Circulant Tensors with Applications to Spectral Hypergraph Theory and Stochastic Process

By Zhongming Chen, Liqun Qi. 2013

47 citations recorded by the source corpus.

Circulant tensors naturally arise from stochastic process and spectral hypergraph theory. The joint moments of stochastic processes are symmetric circulant tensors. The adjacency, Laplacian and signless Laplacian tensors of circulant hypergraphs are also symmetric circulant tensors. The adjacency, Laplacian and signless Laplacian tensors of directed circulant hypergraphs are circulant tensors, but they are not symmetric in general. In this paper, we study spectral properties of circulant tensors and their applications in spectral hypergraph theory and stochastic process. We show that in certain cases, the largest H-eigenvalue of a circulant tensor can be explicitly identified. In particular, the largest H-eigenvalue of a nonnegative circulant tensor can be explicitly identified. This confirms the results in circulant hypergraphs and directed circulant hypergraphs. We prove that an even order circulant B$_0$ tensor is always positive semi-definite. This shows that the Laplacian tensor and the signless Laplacian tensor of a directed circulant even-uniform hypergraph are positive semi-definite. If a stochastic process is $m$th order stationary, where $m$ is even, then its $m$th order moment, which is a circulant tensor, must be positive semi-definite. In this paper, we give various conditions for a circulant tensor to be positive semi-definite.

Subfields: Computational Mathematics, Computational Mechanics, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/6498b26ff2d716c9645f27c680a0741a1c5a2dce

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In Computational Mathematics

1. Haibin Chen
2. Liqun Qi
3. Yisheng Song
4. Chaoqian Li
5. Yaotang Li
6. J. Shao
7. Guanglu Zhou
8. Kelly J. Pearson

In Computational Mechanics

1. Liqun Qi

In Radiology, Nuclear Medicine and Imaging

1. Haibin Chen
2. Liqun Qi
3. Chaoqian Li
4. Yaotang Li
5. Yisheng Song

Citations in the corpus

• Positive Definiteness and Semi-Definiteness of Even Order Symmetric Cauchy Tensors (2014, background): “In recent years, a lot of research papers on structured tensors are presented [2, 3, 4, 8, 13, 14, 15, 17, 18, 21, 22], which studied M-tensors, circulant tensors, completely positive tensors, Hankel tensors, Hilbert tensors, P tensors and B tensors.”