  This paper aims to provide a novel design of a multiscale framelet
convolution for spectral graph neural networks (GNNs). While current spectral
methods excel in various graph learning tasks, they often lack the flexibility
to adapt to noisy, incomplete, or perturbed graph signals, making them fragile
in such conditions. Our newly proposed framelet convolution addresses these
limitations by decomposing graph data into low-pass and high-pass spectra
through a finely-tuned multiscale approach. Our approach directly designs
filtering functions within the spectral domain, allowing for precise control
over the spectral components. The proposed design excels in filtering out
unwanted spectral information and significantly reduces the adverse effects of
noisy graph signals. Our approach not only enhances the robustness of GNNs but
also preserves crucial graph features and structures. Through extensive
experiments on diverse, real-world graph datasets, we demonstrate that our
framelet convolution achieves superior performance in node classification
tasks. It exhibits remarkable resilience to noisy data and adversarial attacks,
highlighting its potential as a robust solution for real-world graph
applications. This advancement opens new avenues for more adaptive and reliable
spectral GNN architectures.
