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

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.

Figures (7)

• Figure 1: A downsized ring graph (left) and the Minnesota graph (right).
• Figure 2: $\nu^2_{k,T}(t)$ plot with fixed $T=10$ and $k=1000$ for ring graph (left) and Minnesota graph (right). Top is for optimal sampling and bottom is for uniform sampling.
• Figure 3: $\sum_{t=0}^{T-1}\nu^2_{k,T}(t)$ plot with various $T=1:1:20$ and $k=100:100:1000$ for ring graph (left) and minnesota graph(right). Top is for optimal sampling and bottom is for uniform sampling.
• Figure 4: Phase transition diagram for the ring graph (top: $T = 1$; bottom: $T = 10$) using optimal (left) and uniform distributions (right). The slopes of boundaries from red to blue are fitted using black dot lines, indicating the critical sampling complexities that ensure recovery with high probability.
• Figure 5: Phase transition diagram for the Minnesota graph (top: $T = 1$; bottom: $T = 10$) using optimal (left) and uniform distribution (right). The slopes of boundaries from red to blue are fitted using black dot lines, indicating the critical sampling complexities that ensure recovery with high probability.

Theorems & Definitions (20)

• Definition 1
• Proposition 1: Lemma 4.3 in huang2020reconstruction
• Definition 2
• Proposition 2
• Definition 3: Dynamic spectral graph weighted coherence for sparse recovery
• Theorem 3: RIP of sampling operator
• proof
• Proposition 4
• proof : Proof of \ref{['thm:cosamp_stable']}
• Remark 1