Graph Distribution-valued Signals: A Wasserstein Space Perspective

2025-09-30

AI Summary

Traditional graph signal processing (GSP) relies on synchronous observations, deterministic modeling, and strict node-wise correspondences—strong assumptions often violated in real-world uncertain graph data. To address these limitations, we propose Graph Distribution-Valued Signals (GDSs), a novel framework that models node signals as probability distributions in the Wasserstein space, thereby generalizing GSP from vector-valued to distribution-valued signals. This paradigm shift unifies and extends classical GSP: deterministic, synchronous signals emerge as special cases; node-level uncertainty and stochasticity are inherently captured; and alignment-free, unpaired distribution filtering becomes feasible. Leveraging optimal transport theory and Wasserstein geometry, we establish a systematic theoretical foundation for distribution-domain graph filtering, representation learning, and inference. Empirical evaluation on prediction tasks demonstrates substantial improvements in filter learning performance and robust modeling of complex, uncertain graph signals.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Federated Web and WoT systems, including distributed, federated and edge-based data processing
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We introduce a novel framework for graph signal processing (GSP) that models signals as graph distribution-valued signals (GDSs), which are probability distributions in the Wasserstein space. This approach overcomes key limitations of classical vector-based GSP, including the assumption of synchronous observations over vertices, the inability to capture uncertainty, and the requirement for strict correspondence in graph filtering. By representing signals as distributions, GDSs naturally encode uncertainty and stochasticity, while strictly generalizing traditional graph signals. We establish a systematic dictionary mapping core GSP concepts to their GDS counterparts, demonstrating that classical definitions are recovered as special cases. The effectiveness of the framework is validated through graph filter learning for prediction tasks, supported by experimental results.

Problem

• Modeling graph signals as distributions to encode uncertainty
• Overcoming limitations of synchronous observations in graph processing
• Generalizing graph filtering beyond strict vertex correspondence

Innovation

• Models graph signals as probability distributions in Wasserstein space
• Encodes uncertainty and stochasticity through distribution-valued signals
• Establishes systematic mapping from classical to distribution-based GSP concepts