Observing and Tracking Bandlimited Graph Processes

One of the most crucial challenges in graph signal processing is the sampling of bandlimited graph signals, i.e., signals that are sparse in a well-defined graph Fourier domain. So far, the prior art is mostly focused on (sub)sampling single snapshots of graph signals ignoring their evolution over time. However, time can bring forth new insights, since many real signals like sensor measurements, biological, financial, and network signals in general, have intrinsic correlations in both domains. In this work, {we fill this lacuna} by jointly considering the graph-time nature of graph signals, named \emph{graph processes} for two main tasks: \emph{i)} observability of graph processes; and \emph{ii)} tracking of graph processes via Kalman filtering; both from a (possibly time-varying) subset of nodes. A detailed mathematical analysis ratifies the proposed methods and provides insights into the role played by the different actors, such as the graph topology, the process bandwidth, and the sampling strategy. Moreover, (sub)optimal sampling strategies that jointly exploit the nature of the graph structure and graph process are proposed. Several numerical tests on both synthetic and real data validate our theoretical findings and illustrate the performance of the proposed methods in coping with time-varying graph signals.