Towards Spectral Sparsification of Simplicial Complexes Based on Generalized Effective Resistance
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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 for the development of 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 emerging from the confluence of computational topology and machine learning. Similar to the challenges faced by massive graphs, computational methods that process simplicial complexes are severely limited by computational costs associated with massive datasets. To apply spectral methods in learning to massive datasets modeled as simplicial complexes, we work towards the sparsification of simplicial complexes based on preserving the spectrum of the associated Laplacian operators. 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 simplexes; 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 demonstrate via experiments the preservation of the up Laplacian during sparsification, as well as the utility of sparsification with respect to spectral clustering.
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