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 . To apply spectral methods in learning to massive datasets modeled as simplicial complexes, we sparsify simplicial complexes , while 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 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.
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 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.
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emerging from the confluence of computational topology and machine learning
for analyzing such complex datasets
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methods in learning
learning methods
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sparsify simplicial complexes , while preserving
develop a method for sparsifying simplicial complexes that preserves
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operators
matrices
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simplexes
simplices