具有广义边界条件的有向图信号处理
Digraph Signal Processing with Generalized Boundary Conditions
摘要：
有向图（有向图）上的信号处理是有问题的，因为图移位以及因此相关联的滤波器通常是不可对角化的。此外，在这种情况下的傅里叶变换现在是从约旦分解获得的，约旦分解对于大型图可能根本不可计算。我们提出了一个新的和一般的解决方案，这个问题的基础上矩阵扰动理论：我们设计了一个算法，增加了少量的边缘到一个给定的有向图破坏非平凡的约旦块。然后得到的有向图是可对角化的，并产生，如我们所示，一个近似的特征基和傅里叶变换的原始有向图。我们解释了为什么以及如何将这种构造视为广义形式的边界条件，这是信号处理中的一种常见做法。我们对随机和真实世界图形的实验表明，我们可以扩展到具有几千个节点的图形，并获得接近正交的傅里叶变换，同时仍然对角化卷积的直观概念。我们的方法与邻接和拉普拉斯移位一起工作，并且可以用作预处理步骤以实现进一步处理，正如我们用原型维纳滤波器应用所示的那样。
英文摘要：
Signal processing on directed graphs (digraphs) is problematic, since the graph shift, and thus associated filters, are in general not diagonalizable. Furthermore, the Fourier transform in this case is now obtained from the Jordan decomposition, which may not be computable at all for large graphs. We propose a novel and general solution for this problem based on matrix perturbation theory: We design an algorithm that adds a small number of edges to a given digraph to destroy nontrivial Jordan blocks. The obtained digraph is then diagonalizable and yields, as we show, an approximate eigenbasis and Fourier transform for the original digraph. We explain why and how this construction can be viewed as generalized form of boundary conditions, a common practice in signal processing. Our experiments with random and real world graphs show that we can scale to graphs with a few thousands nodes, and obtain Fourier transforms that are close to orthogonal while still diagonalizing an intuitive notion of convolution. Our method works with adjacency and Laplacian shift and can be used as preprocessing step to enable further processing as we show with a prototypical Wiener filter application.
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