Signal processing on large networks with group symmetries

Current methods of graph signal processing rely heavily on the specific structure of the underlying network: the shift operator and the graph Fourier transform are both derived directly from a specific graph. In many cases, the network is subject to error or natural changes over time. This motivated a new perspective on GSP, where the signal processing framework is developed for an entire class of graphs with similar structures. This approach can be formalized via the theory of graph limits, where graphs are considered as random samples from a distribution represented by a graphon. When the network under consideration has underlying symmetries, they may be modeled as samples from Cayley graphons. In Cayley graphons, vertices are sampled from a group, and the link probability between two vertices is determined by a function of the two corresponding group elements. Infinite groups such as the 1-dimensional torus can be used to model networks with an underlying spatial reality. Cayley graphons on finite groups give rise to a Stochastic Block Model, where the link probabilities between blocks form a (edge-weighted) Cayley graph. This manuscript summarizes some work on graph signal processing on large networks, in particular samples of Cayley graphons.

Frames for signal processing on Cayley graphs

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 (the `preferred basis'). In this paper, we provide an explicit and detailed representation-theoretic account for the spectral decomposition of the adjacency matrix of a Cayley graph, which results in a preferred basis. Our method applies to all (not necessarily quasi-Abelian) Cayley graphs, and provides descriptions of eigenvalues and eigenvectors based on the coefficient functions of the representations of the underlying group. Next, we use such bases 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.

Frames for Graph Signals on the Symmetric Group: A Representation Theoretic Approach

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.