A Novel Unified Extended Matrix for Graph Signal Processing: Theory and Application

Graph signal processing has become an essential tool for analyzing data structured on irregular domains. While conventional graph shift operators (GSOs) are effective for certain tasks, they inherently lack flexibility in modeling dependencies between non-adjacent nodes, limiting their ability to represent complex graph structures. To address this limitation, this study proposes the unified extended matrix (UEM) framework, which integrates the extended-adjacency matrix and the unified graph representation matrix via parametric design, enabling flexible adaptation to different graph structures and revealing more graph signal information. Theoretical analysis of the UEM is conducted, demonstrating positive semi-definiteness and eigenvalue monotonicity under specific conditions. Then, we proposed a graph Fourier transform based on UEM (UEM-GFT) that adaptively tunes spectral properties to enhance signal processing performance. Experimental results on synthetic and real-world datasets demonstrate that the UEM-GFT outperforms existing GSO-based methods in anomaly detection, achieving superior performance across diverse network topologies.