[论文解读] Graph Signal Processing -- Part II: Processing and Analyzing Signals on Graphs

Ljubiša Stanković, Danilo P. Mandic+4 more|arXiv (Cornell University)|Sep 23, 2019

Advanced Graph Neural Networks参考文献 71被引用 11

一句话总结

本文提出了一套全面的图信号处理与分析框架，基于图信号处理（GSP）原理，核心为基于邻接矩阵和拉普拉斯矩阵的图离散傅里叶变换（GDFT）。该框架建立了频域滤波、降采样、压缩感知、时变信号模型、随机图信号以及顶点-频率分析——包括局部化图傅里叶变换与降低干扰分布——实现了在不规则图域上的局部化、多分辨率信号分析，同时保持了边际性与不确定性特性。

ABSTRACT

The focus of Part I of this monograph has been on both the fundamental properties, graph topologies, and spectral representations of graphs. Part II embarks on these concepts to address the algorithmic and practical issues centered round data/signal processing on graphs, that is, the focus is on the analysis and estimation of both deterministic and random data on graphs. The fundamental ideas related to graph signals are introduced through a simple and intuitive, yet illustrative and general enough case study of multisensor temperature field estimation. The concept of systems on graph is defined using graph signal shift operators, which generalize the corresponding principles from traditional learning systems. At the core of the spectral domain representation of graph signals and systems is the Graph Discrete Fourier Transform (GDFT). The spectral domain representations are then used as the basis to introduce graph signal filtering concepts and address their design, including Chebyshev polynomial approximation series. Ideas related to the sampling of graph signals are presented and further linked with compressive sensing. Localized graph signal analysis in the joint vertex-spectral domain is referred to as the vertex-frequency analysis, since it can be considered as an extension of classical time-frequency analysis to the graph domain of a signal. Important topics related to the local graph Fourier transform (LGFT) are covered, together with its various forms including the graph spectral and vertex domain windows and the inversion conditions and relations. A link between the LGFT with spectral varying window and the spectral graph wavelet transform (SGWT) is also established. Realizations of the LGFT and SGWT using polynomial (Chebyshev) approximations of the spectral functions are further considered. Finally, energy versions of the vertex-frequency representations are introduced.

研究动机与目标

• 开发一个统一的信号处理框架，用于不规则图域上的确定性与随机信号。
• 解决在降维条件下图信号滤波、采样与重构的挑战。
• 通过顶点-频率表示与局部化变换，将经典时频概念扩展至图结构。
• 确保顶点-频率能量分布中的边际性与不确定性特性，以实现精确的信号定位。

提出的方法

• 通过邻接矩阵与图拉普拉斯矩阵的特征分解定义图离散傅里叶变换（GDFT），用于频域分析。
• 在频域中通过传递函数实现图信号滤波，采用切比雪夫多项式近似以实现高效计算。
• 利用支持矩阵与唯一重构条件分析降采样与压缩感知，将其与线性测量及滤波器组相联系。
• 通过扩散过程与Taubin的α–β算法对时变信号进行建模，实现动态图信号跟踪。
• 利用谱域与顶点域窗口的局部化图傅里叶变换（LGFT）构建顶点-频率表示，实现顶点-谱域的联合定位。
• 通过基于核的反演与能量形式推导出降低干扰的顶点-频率分布，满足边际性与平滑性特性。

实验结果

研究问题

• RQ1 如何将经典信号处理概念（如滤波与傅里叶变换）推广至不规则图域中的图信号处理？
• RQ2 在何种条件下可从降采样测量中唯一重构图信号？压缩感知如何应用？
• RQ3 如何形式化地定义在联合顶点-谱域中的局部化分析，以实现在图上的多分辨率信号解释？
• RQ4 图信号顶点-频率表示中的不确定性原理与边际特性是什么？
• RQ5 图上的降低干扰分布如何在最小化交叉项干扰的同时保持能量与平滑性信息？

主要发现

• 基于邻接矩阵与拉普拉斯矩阵的GDFT实现了图信号的频域表示与滤波，其传递函数通过切比雪夫多项式近似，显著提升了计算效率。
• 在特定支持与采样条件下，低通与稀疏图信号的降采样可实现唯一重构，与压缩感知原理紧密关联。
• 局部化图傅里叶变换（LGFT）通过谱域与顶点域窗口支持顶点-谱域定位，其反演可通过求和、带通函数或基于核的方法实现。
• 通过保留顶点与频率边际特性的核构造出降低干扰的顶点-频率分布，其中sinc核可实现平滑且无偏的表示。
• 顶点-频率能量分布满足边际特性，可估计局部平滑性，其基于核的形式推广了经典Cohen类分布。
• 该框架推广了经典时频分析，当图结构为循环图时，退化为标准时频分布，验证了其与既有理论的一致性。