Fully Spectral Graph Scattering Networks With Transformer-Based Component-Adaptive Graph Filter Banks and Spectral Nonlinearity

Graph scattering networks (GSNs) learn multi-scale node representation via graph filter banks along multiple scattering paths of different lengths. However, existing GSNs are still challenged by two problems: i) spectral filtering via single graph filter or filter bank with uniform frequency response to different graph components, and ii) lack of spectral nonlinearity to ensure the invariance to node permutation. To tackle the issues, we propose a novel fully spectral graph scattering network name SpecGSN to allow invariant node representation learning with learnable spectral filtering and nonlinearity. Specifically, for spectral filtering, we design Transformer-based graph filter banks on the orthonormal subspaces induced by graph shift operators to achieve diverse frequency response to different graphs components. Moreover, we develop spectral nonlinear activation via subspace folding to guarantee the commutativity between the nonlinear operator and transform with subspace basis for invariant learnable filtering in the spectral domain. We theoretically demonstrate that SpecGSN is invariant to node permutations and subspace basis rotations. Experimental results show that SpecGSN achieves competitive or state-of-the-art node classification performance on both homophilic and heterophilic graphs with enhanced computation efficiency.