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

TL;DR

This paper introduces the unified extended matrix (UEM) framework for graph signal processing, enabling flexible modeling of complex graph structures and improving signal analysis through a new spectral transform.

Contribution

The paper proposes the UEM framework that integrates extended adjacency and unified graph matrices, with theoretical analysis and a new graph Fourier transform for enhanced processing.

Findings

UEM-GFT outperforms existing methods in anomaly detection.

UEM provides flexible adaptation to various graph structures.

Theoretical properties like positive semi-definiteness are established.

Abstract

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 paper proposes the unified extended matrix (UEM) framework, which integrates the extended-adjacency matrix and the unified graph representation matrix through parametric design, so as to be able to flexibly adapt to different graph structures and reveal more graph signal information. Theoretical analysis of the UEM is conducted, demonstrating positive semi-definiteness and eigenvalue monotonicity under specific conditions. Then, we propose graph Fourier transform based on UEM (UEM-GFT), which can adaptively tune 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 tasks, achieving superior performance across varying network topologies.