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Circulant tensors with applications to spectral hypergraph theory and stochastic process
1. | School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China |
2. | Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong |
References:
[1] |
R. Badeau and R. Boyer, Fast multilinear singular value decomposition for structured tensors,, SIAM J. Matrix Anal. Appl., 30 (2008), 1008.
doi: 10.1137/060655936. |
[2] |
K. C. Chang, K. Pearson and T. Zhang, On eigenvalue problems of real symmetric tensors,, J. Math. Anal. Appl., 350 (2009), 416.
doi: 10.1016/j.jmaa.2008.09.067. |
[3] |
Y. Chen, Y. Dai, D. Han and W. Sun, Positive semidefinite generalized diffusion tensor imaging via quadratic semidefinite programming,, SIAM J. Imaging Sci., 6 (2013), 1531.
doi: 10.1137/110843526. |
[4] |
J. Cooper and A. Dutle, Spectra of uniform hypergraphs,, Linear Algebra Appl., 436 (2012), 3268.
doi: 10.1016/j.laa.2011.11.018. |
[5] |
P. Davis, Circulant Matrices,, Wiley, (1979).
|
[6] |
A. Ducournau and A. Bretto, Random walks in directed hypergraphs and application to semi-supervised image segmentation,, Computer Vision and Image Understanding, 120 (2014), 91.
doi: 10.1016/j.cviu.2013.10.012. |
[7] |
G. Gallo, G. Longo, S. Pallottino and S. Nguyen, Directed hypergraphs and applications,, Discrete Appl. Math., 42 (1993), 177.
doi: 10.1016/0166-218X(93)90045-P. |
[8] |
D. Han and X. Yuan, A note on the alternating direction method of multipliers,, J. Optim. Theory Appl., 155 (2012), 227.
doi: 10.1007/s10957-012-0003-z. |
[9] |
R. Horn and C. Johnson, Matrix Ananlysis,, Cambridge University Press, (1990).
|
[10] |
S. Hu, Z.-H. Huang, H.-Y. Ni and L. Qi, Positive definiteness of diffusion kurtosis imaging,, Inverse Probl. Imaging, 6 (2012), 57.
doi: 10.3934/ipi.2012.6.57. |
[11] |
S. Hu and L. Qi, Algebraic connectivity of an even uniform hypergraph,, J. Comb. Optim., 24 (2012), 564.
doi: 10.1007/s10878-011-9407-1. |
[12] |
S. Hu and L. Qi, The eigenvectors associated with the zero eigenvalues of the Laplacian and signless Laplacian tensors of a uniform hypergraph,, Discrete Appl. Math., 169 (2014), 140.
doi: 10.1016/j.dam.2013.12.024. |
[13] |
S. Hu and L. Qi, The Laplacian of a uniform hypergraph,, J. Comb. Optim., 29 (2015), 331.
doi: 10.1007/s10878-013-9596-x. |
[14] |
S. Hu, L. Qi and J.-Y. Shao, Cored hypergraphs, power hypergraphs and their Laplacian H-eigenvalues,, Linear Algebra Appl., 439 (2013), 2980.
doi: 10.1016/j.laa.2013.08.028. |
[15] |
B. Jiang, S. Ma and S. Zhang, Alternating direction method of multipliers for real and complex polynomial optimization models,, Optimization, 63 (2014), 883.
doi: 10.1080/02331934.2014.895901. |
[16] |
G. Li, L. Qi and G. Yu, The $Z$-eigenvalues of a symmetric tensor and its application to spectral hypergraph theory,, Numer. Linear Algebra Appl., 20 (2013), 1001.
doi: 10.1002/nla.1877. |
[17] |
K. Li and L. Wang, A polynomial time approximation scheme for embedding a directed hypergraph on a ring,, Inform. Process. Lett., 97 (2006), 203.
doi: 10.1016/j.ipl.2005.10.008. |
[18] |
H. Z. Luo, H. X. Wu and G. T. Chen, On the convergence of augmented Lagrangian methods for nonlinear semidefinite programming,, J. Global Optim., 54 (2012), 599.
doi: 10.1007/s10898-011-9779-x. |
[19] |
K. J. Pearson and T. Zhang, On spectral hypergraph theory of the adjacency tensor,, Graphs Combin., 30 (2014), 1233.
doi: 10.1007/s00373-013-1340-x. |
[20] |
J. M. Peña, A class of $P$-matrices with applications to the localization of the eigenvalues of a real matrix,, SIAM J. Matrix Anal. Appl., 22 (2001), 1027.
