The Gromov--Wasserstein (GW) distance provides a framework for comparing metric measure spaces, regardless of their underlying structure or geometry. For network-based data, it enables direct comparisons of graphs with different numbers of nodes, without requiring an embedding or other abstraction. Furthermore, through a variant of GW known as fused Gromov--Wasserstein (fGW), it is also possible to incorporate node features in addition to graph structure. In this work, we implement $k$-nearest neighbors ($k$-NN) classification using the GW and fGW distances. We prove the universal consistency of the GW-$k$-NN classifier on the space of equivalence classes of metric measure spaces with finite support and uniform probability measure. By viewing graphs as finitely supported metric measure spaces equipped with the pairwise distance metric and a uniform probability measure on the nodes, we obtain universal consistency of GW-$k$-NN for the space of graphs. Likewise for fGW-$k$-NN, we prove universal consistency on the space of weak isomorphism classes of structured objects consisting of metric measure spaces with finite support and uniform probability measure and feature maps into Euclidean space, thus establishing universal consistency on the space of node-attributed graphs. Our numerical experiments show that GW-$k$-NN and fGW-$k$-NN consistently perform well across multiple graph datasets, suggesting that metric classifiers such as $k$-NN work well in the GW framework.

翻译：Gromov–Wasserstein（GW）距离为比较度量测度空间提供了一种框架，不受其底层结构或几何形状的限制。对于基于网络的数据，它能够直接比较具有不同节点数的图，无需嵌入或其他抽象表示。此外，通过GW的变体——融合Gromov–Wasserstein（fGW），还可以在图结构之外融入节点特征。本文利用GW和fGW距离实现了$k$近邻（$k$-NN）分类。我们在有限支撑且具有均匀概率测度的度量测度空间等价类上，证明了GW-$k$-NN分类器的一致相合性。通过将图视为配备节点间成对距离度量和节点均匀概率测度的有限支撑度量测度空间，我们得到了图空间上GW-$k$-NN的一致相合性。类似地，对于fGW-$k$-NN，我们在由有限支撑、均匀概率测度的度量测度空间与到欧氏空间的特征映射构成的结构化对象的弱同构类空间上，证明了其一致相合性，从而建立了带节点属性图空间上的fGW-$k$-NN一致相合性。数值实验表明，GW-$k$-NN和fGW-$k$-NN在多个图数据集上均表现良好，这暗示着$k$-NN等度量分类器在GW框架下具有优异性能。