Random space-time sampling and reconstruction of sparse bandlimited graph diffusion field

TL;DR

This paper develops a probabilistic sampling and reconstruction framework for sparse, bandlimited graph signals evolving over time, using a novel dynamic spectral graph coherence measure to optimize sample efficiency.

Contribution

It introduces a randomized space-time sampling scheme for graph signals governed by heat diffusion, with a new coherence measure guiding optimal sampling and reconstruction.

Findings

As few as O(s log(k)) samples suffice for accurate recovery.

The method reduces spatial sampling by leveraging temporal correlations.

Numerical experiments confirm theoretical guarantees and practical effectiveness.

Abstract

In this work, we investigate the sampling and reconstruction of spectrally $s$-sparse bandlimited graph signals governed by heat diffusion processes. We propose a random space-time sampling regime, referred to as {randomized} dynamical sampling, where a small subset of space-time nodes is randomly selected at each time step based on a probability distribution. To analyze the recovery problem, we establish a rigorous mathematical framework by introducing the parameter \textit{the dynamic spectral graph weighted coherence}. This key parameter governs the number of space-time samples needed for stable recovery and extends the idea of variable density sampling to the context of dynamical systems. By optimizing the sampling probability distribution, we show that as few as $\mathcal{O}(s \log(k))$ space-time samples are sufficient for accurate reconstruction in optimal scenarios, where $k$ denotes the bandwidth of the signal. Our framework encompasses both static and dynamic cases, demonstrating a reduction in the number of spatial samples needed at each time step by exploiting temporal correlations. Furthermore, we provide a computationally efficient and robust algorithm for signal reconstruction. Numerical experiments validate our theoretical results and illustrate the practical efficacy of our proposed methods.