  This paper presents a new approach for assembling graph neural networks based
on framelet transforms. The latter provides a multi-scale representation for
graph-structured data. With the framelet system, we can decompose the graph
feature into low-pass and high-pass frequencies as extracted features for
network training, which then defines a framelet-based graph convolution. The
framelet decomposition naturally induces a graph pooling strategy by
aggregating the graph feature into low-pass and high-pass spectra, which
considers both the feature values and geometry of the graph data and conserves
the total information. The graph neural networks with the proposed framelet
convolution and pooling achieve state-of-the-art performance in many types of
node and graph prediction tasks. Moreover, we propose shrinkage as a new
activation for the framelet convolution, which thresholds the high-frequency
information at different scales. Compared to ReLU, shrinkage in framelet
convolution improves the graph neural network model in terms of denoising and
signal compression: noises in both node and structure can be significantly
reduced by accurately cutting off the high-pass coefficients from framelet
decomposition, and the signal can be compressed to less than half its original
size with the prediction performance well preserved.
