Graph learning from incomplete graph signals: From batch to online methods

Inferring graph topologies from data is crucial in many graph-related applications. Existing works typically assume that signals are observed at all nodes, which may not hold due to application-specific constraints. The problem becomes more challenging when data are sequentially available and no delay is tolerated. To address these issues, we propose an approach for learning graphs from incomplete data. First, the problem of learning graphs with missing data is formulated as maximizing the posterior distribution with hidden variables from a Bayesian perspective. Then, we propose an expectation maximization (EM) algorithm to solve the induced problem, in which graph learning and graph signal recovery are jointly performed. Furthermore, we extend the proposed EM algorithm to an online version to accommodate the delay-sensitive situations of sequential data. Theoretically, we analyze the dynamic regret of the proposed online algorithm, illustrating the effectiveness of our algorithm in tracking graphs from partial observations in an online manner. Finally, extensive experiments on synthetic and real data are conducted, and the results corroborate that our approach can learn graphs effectively from incomplete data in both batch and online situations.