Distance measures between graphs are important primitives for a variety of learning tasks. In this work, we describe an unsupervised, optimal transport based approach to define a distance between graphs. Our idea is to derive representations of graphs as Gaussian mixture models, fitted to distributions of sampled node embeddings over the same space. The Wasserstein distance between these Gaussian mixture distributions then yields an interpretable and easily computable distance measure, which can further be tailored for the comparison at hand by choosing appropriate embeddings. We propose two embeddings for this framework and show that under certain assumptions about the shape of the resulting Gaussian mixture components, further computational improvements of this Wasserstein distance can be achieved. An empirical validation of our findings on synthetic data and real-world Functional Brain Connectivity networks shows promising performance compared to existing embedding methods.

翻译：图之间的距离度量是多种学习任务中的重要基础。在本文中，我们提出了一种基于最优传输的无监督方法，用于定义图之间的距离。我们的核心思想是将图表示为高斯混合模型，该模型拟合于在同一空间中采样的节点嵌入的分布。这些高斯混合分布之间的Wasserstein距离随后提供了一种可解释且易于计算的度量标准，并且通过选择合适的嵌入，可以进一步针对当前比较任务进行定制。我们为此框架提出了两种嵌入方法，并表明在关于所得高斯混合分量形状的特定假设下，可以进一步改进这种Wasserstein距离的计算效率。在合成数据和真实世界功能性脑连接网络上的实证验证显示，与现有嵌入方法相比，我们的方法具有有竞争力的性能。