Online Smooth Graph Learning From Incomplete Data

Graphs are essential for extracting crucial information embedded within structured data and are foundational tools across various fields. Predefined graphs, however, cannot adequately capture the intrinsic relationships within data, highlighting the need for learning graphs to construct meaningful representations. Particularly, graph learning is crucial in dynamic scenarios, where graphs evolve in response to streamed signals, requiring real-time adaptation through online methods. Additionally, missing values in sequential data pose challenges that necessitate signal reconstruction techniques to recover incomplete information, ensuring accurate and reliable graph inference. To address such issues, we design a novel online algorithm that achieves joint signal reconstruction and topology inference under smoothness priors. Specifically, the two sub-problems are formulated as a joint optimization task, solvable through alternating minimization. To enable efficient online graph learning with a trade-off in accuracy, the inexact proximal online gradient descent (IPOGD) is incorporated into our algorithm, and a dynamic regret analysis demonstrates a sublinear regret bound. Experimental results on both synthetic and real-world datasets validate its effectiveness in tracking slowly-evolving networks with incomplete data.