In this paper, we propose a topology-aware Kalman filter for hidden dynamics over simplicial complex. Specifically, we consider that the hidden dynamics of a system can be expressed as a simplicial process that respects the structure of the underlying network. And these dynamics are observed through an observation matrix, which can be represented using simplicial convolution filters. This combination allows us to model effectively a broader spectrum of network dynamics than graph-based alternatives, such as edge flow evolution. Additionally, we propose a parametric, structure-aware noise covariance model for the system dynamics. We alternate between estimating the process state using the Kalman filter and updating the parameters through maximum likelihood estimation. The efficacy of the proposed approach is demonstrated through experiments on both real-world and synthetic datasets.

Inference of time varying data over graphs is of importance in real-world applications such as urban water networks, economics, and brain recordings. It typically relies on identifying a computationally affordable joint spatiotemporal method that can leverage the patterns in the data. While this per se is a challenging task, it becomes even more so when the network comes with uncertainties, which, if not accounted for, can lead to unpredictable consequences. To target this setting, we model graph uncertainties as Gaussian noise on the edges and design a stochastic partial differential equation (SPDE) based on it. We use this SPDE as a state equation to model the time varying signal evolution and extend it further to a state-space model where the observations are graph-filtered versions of the state. This allows us to have a joint spatiotemporal expressive kernel that can be estimated online via Kalman filtering and which parameters can also be estimated online via maximum likelihood principles, ultimately, reducing the computational cost. We corroborate the proposed approach on numerical experiments, showing a superior performance to approaches ignoring either the uncertainty or considering a separable spatiotemporal kernel.

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.