用于凯莱图上信号处理的框架
Frames for signal processing on Cayley graphs
AI总结：
本文提出基于表示论的加权凯莱图谱分解方法，并构建Frobenius-Schur框架与凯莱框架，以支持凯莱图上的稳定信号处理。
摘要：
图邻接矩阵的谱分解是图信号处理（GSP）技术设计中的一个基本要素。当邻接矩阵具有多维特征空间时，期望基于更能反映图对称性的特定特征基来构建GSP。在本文中，我们为加权凯莱图的邻接矩阵的谱分解提供了一个明确且详细的表示论解释。我们的方法适用于所有加权凯莱图，无论它们是否为拟阿贝尔的，并提供了从底层群的表示系数函数导出的特征值和特征向量的详细描述。接下来，我们将注意力转向在凯莱图上构建框架。框架是过完备的生成集，确保信号重建的稳定且可能冗余的系统。我们使用我们提出的特征基来构建适用于在凯莱图上开发信号处理的框架。这些是Frobenius-Schur框架和凯莱框架，我们为它们提供了表征和实用的构建方法。
英文摘要：
The spectral decomposition of graph adjacency matrices is an essential ingredient in the design of graph signal processing (GSP) techniques. When the adjacency matrix has multi-dimensional eigenspaces, it is desirable to base GSP constructions on a particular eigenbasis that better reflects the graph's symmetries. In this paper, we provide an explicit and detailed representation-theoretic account for the spectral decomposition of the adjacency matrix of a weighted Cayley graph. Our method applies to all weighted Cayley graphs, regardless of whether they are quasi-Abelian, and offers detailed descriptions of eigenvalues and eigenvectors derived from the coefficient functions of the representations of the underlying group. Next, we turn our attention to constructing frames on Cayley graphs. Frames are overcomplete spanning sets that ensure stable and potentially redundant systems for signal reconstruction. We use our proposed eigenbases to build frames that are suitable for developing signal processing on Cayley graphs. These are the Frobenius--Schur frames and Cayley frames, for which we provide a characterization and a practical recipe for their construction.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
