The vector autoregressive (VAR) model is extensively employed for modelling dynamic processes, yet its scalability is challenged by an overwhelming growth in parameters when dealing with several hundred time series. To overcome this issue, data relations can be leveraged as inductive priors to tackle the curse of dimensionality while still effectively modelling the time series. In this paper, we study the role of simplicial complexes as inductive biases when modelling time series defined on higher-order network structures such as edges and triangles. First, we propose two simplicial VAR models: one that models time series defined on a single simplicial level, such as edge flows, and another that jointly models multiple time series defined across different simplicial levels, ultimately capturing their spatiotemporal interdependencies. The proposed models use simplicial convolutional filters to facilitate parameter sharing and capture structure-aware spatio-temporal dependencies in a multiresolution manner. Second, we develop a joint simplicial-temporal Fourier transform to study the spectral characteristics of the models, depicting them as simplicial-temporal filters. Third, targeting streaming signals, we develop an online algorithm for learning simplicial VAR models. We prove this online learner attains a sublinear dynamic regret bound, ensuring convergence under reasonable assumptions. Finally, we corroborate the proposed approach through experiments on synthetic networks, water distribution networks, and collaborating agents. Our findings show that the proposed models attain competitive signal modelling accuracy with orders of magnitude fewer parameters than the state-of-the-art alternatives.

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.