Spectral Sparsification of Simplicial Complexes for Clustering and Label Propagation

TL;DR

This paper introduces a spectral sparsification method for simplicial complexes that preserves their Laplacian spectrum, enabling scalable higher-order clustering and label propagation techniques for complex datasets.

Contribution

It extends graph sparsification theory to simplicial complexes, introduces a generalized effective resistance, and develops spectral clustering and label propagation methods for higher-order data.

Findings

Spectral sparsification preserves Laplacian spectra of simplicial complexes.

The method improves scalability of higher-order clustering algorithms.

Experimental results demonstrate effectiveness in real datasets.

Abstract

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.