doi: 10.1137/S0895479800370342. |
[21] |
L. Qi, Eigenvalues of a real supersymmetric tensor,, J. Symbolic Comput., 40 (2005), 1302.
doi: 10.1016/j.jsc.2005.05.007. |
[22] |
L. Qi, $H^+$-eigenvalues of Laplacian and signless Laplacian tensors,, Commun. Math. Sci., 12 (2014), 1045.
doi: 10.4310/CMS.2014.v12.n6.a3. |
[23] |
L. Qi, J.-Y. Shao and Q. Wang, Regular uniform hypergraphs, $s$-cycles, $s$-paths and their largest Laplacian H-eigenvalues,, Linear Algebra Appl., 443 (2014), 215.
doi: 10.1016/j.laa.2013.11.008. |
[24] |
L. Qi and Y. Song, An even order symmetric B tensor is positive definite,, Linear Algebra Appl., 457 (2014), 303.
doi: 10.1016/j.laa.2014.05.026. |
[25] |
L. Qi, G. Yu and E. X. Wu, Higher order positive semidefinite diffusion tensor imaging,, SIAM J. Imaging Sci., 3 (2010), 416.
doi: 10.1137/090755138. |
[26] |
L. Qi, G. Yu and Y. Xu, Nonnegative diffusion orientation distribution function,, J. Math. Imaging Vision, 45 (2013), 103.
doi: 10.1007/s10851-012-0346-y. |
[27] |
M. Rezghi and L. Eldén, Diagonalization of tensors with circulant structure,, Linear Algebra Appl., 435 (2011), 422.
doi: 10.1016/j.laa.2010.03.032. |
[28] |
H. Tijms, A First Course in Stochastic Models,, John Wiley, (2003).
doi: 10.1002/047001363X. |
[29] |
Wikipedia, Circulant matrix - wikipedia, the free encyclopedia, 2015,, , (). Google Scholar |
[30] |
J. Xie and A. Chang, H-eigenvalues of signless Laplacian tensor for an even uniform hypergraph,, Front. Math. China, 8 (2013), 107.
doi: 10.1007/s11464-012-0266-6. |
[31] |
J. Xie and A. Chang, On the Z-eigenvalues of the signless Laplacian tensor for an even uniform hypergraph,, Numer. Linear Algebra Appl., 20 (2013), 1030.
doi: 10.1002/nla.1910. |
show all references
References:
[1] |
R. Badeau and R. Boyer, Fast multilinear singular value decomposition for structured tensors,, SIAM J. Matrix Anal. Appl., 30 (2008), 1008.
doi: 10.1137/060655936. |
[2] |
K. C. Chang, K. Pearson and T. Zhang, On eigenvalue problems of real symmetric tensors,, J. Math. Anal. Appl., 350 (2009), 416.
doi: 10.1016/j.jmaa.2008.09.067. |
[3] |
Y. Chen, Y. Dai, D. Han and W. Sun, Positive semidefinite generalized diffusion tensor imaging via quadratic semidefinite programming,, SIAM J. Imaging Sci., 6 (2013), 1531.
doi: 10.1137/110843526. |
[4] |
J. Cooper and A. Dutle, Spectra of uniform hypergraphs,, Linear Algebra Appl., 436 (2012), 3268.
doi: 10.1016/j.laa.2011.11.018. |
[5] |
P. Davis, Circulant Matrices,, Wiley, (1979).
|
[6] |
A. Ducournau and A. Bretto, Random walks in directed hypergraphs and application to semi-supervised image segmentation,, Computer Vision and Image Understanding, 120 (2014), 91.
doi: 10.1016/j.cviu.2013.10.012. |
[7] |
G. Gallo, G. Longo, S. Pallottino and S. Nguyen, Directed hypergraphs and applications,, Discrete Appl. Math., 42 (1993), 177.
doi: 10.1016/0166-218X(93)90045-P. |
[8] |
D. Han and X. Yuan, A note on the alternating direction method of multipliers,, J. Optim. Theory Appl., 155 (2012), 227.
doi: 10.1007/s10957-012-0003-z. |
[9] |
R. Horn and C. Johnson, Matrix Ananlysis,, Cambridge University Press, (1990).
|
[10] |
S. Hu, Z.-H. Huang, H.-Y. Ni and L. Qi, Positive definiteness of diffusion kurtosis imaging,, Inverse Probl. Imaging, 6 (2012), 57.
