论文标题
Graphon过滤器：限制的图形信号处理
Graphon Filters: Graph Signal Processing in the Limit
论文作者
论文摘要
图形信号处理是一个新兴字段，旨在建模网络节点上存在的过程，并通过对该结构的扩散来解释。迄今为止，图形信号处理工程具有对图移动运算符的知识。我们的方法是在图表上研究图形过滤的问题，我们只知道一个模型。为此，我们利用L. Lovasz和B. Szegedy提出的图形理论。我们对图形信号处理的新兴领域做出了三个关键贡献。我们首先显示，从图形绘制的随机图的缩放邻接矩阵定义的过滤器，该图从图形收敛到Fredholm积分运算符上定义的过滤器，并用Graphon作为其内核。其次，利用弗雷德霍尔姆积分方程的数值解的理论利用经典发现，我们定义了傅立叶 - 加密金偏移算子。最后，使用傅里叶 - 加盖金偏移算子，我们得出了仅取决于图形的图形滤波器设计算法，因此仅取决于图形的概率结构，而不是特定的图形本身。通过在各种随机图模型上的模拟中验证了派生的图形过滤算法。
Graph signal processing is an emerging field which aims to model processes that exist on the nodes of a network and are explained through diffusion over this structure. Graph signal processing works have heretofore assumed knowledge of the graph shift operator. Our approach is to investigate the question of graph filtering on a graph about which we only know a model. To do this we leverage the theory of graphons proposed by L. Lovasz and B. Szegedy. We make three key contributions to the emerging field of graph signal processing. We show first that filters defined over the scaled adjacency matrix of a random graph drawn from a graphon converge to filters defined over the Fredholm integral operator with the graphon as its kernel. Second, leveraging classical findings from the theory of the numerical solution of Fredholm integral equations, we define the Fourier-Galerkin shift operator. Lastly, using the Fourier-Galerkin shift operator, we derive a graph filter design algorithm which only depends on the graphon, and thus depends only on the probabilistic structure of the graph instead of the particular graph itself. The derived graphon filtering algorithm is verified through simulations on a variety of random graph models.
