Robust recovery of bandlimited graph signals via randomized dynamical sampling

Heat diffusion processes have found wide applications in modelling dynamical systems over graphs. In this paper, we consider the recovery of a $k$-bandlimited graph signal that is an initial signal of a heat diffusion process from its space-time samples. We propose three random space-time sampling regimes, termed dynamical sampling techniques, that consist in selecting a small subset of space-time nodes at random according to some probability distribution. We show that the number of space-time samples required to ensure stable recovery for each regime depends on a parameter called the spectral graph weighted coherence, that depends on the interplay between the dynamics over the graphs and sampling probability distributions. In optimal scenarios, no more than $\mathcal{O}(k \log(k))$ space-time samples are sufficient to ensure accurate and stable recovery of all $k$-bandlimited signals. In any case, dynamical sampling typically requires much fewer spatial samples than the static case by leveraging the temporal information. Then, we propose a computationally efficient method to reconstruct $k$-bandlimited signals from their space-time samples. We prove that it yields accurate reconstructions and that it is also stable to noise. Finally, we test dynamical sampling techniques on a wide variety of graphs. The numerical results support our theoretical findings and demonstrate the efficiency.