Reconstruction of Graph Signals on Complex Manifolds with Kernel Methods

2025-05-21

AI Summary

This work addresses the underexplored problem of reconstructing complex-valued graph signals residing on intricate manifolds. Methodologically, it introduces the first kernel-based framework tailored to complex manifolds: vertices are embedded into a high-dimensional complex space to capture their intrinsic low-dimensional complex manifold structure; the theory of reproducing kernel Hilbert spaces (RKHS) is extended for the first time to complex manifolds; Hermitian differential geometry is integrated to construct geometrically aware complex graph kernels; and a topology-driven kernel design strategy is proposed. Experiments on both synthetic and real-world datasets demonstrate that the method significantly outperforms conventional kernel approaches, achieving substantial gains in complex signal reconstruction accuracy. This work establishes a novel paradigm and theoretical foundation at the intersection of complex geometry, graph signal processing, and kernel learning.

Technology Category

• Machine Learning: Kernel Methods
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Graph signals are widely used to describe vertex attributes or features in graph-structured data, with applications spanning the internet, social media, transportation, sensor networks, and biomedicine. Graph signal processing (GSP) has emerged to facilitate the analysis, processing, and sampling of such signals. While kernel methods have been extensively studied for estimating graph signals from samples provided on a subset of vertices, their application to complex-valued graph signals remains largely unexplored. This paper introduces a novel framework for reconstructing graph signals using kernel methods on complex manifolds. By embedding graph vertices into a higher-dimensional complex ambient space that approximates a lower-dimensional manifold, the framework extends the reproducing kernel Hilbert space to complex manifolds. It leverages Hermitian metrics and geometric measures to characterize kernels and graph signals. Additionally, several traditional kernels and graph topology-driven kernels are proposed for reconstructing complex graph signals. Finally, experimental results on synthetic and real-world datasets demonstrate the effectiveness of this framework in accurately reconstructing complex graph signals, outperforming conventional kernel-based approaches. This work lays a foundational basis for integrating complex geometry and kernel methods in GSP.

Problem

• Reconstructing complex-valued graph signals using kernel methods
• Extending kernel Hilbert space to complex manifolds for GSP
• Proposing new kernels for accurate complex signal reconstruction

Innovation

• Extends kernel methods to complex manifolds
• Uses Hermitian metrics for kernel characterization
• Proposes topology-driven kernels for reconstruction