doi: 10.3934/ipi.2012.6.57. |
[11] |
S. Hu and L. Qi, Algebraic connectivity of an even uniform hypergraph,, J. Comb. Optim., 24 (2012), 564.
doi: 10.1007/s10878-011-9407-1. |
[12] |
S. Hu and L. Qi, The eigenvectors associated with the zero eigenvalues of the Laplacian and signless Laplacian tensors of a uniform hypergraph,, Discrete Appl. Math., 169 (2014), 140.
doi: 10.1016/j.dam.2013.12.024. |
[13] |
S. Hu and L. Qi, The Laplacian of a uniform hypergraph,, J. Comb. Optim., 29 (2015), 331.
doi: 10.1007/s10878-013-9596-x. |
[14] |
S. Hu, L. Qi and J.-Y. Shao, Cored hypergraphs, power hypergraphs and their Laplacian H-eigenvalues,, Linear Algebra Appl., 439 (2013), 2980.
doi: 10.1016/j.laa.2013.08.028. |
[15] |
B. Jiang, S. Ma and S. Zhang, Alternating direction method of multipliers for real and complex polynomial optimization models,, Optimization, 63 (2014), 883.
doi: 10.1080/02331934.2014.895901. |
[16] |
G. Li, L. Qi and G. Yu, The $Z$-eigenvalues of a symmetric tensor and its application to spectral hypergraph theory,, Numer. Linear Algebra Appl., 20 (2013), 1001.
doi: 10.1002/nla.1877. |
[17] |
K. Li and L. Wang, A polynomial time approximation scheme for embedding a directed hypergraph on a ring,, Inform. Process. Lett., 97 (2006), 203.
doi: 10.1016/j.ipl.2005.10.008. |
[18] |
H. Z. Luo, H. X. Wu and G. T. Chen, On the convergence of augmented Lagrangian methods for nonlinear semidefinite programming,, J. Global Optim., 54 (2012), 599.
doi: 10.1007/s10898-011-9779-x. |
[19] |
K. J. Pearson and T. Zhang, On spectral hypergraph theory of the adjacency tensor,, Graphs Combin., 30 (2014), 1233.
doi: 10.1007/s00373-013-1340-x. |
[20] |
J. M. Peña, A class of $P$-matrices with applications to the localization of the eigenvalues of a real matrix,, SIAM J. Matrix Anal. Appl., 22 (2001), 1027.
doi: 10.1137/S0895479800370342. |
[21] |
L. Qi, Eigenvalues of a real supersymmetric tensor,, J. Symbolic Comput., 40 (2005), 1302.
doi: 10.1016/j.jsc.2005.05.007. |
[22] |
L. Qi, $H^+$-eigenvalues of Laplacian and signless Laplacian tensors,, Commun. Math. Sci., 12 (2014), 1045.
doi: 10.4310/CMS.2014.v12.n6.a3. |
[23] |
L. Qi, J.-Y. Shao and Q. Wang, Regular uniform hypergraphs, $s$-cycles, $s$-paths and their largest Laplacian H-eigenvalues,, Linear Algebra Appl., 443 (2014), 215.
doi: 10.1016/j.laa.2013.11.008. |
[24] |
L. Qi and Y. Song, An even order symmetric B tensor is positive definite,, Linear Algebra Appl., 457 (2014), 303.
doi: 10.1016/j.laa.2014.05.026. |
[25] |
L. Qi, G. Yu and E. X. Wu, Higher order positive semidefinite diffusion tensor imaging,, SIAM J. Imaging Sci., 3 (2010), 416.
doi: 10.1137/090755138. |
[26] |
L. Qi, G. Yu and Y. Xu, Nonnegative diffusion orientation distribution function,, J. Math. Imaging Vision, 45 (2013), 103.
doi: 10.1007/s10851-012-0346-y. |
[27] |
M. Rezghi and L. Eldén, Diagonalization of tensors with circulant structure,, Linear Algebra Appl., 435 (2011), 422.
doi: 10.1016/j.laa.2010.03.032. |
[28] |
H. Tijms, A First Course in Stochastic Models,, John Wiley, (2003).
doi: 10.1002/047001363X. |
[29] |
Wikipedia, Circulant matrix - wikipedia, the free encyclopedia, 2015,, , (). Google Scholar |
[30] |
J. Xie and A. Chang, H-eigenvalues of signless Laplacian tensor for an even uniform hypergraph,, Front. Math. China, 8 (2013), 107.
doi: 10.1007/s11464-012-0266-6. |
[31] |
J. Xie and A. Chang, On the Z-eigenvalues of the signless Laplacian tensor for an even uniform hypergraph,, Numer. Linear Algebra Appl., 20 (2013), 1030.
doi: 10.1002/nla.1910. |
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