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(Sep. 13) Optimization Methods for Tensor Eigenvalues with Applications in Hypergraphs

Last updated :2018-09-13

Topic: Optimization Methods for Tensor Eigenvalues with Applications in Hypergraphs
Speaker: Dr. CHEN Yannan
(The Hong Kong Polytechnic University)
Time: 15:00-17:00, Thursday, September 13, 2018
Venue: Meeting Room, 1st Floor, No. 6 Red House, Zhuhai Campus, SYSU

Abstract:
The spectral theory of 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. Then, we develop a first-order optimization algorithm for computing H- and Z-eigenvalues of these large scale sparse tensors (CEST). Numerical experiments illustrate that CEST is capable of computing eigenvalues of tensors related to a hypergraph with millions of vertices.

Whereafter, we explore the Fiedler vector of an even-uniform hypergraph, which is the Z-eigenvector associated with the second smallest Z-eigenvalue of a normalized Laplacian tensor arising from the hypergraph. Based on the Fiedler vector, a novel tensor-based spectral method is established for partitioning vertices of the hypergraph. Then, we establish a feasible optimization algorithm to compute the Fiedler vector. Finally, preliminary numerical experiments illustrate that the new approach based on a hypergraph-based Fiedler vector is effective and promising.