Frames for Graph Signals on the Symmetric Group: A Representation Theoretic Approach
This work addresses a domain-specific problem in graph signal processing for ranked data analysis, representing an incremental extension of prior methods.
The paper tackles the problem of constructing overcomplete dictionaries (frames) for graph signals on the symmetric group, which is relevant for ranked data analysis, by generalizing Frobenius-Schur frames from the permutahedron to any inverse-closed generating set.
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.