发表机构
City University of Hong Kong; Chiang Mai University; Claremont Graduate University（香港城市大学; 清迈大学; 克莱蒙特研究生大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出将有向图表示为Hermitian矩阵，基于谱分解和k-means递归实现层次聚类，并利用簇的入/出度构造多尺度样条拟插值，实现有向图信号去噪，实验验证了其优越性。
AI 中文摘要
在本文中，我们提出将有向图（digraph）表示为从其邻接矩阵导出的一个Hermitian矩阵。该表示同时刻画了有向图的连通性和边的方向。基于该Hermitian矩阵的谱分解，我们引入了一种结合k-means的有向图聚类算法，以在图上产生一个划分。将该算法（自底向上）递归应用于具有部分标记顶点的有向图，可得到一种谱层次有向图聚类（\\(\myproj\\)）算法，该算法产生有向图的一致嵌套划分，或等价地，产生一棵树结构。此外，基于有向图聚类中每个簇的入度和出度，可以以自顶向下的方式导出一对层次区间划分（过滤），从而产生一对嵌套的节点序列。这些节点序列有助于构造多尺度样条拟插值，使得带噪图信号能够被分解为粗略逼近和层间细节，随后进行自适应阈值处理和重构。在合成和真实世界有向图上的实验表明，我们的\\(\myproj\\)算法在多种图结构属性（同质性和异质性）及监督设置下，在有向图聚类方面具有优越性。此外，使用多尺度样条拟插值进行有向图信号处理的实验进一步证明了在RMSE和SNR方面对有向图信号恢复的有效性。
英文摘要
In this paper, we propose a representation of a digraph (directed graph) as a Hermitian matrix derived from its adjacency matrix. This representation characterizes both the connectivity and the edge orientation of the digraph. Based on the spectral decomposition of the Hermitian matrix, a digraph clustering algorithm with $k$-means is introduced to produce a partition on the graph. Applying this algorithm (bottom-up) recursively to a digraph with partially labeled vertices yields a spectral hierarchical digraph clustering (\myproj) algorithm that produces consistent nested partitions of the digraph, or equivalently, a tree structure. Furthermore, based on the in-degree and out-degree of each cluster in the digraph clustering, a pair of hierarchical interval partitions (filtrations) can be derived in a top-down manner to produce a pair of nested knot sequences. These knot sequences facilitate the construction of multilevel spline quasi-interpolants, enabling a noisy graph signal to be decomposed into a coarse approximation and inter-level details, followed by adaptive thresholding and reconstruction. Experiments on synthetic and real-world digraphs demonstrate the superiority of our {\myproj} algorithm for digraph clustering across diverse graph structural properties (homophily and heterophily) and supervision settings. Moreover, experiments on digraph signal processing using multilevel spline quasi-interpolants further demonstrate the effectiveness of signal recovery on digraphs in terms of RMSE and SNR.
