  As large-scale graphs become increasingly more prevalent, it poses
significant computational challenges to process, extract and analyze large
graph data. Graph coarsening is one popular technique to reduce the size of a
graph while maintaining essential properties. Despite rich graph coarsening
literature, there is only limited exploration of data-driven methods in the
field. In this work, we leverage the recent progress of deep learning on graphs
for graph coarsening. We first propose a framework for measuring the quality of
coarsening algorithm and show that depending on the goal, we need to carefully
choose the Laplace operator on the coarse graph and associated projection/lift
operators. Motivated by the observation that the current choice of edge weight
for the coarse graph may be sub-optimal, we parametrize the weight assignment
map with graph neural networks and train it to improve the coarsening quality
in an unsupervised way. Through extensive experiments on both synthetic and
real networks, we demonstrate that our method significantly improves common
graph coarsening methods under various metrics, reduction ratios, graph sizes,
and graph types. It generalizes to graphs of larger size ($25\times$ of
training graphs), is adaptive to different losses (differentiable and
non-differentiable), and scales to much larger graphs than previous work.
