Randomized Space-Time Sampling for Affine Graph Dynamical Systems
This work addresses a fundamental challenge in graph signal processing for systems with constant forcing terms, offering a novel sampling framework with theoretical guarantees, though it is incremental in extending prior homogeneous dynamics work.
The paper tackles the problem of recovering bandlimited initial states and constant source terms in graph dynamical systems with non-orthogonal-diagonalizable matrices, establishing sampling complexity bounds and a robust recovery algorithm with provable error guarantees validated on synthetic and real-world datasets.
This paper investigates the problem of dynamical sampling for graph signals influenced by a constant source term. We consider signals evolving over time according to a linear dynamical system on a graph, where both the initial state and the source term are bandlimited. We introduce two random space-time sampling regimes and analyze the conditions under which stable recovery is achievable. While our framework extends recent work on homogeneous dynamics, it addresses a fundamentally different setting where the evolution includes a constant source term. This results in a non-orthogonal-diagonalizable system matrix, rendering classical spectral techniques inapplicable and introducing new challenges in sampling design, stability analysis, and joint recovery of both the initial state and the forcing term. A key component of our analysis is the spectral graph weighted coherence, which characterizes the interplay between the sampling distribution and the graph structure. We establish sampling complexity bounds ensuring stable recovery via the Restricted Isometry Property (RIP), and develop a robust recovery algorithm with provable error guarantees. The effectiveness of our method is validated through extensive experiments on both synthetic and real-world datasets.