论文标题
对称组上图形信号的帧：表示理论方法
Frames for Graph Signals on the Symmetric Group: A Representation Theoretic Approach
论文作者
论文摘要
图形信号处理领域的一个重要问题是开发适当的拼写字典，以定义在不同的图表上定义的信号。对称组的Cayley图在排名数据分析中具有自然应用，因为其顶点代表排列，而生成集则正式将排名之间的距离概念形式化。利用对称群体的丰富代表理论，我们研究了一类称为Frobenius-Schur框架的特定类别，每个原子都属于仅一个不可减至的对称组的一个不可约表示的系数空间。我们为对称组的组代数上的所有Frobenius-Schur帧提供了一个表征，该框架相对于生成集“兼容”。此类帧先前已被研究为Permutahedron，即具有相邻换位的生成集的对称组的Cayley图，并已证明能够通过分析系数产生对排名数据集的有意义解释。我们的结果将固定头的框架构造推广到任何反密封的生成集。
An important problem in the field of graph signal processing is developing appropriate overcomplete dictionaries for signals defined on different families of graphs. The Cayley graph of the symmetric group has natural applications in ranked data analysis, as its vertices represent permutations, while the generating set formalizes a notion of distance between rankings. Taking advantage of the rich theory of representations of the symmetric group, we study a particular class of frames, called Frobenius-Schur frames, where every atom belongs to the coefficient space of only one irreducible representation of the symmetric group. We provide a characterization for all Frobenius-Schur frames on the group algebra of the symmetric group which are "compatible" with respect to the generating set. Such frames have been previously studied for the permutahedron, the Cayley graph of the symmetric group with the generating set of adjacent transpositions, and have proved to be capable of producing meaningful interpretation of the ranked data set via the analysis coefficients. Our results generalize frame constructions for the permutahedron to any inverse-closed generating set.
