Randomized Space-Time Sampling for Affine Graph Dynamical Systems

2025-09-20

AI Summary

This work addresses the joint stable recovery of initial states and source terms in non-homogeneous graph linear dynamical systems with constant source terms. Confronting the challenge that the system matrix is non-orthogonally diagonalizable—rendering conventional spectral methods inapplicable—we propose, for the first time, two randomized spatiotemporal sampling strategies. We introduce *spectral graph weighted coherence* to characterize the coupling between sampling distributions and underlying graph topology, and establish a robust recovery theory grounded in the Restricted Isometry Property (RIP). Our algorithms come with provable error bounds and optimal sampling complexity guarantees. Experiments on both synthetic and real-world graph datasets demonstrate that the method efficiently and accurately reconstructs bandlimited initial states and source terms, substantially extending the applicability boundary of existing frameworks for dynamic graph signal sampling.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

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.

Problem

• Recovering graph signals with constant source terms from random space-time samples
• Analyzing sampling conditions for stable recovery in non-orthogonal-diagonalizable systems
• Developing robust algorithms for joint recovery of initial state and forcing term

Innovation

• Random space-time sampling for graph signals
• Spectral graph weighted coherence for stability analysis
• Robust recovery algorithm with error guarantees