  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.
