AI 中文总结
该研究引入图分布值信号框架处理图信号，解决经典GSP局限，为依赖信号的图结构提供方法，建立GSP概念与GDS对应关系，给出理论连续性结果，通过示例应用与实证研究验证了方法的实用性和有效性。
AI 中文摘要
我们引入了一种用于图信号处理（GSP）的框架，其中信号被表示为图分布值信号（GDS），即在瓦瑟斯坦空间中的概率测度。此视角解决了基于经典向量的GSP的基本局限性，包括顶点间完全同步观测的要求以及观测滤波器输入 - 输出对中严格时间对应性的需求。通过将图结构建模为依赖信号实现的分布，我们为实际应用中常见的依赖信号的图结构提供了一种有原则的方法，同时明确编码图拓扑中的不确定性。我们的框架固有地捕捉不确定性和随机性，严格推广了传统图信号（可解释为狄拉克δ测度）。我们建立了基础GSP概念与其GDS类似物之间的系统对应关系，表明经典公式是我们框架的特殊情况。我们为GDS变换建立了理论连续性结果，为输入扰动和分布近似提供了稳定性保证。我们通过示例应用（包括图滤波器学习和异常检测）展示了该方法的实用性，并通过实证研究验证了其有效性。
英文摘要
We introduce a framework for graph signal processing (GSP) in which signals are represented as graph distribution-valued signals (GDSs), i.e., probability measures in a Wasserstein space. This perspective addresses fundamental limitations of classical vector-based GSP, including the requirement for complete synchronous observations across vertices and the need for strict temporal correspondence in observed filter input--output pairs. Furthermore, by modeling the graph structure as a distribution conditioned on signal realizations, we provide a principled approach to signal-dependent graph structures, which are common in real-world applications, while explicitly encoding uncertainty in graph topology. Our framework inherently captures uncertainty and stochasticity while strictly generalizing traditional graph signals, which can be interpreted as Dirac delta measures. We develop a systematic correspondence between foundational GSP concepts and their GDS analogs, showing that classical formulations emerge as special cases of our framework. We establish theoretical continuity results for GDS transforms, providing stability guarantees for input perturbations and distribution approximations. We demonstrate the utility of this approach through example applications, including graph filter learning and anomaly detection, and validate its effectiveness through empirical studies.
CommentsSubmitted to IEEE Transactions on Signal Processing
