  Graph pooling has been increasingly considered for graph neural networks
(GNNs) to facilitate hierarchical graph representation learning. Existing graph
pooling methods commonly consist of two stages, i.e., selecting the top-ranked
nodes and removing the rest nodes to construct a coarsened graph
representation. However, local structural information of the removed nodes
would be inevitably dropped in these methods, due to the inherent coupling of
nodes (location) and their features (signals). In this paper, we propose an
enhanced three-stage method via lifting, named LiftPool, to improve
hierarchical graph representation by maximally preserving the local structural
information in graph pooling. LiftPool introduces an additional stage of graph
lifting before graph coarsening to preserve the local information of the
removed nodes and decouple the processes of node removing and feature
reduction. Specifically, for each node to be removed, its local information is
obtained by subtracting the global information aggregated from its neighboring
preserved nodes. Subsequently, this local information is aligned and propagated
to the preserved nodes to alleviate information loss in graph coarsening.
Furthermore, we demonstrate that the proposed LiftPool is localized and
permutation-invariant. The proposed graph lifting structure is general to be
integrated with existing downsampling-based graph pooling methods. Evaluations
on benchmark graph datasets show that LiftPool substantially outperforms the
state-of-the-art graph pooling methods in the task of graph classification.
