A Gromov--Wasserstein Geometric View of Spectrum-Preserving Graph Coarsening
This work addresses graph coarsening for machine learning applications, offering a geometric perspective that is incremental but provides specific gains in preserving spectral properties.
The paper tackles the problem of graph coarsening by developing a theory based on preserving graph distances using the Gromov-Wasserstein metric, and proposes a method that improves existing spectrum-preserving techniques with weighted kernel K-means, showing enhanced performance in tasks like graph classification and regression.
Graph coarsening is a technique for solving large-scale graph problems by working on a smaller version of the original graph, and possibly interpolating the results back to the original graph. It has a long history in scientific computing and has recently gained popularity in machine learning, particularly in methods that preserve the graph spectrum. This work studies graph coarsening from a different perspective, developing a theory for preserving graph distances and proposing a method to achieve this. The geometric approach is useful when working with a collection of graphs, such as in graph classification and regression. In this study, we consider a graph as an element on a metric space equipped with the Gromov--Wasserstein (GW) distance, and bound the difference between the distance of two graphs and their coarsened versions. Minimizing this difference can be done using the popular weighted kernel $K$-means method, which improves existing spectrum-preserving methods with the proper choice of the kernel. The study includes a set of experiments to 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 available at https://github.com/ychen-stat-ml/GW-Graph-Coarsening .