The myriad complex systems with multiway interactions motivate the extension of graph-based pairwise connections to higher-order relations. In particular, the simplicial complex has inspired generalizations of graph neural networks (GNNs) to simplicial complex-based models. Learning on such systems requires large amounts of data, which can be expensive or impossible to obtain. We propose data augmentation of simplicial complexes through both linear and nonlinear mixup mechanisms that return mixtures of existing labeled samples. In addition to traditional pairwise mixup, we present a convex clustering mixup approach for a data-driven relationship among several simplicial complexes. We theoretically demonstrate that the resultant synthetic simplicial complexes interpolate among existing data with respect to homomorphism densities. Our method is demonstrated on both synthetic and real-world datasets for simplicial complex classification.

翻译：具有多向交互作用的复杂系统促使基于图的成对连接向高阶关系扩展。特别地，单纯复形启发了图神经网络（GNNs）向基于单纯复形模型的泛化。在此类系统上进行学习需要大量数据，而这些数据的获取可能代价高昂或根本无法实现。我们提出通过线性和非线性混合机制对单纯复形进行数据增广，该方法返回现有带标签样本的混合体。除传统成对混合外，我们提出一种凸聚类混合方法，用于处理多个单纯复形间数据驱动的关联。我们从理论上证明，就同态密度而言，合成得到的单纯复形是现有数据的插值结果。我们在合成数据集和真实数据集上验证了该方法在单纯复形分类中的有效性。