提出统一扩展矩阵框架，提升图信号处理对复杂结构的建模能力。
A Novel Unified Extended Matrix for Graph Signal Processing: Theory and Application
- 通过参数化设计融合扩展邻接矩阵与统一图表示矩阵
- 在异常检测任务中优于现有方法，适应不同网络拓扑
- 适合研究图神经网络与信号处理交叉方向的学者
图信号处理已成为分析不规则域数据的重要工具。尽管传统图移位算子（GSO）在特定任务中有效，但其在建模非相邻节点间依赖关系方面固有灵活性不足，限制了对复杂图结构的表达能力。为此，本文提出统一扩展矩阵（UEM）框架，通过参数化设计整合扩展邻接矩阵与统一图表示矩阵，可灵活适配不同图结构并揭示更多图信号信息。对UEM进行理论分析，证明在特定条件下具有半正定性与特征值单调性。进一步提出基于UEM的图傅里叶变换（UEM-GFT），可自适应调节谱特性以提升信号处理性能。在合成与真实数据集上的实验表明，UEM-GFT在异常检测任务中优于现有基于GSO的方法，且在不同网络拓扑下均表现优异。
原文摘要 · Abstract (English)
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.
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