A novel method for comparing non-aligned graphs of various sizes is proposed, based on the Wasserstein distance between graph signal distributions induced by the respective graph Laplacian matrices. Specifically, a new formulation for the one-to-many graph alignment problem is casted, which aims at matching a node in the smaller graph with one or more nodes in the larger graph. By incorporating optimal transport into our graph comparison framework, a structurally-meaningful graph distance, and a signal transportation plan that models the structure of graph data are generated. The resulting alignment problem is solved with stochastic gradient descent, where a novel Dykstra operator is used to ensure that the solution is a one-to-many (soft) assignment matrix. The performance of our novel framework is demonstrated on graph alignment, graph classification and graph signal transportation. Our method is shown to lead to significant improvements with respect to the state-of-the-art algorithms on each ofthese tasks.

中文翻译：

---

基于 Wasserstein 的图对齐

基于由各个图拉普拉斯矩阵引起的图信号分布之间的Wasserstein距离，提出了一种比较各种大小的非对齐图的新方法。具体来说，提出了一种针对一对多图对齐问题的新公式，旨在将较小图中的节点与较大图中的一个或多个节点进行匹配。通过将最优传输结合到我们的图比较框架中，可以生成结构上有意义的图距离和对图数据结构进行建模的信号传输计划。由此产生的对齐问题通过随机梯度下降来解决，其中使用了一种新颖的 Dykstra 算子来确保解决方案是一对多（软）分配矩阵。我们的新框架的性能在图对齐上得到证明，图分类和图信号传输。我们的方法被证明可以显着改进每项任务的最新算法。