Kalman Filtering for Simplicial Processes

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.