Graph-Time Convolutional Neural Networks: Architecture and Theoretical Analysis

Devising and analysing learning models for spatiotemporal network data is of importance for tasks including forecasting, anomaly detection, and multi-agent coordination, among others. Graph Convolutional Neural Networks (GCNNs) are an established approach to learn from time-invariant network data. The graph convolution operation offers a principled approach to aggregate information and offers mathematical analysis by exploring tools from graph signal processing. This analysis provides insights into the equivariance properties of GCNNs; spectral behaviour of the learned filters; and the stability to graph perturbations, which arise from support perturbations or uncertainties. However, extending the convolutional learning and respective analysis to the spatiotemporal domain is challenging because spatiotemporal data have more intrinsic dependencies. Hence, a higher flexibility to capture
jointly
the spatial and temporal dependencies is required to learn meaningful higher-order representations. Here, we leverage product graphs to represent the spatiotemporal dependencies in the data and introduce Graph-Time Convolutional Neural Networks (GTCNNs) as a principled architecture. We also introduce a parametric product graph to learn the spatiotemporal coupling. The convolution principle further allows a similar mathematical tractability as for GCNNs. In particular, the stability result shows GTCNNs are stable to spatial perturbations. owever, there is an implicit trade-off between discriminability and robustness; i.e., the more complex the model, the less stable. Extensive numerical results on benchmark datasets corroborate our findings and show the GTCNN compares favorably with state-of-the-art solutions. We anticipate the GTCNN to be a starting point for more sophisticated models that achieve good performance but are also fundamentally grounded.