Dirichlet Energy Enhancement of Graph Neural Networks by Framelet
  Augmentation
Jialin Chen Yuelin Wang Cristian Bodnar Rex Ying Pietro Lio Yu Guang Wang
Graph convolutions have been a pivotal element in learning graph
representations. However, recursively aggregating neighboring information with
graph convolutions leads to indistinguishable node features in deep layers,
which is known as the over-smoothing issue. The performance of graph neural
networks decays fast as the number of stacked layers increases, and the
Dirichlet energy associated with the graph decreases to zero as well. In this
work, we introduce a framelet system into the analysis of Dirichlet energy and
take a multi-scale perspective to leverage the Dirichlet energy and alleviate
the over-smoothing issue. Specifically, we develop a Framelet Augmentation
strategy by adjusting the update rules with positive and negative increments
for low-pass and high-passes respectively. Based on that, we design the Energy
Enhanced Convolution (EEConv), which is an effective and practical operation
that is proved to strictly enhance Dirichlet energy. From a message-passing
perspective, EEConv inherits multi-hop aggregation property from the framelet
transform and takes into account all hops in the multi-scale representation,
which benefits the node classification tasks over heterophilous graphs.
Experiments show that deep GNNs with EEConv achieve state-of-the-art
performance over various node classification datasets, especially for
heterophilous graphs, while also lifting the Dirichlet energy as the network
goes deeper.