NDLI: Computing Eigenvalues of Large Scale Sparse Tensors Arising from a Hypergraph

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Qi, Liqun Chang, Jingya Chen, Yannan
Copyright Year	2016
Abstract	The spectral theory of higher-order symmetric tensors is an important tool for revealing 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. By 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 ojasiewicz inequality, we prove that the sequence of iterates generated by CEST converges to an eigenvector of the tensor. When CEST is started from multiple random 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 related to a hypergraph with millions of vertices.
Sponsorship	Research Grants Council, University Grants Committee. National Natural Science Foundation of China
Starting Page	A3618
Ending Page	A3643
Page Count	26
File Format	PDF
ISSN	10648275
DOI	10.1137/16M1060224
e-ISSN	10957197
Issue Number	6
Volume Number	38
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2016-11-15
Access Restriction	Subscribed
Subject Keyword	Multilinear algebra, tensor products Programming involving graphs or networks Eigenvalues, singular values, and eigenvectors hypergraph Methods of quasi-Newton type Mathematical programming methods L-BFGS sparse tensor ojasiewicz inequality Eigenvalues, eigenvectors Laplacian tensor Hypergraphs eigenvalue large scale tensor spherical optimization
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

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