  Spectral-based graph neural networks (SGNNs) have been attracting increasing
attention in graph representation learning. However, existing SGNNs are limited
in implementing graph filters with rigid transforms (e.g., graph Fourier or
predefined graph wavelet transforms) and cannot adapt to signals residing on
graphs and tasks at hand. In this paper, we propose a novel class of graph
neural networks that realizes graph filters with adaptive graph wavelets.
Specifically, the adaptive graph wavelets are learned with neural
network-parameterized lifting structures, where structure-aware attention-based
lifting operations (i.e., prediction and update operations) are developed to
jointly consider graph structures and node features. We propose to lift based
on diffusion wavelets to alleviate the structural information loss induced by
partitioning non-bipartite graphs. By design, the locality and sparsity of the
resulting wavelet transform as well as the scalability of the lifting structure
are guaranteed. We further derive a soft-thresholding filtering operation by
learning sparse graph representations in terms of the learned wavelets,
yielding a localized, efficient, and scalable wavelet-based graph filters. To
ensure that the learned graph representations are invariant to node
permutations, a layer is employed at the input of the networks to reorder the
nodes according to their local topology information. We evaluate the proposed
networks in both node-level and graph-level representation learning tasks on
benchmark citation and bioinformatics graph datasets. Extensive experiments
demonstrate the superiority of the proposed networks over existing SGNNs in
terms of accuracy, efficiency, and scalability.
