Joint graph learning from Gaussian observations in the presence of hidden nodes
This addresses a practical challenge in network analysis where only partial node observations are available, but the approach is incremental as it builds on existing graph learning frameworks.
The paper tackled the problem of learning multiple related graphs from Gaussian observations when some nodes are hidden, by proposing a joint graph learning method that models graph similarity among hidden nodes and solves a convex optimization problem for a regularized maximum likelihood estimator, achieving improved performance over baselines on synthetic and real-world graphs.
Graph learning problems are typically approached by focusing on learning the topology of a single graph when signals from all nodes are available. However, many contemporary setups involve multiple related networks and, moreover, it is often the case that only a subset of nodes is observed while the rest remain hidden. Motivated by this, we propose a joint graph learning method that takes into account the presence of hidden (latent) variables. Intuitively, the presence of the hidden nodes renders the inference task ill-posed and challenging to solve, so we overcome this detrimental influence by harnessing the similarity of the estimated graphs. To that end, we assume that the observed signals are drawn from a Gaussian Markov random field with latent variables and we carefully model the graph similarity among hidden (latent) nodes. Then, we exploit the structure resulting from the previous considerations to propose a convex optimization problem that solves the joint graph learning task by providing a regularized maximum likelihood estimator. Finally, we compare the proposed algorithm with different baselines and evaluate its performance over synthetic and real-world graphs.