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Graph coarsening is a technique for solving large-scale graph problems byworking on a smaller version of the original graph, and possibly interpolatingthe results back to the original graph. It has a long history in scientificcomputing and has recently gained popularity in machine learning, particularlyin methods that preserve the graph spectrum. This work studies graph coarseningfrom a different perspective, developing a theory for preserving graphdistances and proposing a method to achieve this. The geometric approach isuseful when working with a collection of graphs, such as in graphclassification and regression. In this study, we consider a graph as an elementon a metric space equipped with the Gromov--Wasserstein (GW) distance, andbound the difference between the distance of two graphs and their coarsenedversions. Minimizing this difference can be done using the popular weightedkernel $K$-means method, which improves existing spectrum-preserving methodswith the proper choice of the kernel. The study includes a set of experimentsto support the theory and method, including approximating the GW distance,preserving the graph spectrum, classifying graphs using spectral information,and performing regression using graph convolutional networks. Code is availableat https://github.com/ychen-stat-ml/GW-Graph-Coarsening .

A Gromov--Wasserstein Geometric View of Spectrum-Preserving Graph Coarsening