趋同性保证的新兴工作线。我们通过论证从EM更新中获取的某种多元性方程式系统具有独特的非三维的解决方案,来显示我们关于日志类似值非边定点的结果。