Learning Gaussian Latent Graphical Models Via Partial Convex Optimization

Abstract

Latent Gaussian graphical models are very useful in probabilistic modeling to measure the statistical relationships between different variables and present them in the form of a graph. These models are applied to a variety of domains, such as economics, social network modeling and natural sciences. However, traditional ways of learning latent Gaussian models have some weakness. For example, algorithms for latent tree graphical models usually cannot handle a complex model as they have strict presumptions about graph structure. The presumptions of tree graphical models do not perform well in a lot of real data. Besides, some use the convex optimization of maximum likelihood estimator (MLE) to solve the model. However, the computation of convex optimization increases exponentially while the number of variables increases. Thus, we come up with a fast, local-based algorithm that combines convex optimization with latent tree graphical models to save a lot computation while also handle complex models. This algorithm first applies ChowLiu Recursive Grouping to find the skeleton of the final graph and learn the hidden variables in the tree-like part of the graph. Then, for the part that variables have a complex relationship with each other, we propose a local convex optimization to compute the structure. After combining structures for two different parts, the algorithm generates a result that performs better than latent tree graphical models measured by loglikelihood and has much less computation than traditional convex optimization.