arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2003.05030eess.SP
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图on信号处理(Graphon Signal Processing)

Graphon Signal Processing

Luana Ruiz, Luiz F. O. Chamon, Alejandro Ribeiro

AI总结:

本文提出graphon信号处理理论,证明图傅里叶变换、图滤波器响应在图序列收敛时的极限性质,将图信号处理拓展至大规模节点图与动态图场景。

摘要:

Graphon(图极限)是一种无穷维对象,代表节点数趋于无穷时收敛图序列的极限。本文提出了一套以graphon傅里叶变换、线性平移不变graphon滤波器为核心的graphon信号处理理论,二者分别对应图傅里叶变换与图滤波器的graphon版本。研究证明,对于收敛的图序列及关联的图信号:(i)当graphon信号为带限信号时,图傅里叶变换收敛于graphon傅里叶变换;(ii)图滤波器的谱响应与顶点响应,收敛于系数相同的graphon滤波器的谱响应与顶点响应。这些定理表明,对于属于特定族的图(即收敛于某一graphon的图序列中的图),图傅里叶分析与图滤波器设计具有明确定义的极限。相应地,这些结论将图信号处理的适用性拓展至大规模节点图——因为针对极限graphon设计的信号处理流程可应用于有限图;同时也拓展至动态图——因为我们可关联针对同一收敛图序列中不同图设计的信号处理流程的结果。

英文摘要:

Graphons are infinite-dimensional objects that represent the limit of convergent sequences of graphs as their number of nodes goes to infinity. This paper derives a theory of graphon signal processing centered on the notions of graphon Fourier transform and linear shift invariant graphon filters, the graphon counterparts of the graph Fourier transform and graph filters. It is shown that for convergent sequences of graphs and associated graph signals: (i) the graph Fourier transform converges to the graphon Fourier transform when the graphon signal is bandlimited; (ii) the spectral and vertex responses of graph filters converge to the spectral and vertex responses of graphon filters with the same coefficients. These theorems imply that for graphs that belong to certain families, i.e., that are part of sequences that converge to a certain graphon, graph Fourier analysis and graph filter design have well defined limits. In turn, these facts extend applicability of graph signal processing to graphs with large number of nodes -- since signal processing pipelines designed for limit graphons can be applied to finite graphs -- and to dynamic graphs -- since we can relate the result of SP pipelines designed for different graphs from the same convergent graph sequence.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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