Computing Eigenvalues of Large Scale Sparse Tensors Arising from a Hypergraph

TL;DR

This paper introduces an efficient optimization-based method, CEST, for computing eigenvalues of large sparse tensors from hypergraphs, demonstrating high efficiency and scalability through numerical experiments.

Contribution

It develops a first-order optimization algorithm leveraging L-BFGS and Kurdyka-Łojasiewicz property for large-scale sparse tensor eigenvalue computation, a novel approach in hypergraph spectral analysis.

Findings

CEST efficiently computes eigenvalues of large hypergraph tensors.

The method converges to eigenvectors using the Kurdyka-Łojasiewicz property.

Numerical experiments show CEST's scalability to hypergraphs with millions of vertices.

Abstract

The spectral theory of higher-order symmetric tensors is an important tool to reveal some important properties of a hypergraph via its adjacency tensor, Laplacian tensor, and signless Laplacian tensor. Owing to the sparsity of these tensors, we propose an efficient approach to calculate products of these tensors and any vectors. Using the state-of-the-art L-BFGS approach, we develop a first-order optimization algorithm for computing H- and Z-eigenvalues of these large scale sparse tensors (CEST). With the aid of the Kurdyka-{\L}ojasiewicz property, we prove that the sequence of iterates generated by CEST converges to an eigenvector of the tensor.When CEST is started from multiple randomly initial points, the resulting best eigenvalue could touch the extreme eigenvalue with a high probability. Finally, numerical experiments on small hypergraphs show that CEST is efficient and promising. Moreover, CEST is capable of computing eigenvalues of tensors corresponding to a hypergraph with millions of vertices.