论文标题
Graphon信号处理
Graphon Signal Processing
论文作者
论文摘要
图形是无限维对物体，代表图形的收敛序列的极限，因为它们的节点数量为无穷大。本文得出了Graphon信号处理的理论，该理论以Graphon Fourier变换和线性移位不变型Graphon滤波器（Graphon formainiant Graghon过滤器）的概念（图形傅立叶变换和图形过滤器）的概念为中心。结果表明，对于图形和关联的图形信号的收敛序列：（i）当图形信号被限制时，图傅立叶变换会收敛到图形傅里叶变换； （ii）图形过滤器的光谱和顶点响应以相同的系数收敛到Graphon滤波器的光谱和顶点响应。这些定理暗示，对于属于某些家族的图，即是收敛到某个图形子的序列的一部分，图形傅立叶分析和图形滤波器设计具有明确的限制。反过来，这些事实将图形信号处理的适用性扩展到具有大量节点的图形 - 因为我们可以将设计用于极限图形的信号处理管道应用于有限的图形以及动态图 - 因为我们可以将设计用于从同一收敛图序列的不同图形的SP管道的结果相关联。
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.
