Reconstruction of Graph Signals on Complex Manifolds with Kernel Methods

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 metric-induced and graph fractional spectral 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.