Graph learning is the fundamental task of estimating unknown graph connectivity from available data. Typical approaches assume that not only is all information available simultaneously but also that all nodes can be observed. However, in many real-world scenarios, data can neither be known completely nor obtained all at once. We present a novel method for online graph estimation that accounts for the presence of hidden nodes. We consider signals that are stationary on the underlying graph, which provides a model for the unknown connections to hidden nodes. We then formulate a convex optimization problem for graph learning from streaming, incomplete graph signals. We solve the proposed problem through an efficient proximal gradient algorithm that can run in real-time as data arrives sequentially. Additionally, we provide theoretical conditions under which our online algorithm is similar to batch-wise solutions. Through experimental results on synthetic and real-world data, we demonstrate the viability of our approach for online graph learning in the presence of missing observations.

翻译：图学习是从可用数据中估计未知图连接性的基本任务。典型方法不仅假设所有信息可同时获取，而且所有节点均可观测。然而，在许多现实场景中，数据既无法完全获知，也不能一次性获得。本文提出一种新颖的在线图估计方法，能够处理隐藏节点的存在。我们考虑在底层图上具有平稳性的信号，这为未知的隐藏节点连接提供了模型。随后，我们针对流式不完整图信号构建了图学习的凸优化问题。通过一种可在数据顺序到达时实时运行的高效近端梯度算法求解该问题。此外，我们提供了理论条件以证明在线算法与批处理解的近似性。通过合成数据与真实数据的实验结果，验证了所提方法在存在缺失观测情况下进行在线图学习的可行性。