  The vanilla Graph Convolutional Network (GCN) uses a low-pass filter to
extract low-frequency signals from graph topology, which may lead to the
over-smoothing problem when GCN goes deep. To this end, various methods have
been proposed to create an adaptive filter by incorporating an extra filter
(e.g., a high-pass filter) extracted from the graph topology. However, these
methods heavily rely on topological information and ignore the node attribute
space, which severely sacrifices the expressive power of the deep GCNs,
especially when dealing with disassortative graphs. In this paper, we propose a
cross-space adaptive filter, called CSF, to produce the adaptive-frequency
information extracted from both the topology and attribute spaces.
Specifically, we first derive a tailored attribute-based high-pass filter that
can be interpreted theoretically as a minimizer for semi-supervised kernel
ridge regression. Then, we cast the topology-based low-pass filter as a
Mercer's kernel within the context of GCNs. This serves as a foundation for
combining it with the attribute-based filter to capture the adaptive-frequency
information. Finally, we derive the cross-space filter via an effective
multiple-kernel learning strategy, which unifies the attribute-based high-pass
filter and the topology-based low-pass filter. This helps to address the
over-smoothing problem while maintaining effectiveness. Extensive experiments
demonstrate that CSF not only successfully alleviates the over-smoothing
problem but also promotes the effectiveness of the node classification task.
