The present paper is devoted to clustering geometric graphs. While the standard spectral clustering is often not effective for geometric graphs, we present an effective generalization, which we call higher-order spectral clustering. It resembles in concept the classical spectral clustering method but uses for partitioning the eigenvector associated with a higher-order eigenvalue. We establish the weak consistency of this algorithm for a wide class of geometric graphs which we call Soft Geometric Block Model. A small adjustment of the algorithm provides strong consistency. We also show that our method is effective in numerical experiments even for graphs of modest size.

本文致力于聚类几何图形。虽然标准频谱聚类通常对几何图形无效，但我们提出了一种有效的概括，我们称之为高阶频谱聚类。它在概念上类似于经典的频谱聚类方法，但用于划分与高阶特征值相关的特征向量。我们为一类称为软几何块模型的几何图形建立了该算法的弱一致性。对该算法进行少量调整即可提供强大的一致性。我们还表明，即使对于中等大小的图形，我们的方法在数值实验中也是有效的。