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 .

翻译：图粗化是一种通过处理原始图的简化版本（并可能将结果插值回原始图）来解决大规模图问题的技术。它在科学计算领域历史悠久，并近年来在机器学习中受到关注，特别是在保持图谱的方法中。本研究从不同视角探讨图粗化，发展了保持图距离的理论，并提出了一种实现该目标的方法。当处理图集合（如图分类和回归）时，这种几何方法尤其有用。本研究中，我们将图视为配备有格罗莫夫-瓦瑟斯坦（GW）距离的度量空间中的元素，并限定两个图之间的距离与其粗化版本之间的差异。最小化该差异可通过流行的加权核$K$-均值方法实现，该方法通过恰当选择核函数改进了现有的谱保持方法。本研究包含一组实验以支持理论和方法，包括近似GW距离、保持图谱、利用谱信息进行图分类以及使用图卷积网络进行回归。代码可访问 https://github.com/ychen-stat-ml/GW-Graph-Coarsening 获取。