Unified Fourier bases for signals on random graphs with group symmetries (2406.06306v2)

Abstract: We consider a recently proposed approach to graph signal processing (GSP) based on graphons. We show how the graphon-based approach to GSP applies to graphs sampled from a stochastic block model derived from a weighted Cayley graph. When SBM block sizes are equal, a nice Fourier basis can be derived from the representation theory of the underlying group. We explore how the SBM Fourier basis is affected when block sizes are not uniform. When block sizes are nearly uniform, we demonstrate that the group Fourier basis closely approximates the SBM Fourier basis. More specifically, we quantify the approximation error using matrix perturbation theory. When block sizes are highly non-uniform, the group-based Fourier basis can no longer be used. However, we show that partial information regarding the SBM Fourier basis can still be obtained from the underlying group.