Learning Time-Varying Graph Signals via Koopman

A wide variety of real-world data, such as sea measurements, e.g., temperatures collected by distributed sensors and multiple unmanned aerial vehicles (UAV) trajectories, can be naturally represented as graphs, often exhibiting non-Euclidean structures. These graph representations may evolve over time, forming time-varying graphs. Effectively modeling and analyzing such dynamic graph data is critical for tasks like predicting graph evolution and reconstructing missing graph data. In this paper, we propose a framework based on the Koopman autoencoder (KAE) to handle time-varying graph data. Specifically, we assume the existence of a hidden non-linear dynamical system, where the state vector corresponds to the graph embedding of the time varying graph signals. To capture the evolving graph structures, the graph data is first converted into a vector time series through graph embedding, representing the structural information in a finite-dimensional latent space. In this latent space, the KAE is applied to learn the underlying non-linear dynamics governing the temporal evolution of graph features, enabling both prediction and reconstruction tasks.