Graphon Signal Processing
Graphon Signal Processing
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DOI：
10.1109/tsp.2021.3106857
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发表时间：
2020-03
影响因子：
5.4
通讯作者：
Luana Ruiz;Luiz F. O. Chamon;Alejandro Ribeiro
中科院分区：
文献类型：
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作者：
Luana Ruiz;Luiz F. O. Chamon;Alejandro Ribeiro
图子是无限维的对象，表示图的收敛序列的极限，因为它们的节点数趋于无穷大。本文以图子傅立叶变换和线性移位不变图子滤波器的概念为中心，导出了图子信号处理理论，图子傅立叶变换和图滤波器的对应物是图子。结果表明，对于收敛的图序列和相应的图信号：（i）当图子信号是带限的时，图傅里叶变换收敛到图子傅里叶变换;（ii）图滤波器的谱响应和顶点响应收敛到具有相同系数的图子滤波器的谱响应和顶点响应.这些定理意味着，对于属于某些族的图，即，它们是收敛到某个图子的序列的一部分，图傅立叶分析和图滤波器设计具有明确的限制。反过来，这些事实将图信号处理的适用性扩展到具有大量节点的图-因为为极限图子设计的信号处理流水线可以应用于有限图-以及动态图-因为我们可以将为来自相同收敛图序列的不同图设计的SP流水线的结果相关联。
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.