Summary: As a generalization of the use of graphs to describe pairwise interactions, simplicial complexes can be used to model higher order interactions between three or more objects in complex systems. There has been a recent surge in activity to develop data analysis methods applicable to simplicial complexes, including techniques based on computational topology, higher order random processes, generalized Cheeger inequalities, isoperimetric inequalities, and spectral methods. In particular, spectral learning methods (e.g., label propagation and clustering) that directly operate on simplicial complexes represent a new direction for analyzing such complex datasets.
To apply spectral learning methods to massive datasets modeled as simplicial complexes, we develop a method for sparsifying simplicial complexes that preserves the spectrum of the associated Laplacian matrices. We show that the theory of Spielman and Srivastava for the sparsification of graphs extends to simplicial complexes via the up Laplacian. In particular, we introduce a generalized effective resistance for simplices, provide an algorithm for sparsifying simplicial complexes at a fixed dimension, and give a specific version of the generalized Cheeger inequality for weighted simplicial complexes. Finally, we introduce higher order generalizations of spectral clustering and label propagation for simplicial complexes and demonstrate via experiments the utility of the proposed spectral sparsification method for these applications.