Learning Time-Varying Graph Signals via Koopman

TL;DR

This work tackles time-varying graphs by embedding each graph instance into a latent state ${\bf g}(t)$ and learning linear Koopman dynamics in that embedding via a graph Koopman autoencoder (GKAE). The model combines a graph encoder, a Koopman encoder, a learnable Koopman matrix ${\bf K}$, a Koopman decoder, and a graph decoder to enable accurate reconstruction and long-horizon prediction across graphs whose structure, weights, and signals evolve. A latent-consistency extension via a latent-consistency autoencoder (LC-autoencoder) enables transductive reconstruction from partially observed data, including completely masked nodes. Across Type-1 and Type-3 datasets, GKAE demonstrates superior reconstruction accuracy and improved long-term prediction relative to state-of-the-art GSP/GNN and Koopman-based baselines, highlighting its ability to capture non-linear dynamics through linear latent embeddings and to handle dynamic graph structures in a unified framework.

Abstract

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.

Figures (10)

• Figure 1: Types of time-varying graphs: Type-1 varies in signals only, Type-2 in signals and edge weights, and Type-3 in signals, edge weights, and structure.
• Figure 2: Schematic of the proposed Graph Koopman Autoencoder (GKAE) framework. The graph encoder aggregates (${\cal A}$) and updates graph signals based on neighbors to generate a graph embedding. The Koopman autoencoder predicts future embeddings linearly, and the graph decoder reconstructs the graph in its original space.
• Figure 3: Application of the GKAE for prediction and reconstruction using the LC-autoencoder, which processes sampled graph data while maintaining latent space consistency to reconstruct the original graph signals.
• Figure 4: Loss convergence over varying dimensions for different Koopman matrix dimensions $M$.
• Figure 5: Performance comparisons over three dataset types over varying number of nodes $N$, compared for (top row) RMSE, (bottom row) MAE. The best three models have been compared.