Graph-Time Convolutional Neural Networks: Architecture and Theoretical Analysis.

Abstract

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 multi-resolution information in each layer and offers some degree of mathematical analysis by exploring tools from graph signal processing. This analysis provides insights on the equivariance properties of GCNNs; spectral behaviour of the learned filters; and the stability to perturbations in the graph topology, which arises because of support perturbations or uncertainties. However, extending the convolution-principled 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 the 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 to aid learning. The proposed approach can work with any type of product graph and we also introduce a parametric product graph to learn also 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 but 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 favourably 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.