Many dynamical processes over graphs are either exact or well-approximated by linear dynamical systems, such as random walks, heat diffusion processes, or more generally graph filtering processes, with applications in modeling traffic flow, temperature variation, and brain activities over graph networks.  It is of practical interest to estimate initial signal (source localization) and even the evolution operator (relevant to graph topology) from samples of evolving graph signals. This is not only for enhancing graph data processing tasks but also for data interpretability.

In this talk, we consider a challenging scenario where only partial observations of evolving states are available. We will present some progress on the theoretical and algorithmic aspects for recovering the initial state and even the evolution operators from sparse space-time samples, levering ideas from dynamical sampling, compressed sensing, super-resolution, and matrix completion.  Our results shed light on the interplay of sampling, graph topology, temporal dynamics and well-posedness of the inverse problems. It is based on the joint work with Jiahui Cheng, Longxiu Huang,  Christian Kümmerle, Mauro Maggioni, and Deanna Needell.